Margins Study Assistant

Chapter 1

Sets

Summary

A set is a collection you can decide membership for. The chapter covers roster and set-builder notation, the empty set, subsets, Venn diagrams, complements, and how to combine two sets. The size of a union is n(A) + n(B) − n(A ∩ B), because the overlap would otherwise be counted twice.

Key terms

Each term has a plain-language definition written for this guide. The book’s own key-terms page mostly names the word and sends you back to the section.

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1.1 Basic Set Concepts

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

set: A set is a collection of objects treated as one group. For the set to be useful, you have to be able to decide whether any given object belongs to it.
elements: Elements are the individual objects inside a set. If 3 belongs to A, write 3 ∈ A and say that 3 is an element of A.
well-defined set: A set is well defined when every object is clearly in or clearly out, with no personal judgment. “Integers greater than 10” is well defined; “the tall buildings” is not, because people can disagree.
empty set: The empty set has no elements. It is written ∅ or { }, and it is a real set: it is not the same thing as the number 0.
roster method: The roster method names a set by listing its elements inside braces, as in {2, 4, 6}. Order and repeated names do not change the set, so {2, 4, 6} and {6, 2, 4, 2} are the same set.
finite set: A finite set comes to an end. You can count its elements and finish at a whole number, even when that number is large.
infinite set: An infinite set does not come to an end. The natural numbers are infinite because every number has a next one.
natural numbers: The natural numbers are the counting numbers 1, 2, 3, 4, and so on. They are the numbers used to count the elements of a set.
integer: Integers are the whole numbers together with their opposites and zero: …, −2, −1, 0, 1, 2, …. A fraction or a decimal that is not a whole number is not an integer.
set-builder notation: Set-builder notation describes a set by a rule instead of a list. {x | x is an even integer} is read “the set of all x such that x is an even integer.”
cardinality of a set: Cardinality is the number of elements in a set, written n(A) or |A|. The set {a, b, c} has cardinality 3. Repeated listings of the same element still count once.
countably infinite: A set is countably infinite when its elements can be matched one-to-one with the natural numbers, so they can be listed as a first, a second, a third, and so on. The integers are countably infinite even though they run in both directions.
equal sets: Two sets are equal when they contain exactly the same elements, nothing more and nothing missing. {1, 2} and {2, 1} are equal.
equivalent sets: Two sets are equivalent when they have the same number of elements, so each element of one can be paired with exactly one element of the other. The objects themselves do not have to be the same.

1.2 Subsets

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

subset: A is a subset of B, written A ⊆ B, when every element of A is also an element of B. Every set is a subset of itself, and the empty set is a subset of every set.
proper subset: A is a proper subset of B, written A ⊂ B, when A is a subset of B but the two sets are not equal. B has at least one element that A does not have.
equivalent subsets: Subsets are equivalent when they have the same cardinality. You can match their elements one-to-one even if the elements are different objects.
exponential notation: A set with n elements has 2ⁿ subsets, counting the set itself and the empty set. Exponential notation writes that count as a power of 2, so a set of 3 elements has 2³ = 8 subsets.

1.3 Understanding Venn Diagrams

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Venn diagram: A Venn diagram draws each set as a loop, usually a circle, inside a rectangle. The overlap shows objects in more than one set, and the region outside a loop shows what is not in that set.
universal set: The universal set, U, is the whole collection you have agreed to consider for the problem. Every set in the diagram is inside U, and the rectangle stands for U.
disjoint set: Two sets are disjoint when they share no elements. Their loops do not overlap, and their intersection is the empty set.
complement of a set: The complement of A, written A′ or Aᶜ, is everything in the universal set that is not in A. If U = {1, 2, 3, 4} and A = {1, 2}, then A′ = {3, 4}.

1.4 Set Operations with Two Sets

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

intersection of two sets: The intersection A ∩ B contains only the elements that are in both A and B. If the sets share nothing, the intersection is empty.
union of two sets: The union A ∪ B contains every element that is in A, in B, or in both. Its size is n(A) + n(B) − n(A ∩ B), because the overlap would otherwise be counted twice.

Chapter 2

Logic

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Summary

A statement is a sentence that is true or false. The chapter builds compound statements with and, or, if-then, and if-and-only-if, then tests them with truth tables. A conditional is false only when the hypothesis is true and the conclusion is false, and it is equivalent to its contrapositive, not its converse. De Morgan’s laws say how to negate and and or.

Key terms

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2.1 Statements and Quantifiers

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

logic: Logic studies statements and whether a conclusion really follows from the reasons offered for it. A claim is treated as true or false, and symbols make the structure of the claim easier to check.
logical statement: A logical statement is a sentence that is either true or false, but not both at the same time. Questions, commands, and vague opinions are not statements.
truth values: The truth value of a statement is true or false, often marked T or F. A compound statement gets its truth value from the truth values of its parts and from the connective that joins them.
symbolic form: Symbolic form replaces statements with letters such as p and q, and replaces connecting words with symbols such as ∧, ∨, and →. “If it rains, then the game is canceled” becomes p → q.
negation of a logical statement: The negation of a statement says the opposite and has the opposite truth value. The negation of “The door is open” is “The door is not open,” written ~p.
quantifier: A quantifier says how many cases a statement covers. “All,” “every,” and “no” are universal; “some” and “there exists” are existential. Negating one kind turns it into the other kind.
premises: Premises are the statements offered as reasons in an argument. The argument claims that if you accept the premises, you should accept the conclusion.
conclusion: The conclusion is the statement the premises are supposed to support. It is often signaled by a word such as “therefore.”
inductive logical arguments: An inductive argument reaches a general claim from examples or patterns. The conclusion can be reasonable without being guaranteed, even when every example given is true.

2.2 Compound Statements

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compound statement: A compound statement joins two or more statements. Whether it is true depends on the truth of the parts and on which connective joins them.
connective: A connective is a word or symbol that joins statements, such as “and,” “or,” “if…then,” and “if and only if.” Each connective has its own rule for when the whole statement is true.
conjunction: A conjunction is an “and” statement, written p ∧ q. It is true only when both parts are true, and it is false if either part is false.
disjunction: A disjunction is an “or” statement, written p ∨ q. In this course it is true when at least one part is true, including the case when both are true.
conditional: A conditional is an “if…then” statement, written p → q. It is false only when p is true and q is false. In the other three cases it is true, including when p is false.
hypothesis: In the conditional p → q, the hypothesis is p, the condition after “if.” It is also called the antecedent.
conclusion: In the conditional p → q, the conclusion is q, the claim after “then.” It is also called the consequent. This is the conclusion of one statement, not of a whole argument.
biconditional: A biconditional, written p ↔ q, means “p if and only if q.” It is true when both parts have the same truth value: both true, or both false.
dominance of connectives: Dominance is the order used when a symbolic statement has no parentheses. Work negation first, then “and,” then “or,” then a conditional, and a biconditional last. Parentheses override that order.

2.3 Constructing Truth Tables

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Truth table: A truth table lists every way the component statements can be true or false, one combination per row, and shows the truth value of the whole statement in that row. One letter needs 2 rows, two letters need 4, and three letters need 8.
Multiplication principle: Each new independent statement doubles the number of rows, because that statement can be true or false. A table for n simple statements has 2ⁿ rows.
Valid: An argument is valid when the conclusion must be true whenever all of the premises are true. Validity is about the form. A valid argument can still have a false conclusion if one of the premises is false.

2.5 Equivalent Statements

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logically equivalent: Two statements are logically equivalent when they have the same truth value in every row of a truth table. One may replace the other without changing the truth of a larger statement.
tautology: A tautology is a statement that is true in every row of its truth table. “p or not p” is a tautology, because one of those two claims is always true.
inverse: The inverse of p → q is ~p → ~q. It negates both parts. It is not always equivalent to the original conditional.
converse: The converse of p → q is q → p. It swaps the hypothesis and the conclusion. A true conditional can have a false converse.
contrapositive: The contrapositive of p → q is ~q → ~p. It swaps the parts and negates both. It is logically equivalent to the original conditional, so it is true exactly when the original is true.

2.6 De Morgan’s Laws

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Boolean logic: Boolean logic combines truth values with “and,” “or,” and “not.” De Morgan’s laws say how “not” crosses the other two: the negation of p ∧ q is ~p ∨ ~q, and the negation of p ∨ q is ~p ∧ ~q.
negation of a conditional: The negation of “if p then q” is not another “if…then.” It is “p and not q”: the hypothesis happened, and the promised result did not.

2.7 Logical Arguments

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

sound: A sound argument is valid and also has true premises. Its conclusion is true. A valid argument with a false premise is not sound.
fallacy: A fallacy is a break in the reasoning. The conclusion does not follow from the premises, even if each sentence sounds plausible on its own.
deductive arguments: A deductive argument claims that the conclusion follows with certainty from the premises. If the form is valid and the premises are true, the conclusion cannot be false.
law of detachment: The law of detachment says that from p → q and p, you may conclude q. Affirming the “if” part lets you affirm the “then” part. This form is also called modus ponens.
law of denying the consequent: From p → q and ~q, you may conclude ~p. If the result did not happen, the condition that would have forced it did not happen either. Denying p, or affirming q, is not a valid step by itself.
chain rule for conditional arguments: From p → q and q → r, you may conclude p → r. The middle statement links the first hypothesis to the last conclusion, so it can drop out of the final conditional.

Chapter 3

Real numbers and number theory

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Summary

The chapter places integers, rationals, and irrationals inside the real numbers, then turns to divisibility. A prime has exactly two positive divisors, so 1 is neither prime nor composite and 2 is the only even prime. Greatest common divisors, least common multiples, and prime factorizations are the tools for comparing whole numbers.

Key terms

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3.1 Prime and Composite Numbers

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

natural numbers: Here the natural numbers are the counting numbers 1, 2, 3, 4, and so on. Factors, multiples, primes, and composites are all discussed inside this set.
factor of a number: A factor of a number divides it with no remainder. 3 is a factor of 12 because 12 ÷ 3 = 4 exactly. 1 and the number itself are always factors.
multiple of a number: A multiple of a number is the result of multiplying it by a natural number. The multiples of 4 begin 4, 8, 12, 16, and they go on without end.
prime number: A prime number is a natural number greater than 1 whose only factors are 1 and itself. 2, 3, 5, 7, and 11 are prime. 1 is not prime, and 2 is the only even prime.
composite number: A composite number is a natural number greater than 1 that is not prime. It has a factor besides 1 and itself. 4, 6, 8, 9, and 12 are composite.
prime factorization: Prime factorization writes a number as a product of prime factors, often using exponents. For example, 60 = 2² · 3 · 5. Except for the order of the factors, a number has only one prime factorization.
greatest common divisor (GCD): The greatest common divisor of two numbers is the largest natural number that divides both of them. From the prime factorizations, keep the shared primes and use the smaller power of each.
least common multiple (LCM): The least common multiple of two numbers is the smallest natural number that both of them divide. From the prime factorizations, use every prime that appears, raised to the larger power.

3.2 The Integers

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integer: The integers are …, −3, −2, −1, 0, 1, 2, 3, …. They include the natural numbers, zero, and the negatives of the natural numbers, and they do not include fractions.
absolute value: The absolute value of a number is its distance from 0 on the number line, so it is never negative. |−7| = 7 and |7| = 7. Distance does not have a sign.
average of a set of numbers: The average, or mean, is the sum of the numbers divided by how many numbers there are. The average of 4, 8, and 9 is (4 + 8 + 9) / 3 = 7.

3.3 Order of Operations

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order of operations: The order of operations is the agreed sequence for evaluating an expression: grouping symbols first, then exponents, then multiplication and division from left to right, then addition and subtraction from left to right.
PEMDAS: PEMDAS is a memory aid: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction. Multiplication does not always come before division; they are one step and are done left to right. Addition and subtraction work the same way.

3.4 Rational Numbers

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density property of rational numbers: Between any two rational numbers there is another rational number. You can always average them, or find another fraction strictly between them, so the rationals have no gaps of that kind.
improper fraction: An improper fraction has a numerator greater than or equal to its denominator, such as 7/4 or 5/5. Its value is at least 1, and it can be written as a mixed number when the numerator is larger.
lowest terms: A fraction is in lowest terms when the numerator and the denominator have no common factor other than 1. Divide both by their greatest common divisor: 6/8 becomes 3/4.
mixed number: A mixed number writes a value as a whole number plus a proper fraction, such as 1 3/4. That mixed number equals the improper fraction 7/4.
rational number: A rational number can be written as a ratio of two integers a/b with b not zero. Every integer is rational, and every rational number has a decimal that either ends or repeats.
repeating decimal: A repeating decimal has a digit or a block of digits that continues forever, such as 0.333… = 1/3. A bar over the repeating block marks the part that does not stop.
terminating decimal: A terminating decimal ends, such as 0.5 or 0.25. Written in lowest terms, its denominator has no prime factors other than 2 and 5.

3.5 Irrational Numbers

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conjugate numbers: The conjugate of a two-term expression changes the sign of the second term. The conjugate of a + b√c is a − b√c. Multiplying by the conjugate produces a difference of squares and can clear a radical from a denominator.
difference of squares: A difference of squares has the form a² − b², and it factors as (a − b)(a + b). That identity is why a binomial times its conjugate has no middle term.
irrational numbers: An irrational number cannot be written as a ratio of two integers. Its decimal never ends and never settles into a repeating block. √2 and π are irrational.
lowest terms: A radical expression is simplified when the number under the radical has no perfect-square factor other than 1, and no radical remains in a denominator you were asked to clear. √12 = √(4 · 3) = 2√3.
rationalize the denominator: Rationalizing the denominator rewrites a fraction so the denominator has no radical. Multiply the top and bottom by the radical itself, or by the conjugate when the denominator has two terms. The value of the fraction does not change.

3.6 Real Numbers

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

complex number: A complex number has the form a + bi, where a and b are real and i is the imaginary unit with i² = −1. The real numbers are the complex numbers in which b = 0.
imaginary number: An imaginary number uses i, defined by i² = −1. A pure imaginary number has the form bi with b not zero. For example, √−4 = 2i.
real number: A real number is a point on the continuous number line. The real numbers include every rational and irrational number, and they do not include a nonzero imaginary part.

3.7 Clock Arithmetic

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clock arithmetic: Clock arithmetic counts around a circle and starts over after the last mark. Only the remainder matters. On a 12-hour clock, 10 + 5 lands on 3, not on 15.
modulo 7: Modulo 7 keeps the remainder after division by 7. The only possible results are 0, 1, 2, 3, 4, 5, and 6. Weekdays work this way: 8 days after Tuesday is the same weekday as 1 day after Tuesday.
modulo 12: Modulo 12 keeps the remainder after division by 12, so the possible results are 0 through 11. It is the arithmetic behind a 12-hour clock.

3.8 Exponents

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base: In bⁿ, the base is the factor being repeated. In 5³ the base is 5, the number that is multiplied.
exponent: The exponent says how many times the base is used as a factor. 5³ = 5 · 5 · 5 = 125. A zero exponent on a nonzero base equals 1, and a negative exponent means a reciprocal.

3.9 Scientific Notation

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scientific notation: Scientific notation writes a number as a × 10ⁿ, where a is at least 1 and less than 10, and n is an integer. 45,000 = 4.5 × 10⁴, and 0.006 = 6 × 10⁻³.
standard notation: Standard notation is the ordinary decimal form of a number, not a power of 10. 4.5 × 10⁴ written in standard notation is 45,000.

3.10 Arithmetic Sequences

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sequence: A sequence is an ordered list of numbers that follows a rule. Position matters: the first term, the second term, and the rest are a specific order, not just a set.
term of a sequence: A term is one number in a sequence. The nth term, written aₙ, is the number in position n.
arithmetic sequence: An arithmetic sequence adds the same number each time to get the next term. In 3, 7, 11, 15 the difference is 4. The nth term is aₙ = a₁ + (n − 1)d.
first term: The first term, a₁, is the starting number of the sequence. Later terms are built from it by repeatedly adding the common difference or multiplying by the common ratio.
constant difference: The constant difference, also called the common difference, is the fixed amount added to each term of an arithmetic sequence. Subtract a term from the next one to find it. A negative difference makes the sequence decrease.

3.11 Geometric Sequences

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geometric sequence: A geometric sequence multiplies by the same number each time to get the next term. In 2, 6, 18, 54 the multiplier is 3. The nth term is aₙ = a₁ · rⁿ⁻¹.
common ratio: The common ratio is the fixed multiplier of a geometric sequence. Divide any term by the term before it to find the ratio. A ratio whose absolute value is less than 1 pulls later terms toward 0.

Chapter 4

Number representation

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Summary

A number and the marks used to write it are different things. Place value depends on the base: each step left in base 2 is worth twice as much, and hexadecimal continues past 9 with the digits A through F. The chapter also uses older systems, such as Roman numerals, to show how a numeral is decoded.

Key terms

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4.1 Hindu-Arabic Positional System

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numeral: A numeral is a written symbol for a number. “12,” “XII,” and a base-2 numeral can name related values, but they are different numerals.
number: A number is the quantity itself, the amount or the position you mean. Numerals are the different ways of writing that quantity down.
exponential expression: An exponential expression is a base with an exponent, such as 10³. In a positional system, each place value is an exponential expression.
base: The base of a numeration system is how many digit symbols it uses, including a symbol for zero. Base 10 uses the digits 0 through 9, and each place is worth a power of 10.
exponent: The exponent on a place value tells how far that place sits from the ones place. In 10⁴, the exponent 4 means four places to the left of the ones place.
place value: Place value is what a digit is worth because of the column it occupies. In 3,482 the digit 4 sits in the hundreds place, so it contributes 4 × 100.
base 10 system: The base 10 system, or decimal system, builds numbers from powers of 10. Each move of one place to the left multiplies the place value by 10.
Hindu-Arabic numeration system: The Hindu-Arabic system is the base 10 positional system used in ordinary writing. It has a symbol for zero, and the value of a digit depends on its place.
expanded form: Expanded form writes a number as the sum of each digit times its place value. 3,482 = 3 × 10³ + 4 × 10² + 8 × 10¹ + 2 × 10⁰.

4.2 Early Numeration Systems

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additive system of numbers: In an additive system, each symbol has one fixed value, and you add the symbols you see. Position does not multiply a symbol’s value. Egyptian hieroglyphic numerals work this way.
positional system of numbers: In a positional system, the same symbol can stand for different amounts depending on which place it occupies. The base tells how the place values grow.
Babylonian system of numbers: The Babylonian system is positional and based on 60. Places stand for ones, sixties, sixty squared, and so on. That base is why hours and angle degrees are still divided into 60s.
Mayan system of numbers: The Mayan system is positional and mostly based on 20, with a modified place used for the calendar. A shell stands for zero, and combinations of dots and bars build the other digits.
Roman system of numbers: The Roman system uses letters such as I, V, X, L, C, D, and M. Values are usually added, but a smaller value written before a larger one is subtracted, as in IV for 4. There is no symbol for zero and no place value in the modern sense.

4.3 Converting with Base Systems

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base 10: Base 10 is the decimal system, the system used for the starting number or the result when a problem says to convert. A number written with no base label is read as base 10.
remainder: The remainder is what is left when a division does not come out even. It is a whole number smaller than the divisor. When you convert from base 10, the remainders become the digits of the new numeral, last remainder first.
dividend: The dividend is the number being divided. In 23 ÷ 5, the dividend is 23.
divisor: The divisor is the number you divide by. In 23 ÷ 5, the divisor is 5. When you change base, the divisor is the base you are converting into.
quotient: The quotient is the whole-number result of a division, not counting the remainder. 23 = 5 · 4 + 3, so the quotient is 4 and the remainder is 3. You divide the quotient again to get the next digit.

Chapter 5

Algebra

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Summary

This chapter is the algebra a liberal-arts course actually uses: linear equations, proportions, percent sentences, and the line y = mx + b. The slope is the coefficient of x and the constant term is the y-intercept. Those tools come back in the money, data, and geometry chapters.

Key terms

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5.1 Algebraic Expressions

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variable: A variable is a letter that stands for a number you do not know yet, or for a number that can change. In 3x + 2, the variable is x.
constant: A constant is a fixed number in an expression. It does not change when the variable does. In 3x + 2, the 2 is a constant.
expression: An expression is a mathematical phrase made of numbers, variables, and operations. It has no equal sign. 3x + 2 is an expression, and it has a value once you know x.
equation: An equation says two expressions have the same value. It contains an equal sign. 3x + 2 = 14 is an equation, and solving it means finding the value of x that makes it true.
equal sign: The equal sign, =, claims that the value on the left is the same as the value on the right. It is a statement of equality, not an arrow telling you to compute one side only.
term: A term is one piece of an expression, separated from the other pieces by addition or subtraction. In 4x² − 3x + 7, the terms are 4x², −3x, and 7.
coefficient: The coefficient is the number multiplied by the variable part of a term. In −5x², the coefficient is −5. If no number is written, as in x, the coefficient is 1.
like terms: Like terms have the same variables raised to the same powers. 4x and −7x are like terms and combine to −3x. 4x and 4x² are not like terms, because the powers differ.
Distributive Property: The distributive property says a(b + c) = ab + ac. Multiply the outside factor by each term inside the parentheses, then add those products. It also runs backward when you factor out a common factor.

5.2 Linear Equations in One Variable with Applications

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linear equation: A linear equation in one variable can be written so the variable appears only to the first power, as in ax + b = c with a not zero. It has one solution. Undo the operations on both sides, in reverse order, until the variable stands alone.

5.3 Linear Inequalities in One Variable with Applications

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linear inequality: A linear inequality uses <, >, ≤, or ≥ instead of an equal sign. The solution is usually a whole interval of numbers, which you can draw as a ray on a number line. An open circle marks < or >; a closed circle marks ≤ or ≥.
Addition and Subtraction Property of Linear Inequalities: Adding or subtracting the same number on both sides of an inequality keeps the inequality sign the same. If x − 3 > 5, then x > 8.
Multiplication and Division Property of Linear Inequalities: Multiplying or dividing both sides by a positive number keeps the inequality sign. Multiplying or dividing by a negative number reverses the sign. Dividing −2x ≥ 6 by −2 gives x ≤ −3.

5.4 Ratios and Proportions

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ratio: A ratio compares two quantities by division. 3 to 5, 3:5, and 3/5 are the same comparison. Keep the order the problem asks for, and name the units until you know they cancel.
proportion: A proportion says two ratios are equal, such as a/b = c/d with neither denominator zero. The cross products are equal: a · d = b · c. That equation is what you solve for a missing term.
constant of proportionality: When two quantities are proportional, y = kx, the constant of proportionality is the fixed multiplier k. If 5 pounds cost $8, then k = 8/5 dollars per pound, and any other weight is multiplied by that same k.
scale: A scale is the ratio between a length on a model, map, or drawing and the matching length on the real object. A scale of 1 inch to 4 feet means every inch on the drawing stands for 4 feet of the object.
construct ratios: Constructing a ratio means writing the two quantities as a fraction in the requested order, with units attached. “3 teachers for 60 students” is 3 teachers / 60 students, which simplifies to 1/20.
solve proportions: To solve a proportion, write the cross products and solve the resulting equation, or scale both parts of a ratio by the same number. If 2/5 = x/20, then 5x = 40, so x = 8.
use proportions to solve scaling problems: A scaling problem uses a known ratio to find a missing length, cost, or count. Place matching units in the same position in both ratios, such as inches over inches, then solve the proportion.

5.5 Graphing Linear Equations and Inequalities

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ordered pair: An ordered pair (x, y) locates a point by its horizontal coordinate first and its vertical coordinate second. (2, 5) and (5, 2) are different points.
origin: The origin is the point (0, 0), where the horizontal axis and the vertical axis cross.
points on the axes: A point on the x-axis has y-coordinate 0, such as (4, 0). A point on the y-axis has x-coordinate 0, such as (0, −3). The origin is the only point on both axes.
linear equation in two variables: A linear equation in two variables can be written as Ax + By = C, with A and B not both zero. Its graph is a straight line, and every point on that line is a solution.
standards form of a linear equation: Standard form places both variable terms on one side: Ax + By = C. A, B, and C are real numbers, and A and B are not both zero. The book’s key-term list spells this label “standards form.”
solution: A solution of an equation in two variables is any ordered pair that makes the equation true when the coordinates are substituted. A line has infinitely many solutions.
linear inequality in two variables: A linear inequality in two variables, such as y > 2x + 1, describes a region of the plane rather than only a line. Replace the inequality sign with = to get the boundary, then shade the side that makes the inequality true.
solution to a linear inequality: A solution is any ordered pair that makes the inequality true. On the graph it lies in the shaded region. It lies on the boundary line only when the sign is ≤ or ≥.
boundary line: The boundary line is the graph of the related equation. Draw it solid for ≤ or ≥, because the line is included, and dashed for < or >, because the line is not included. A test point tells you which side to shade.

5.6 Quadratic Equations In One Variable with Applications

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

monomial: A monomial is one term: a number, a variable, or a product of a number and variables whose exponents are whole numbers. Examples are 7, −3x, and 4x²y.
polynomial: A polynomial is a sum of monomials, such as 4x² − x + 5. Its degree is the largest exponent of the variable in any term.
binomial: A binomial is a polynomial with exactly two terms, such as x + 4 or x² − 9.
trinomial: A trinomial is a polynomial with exactly three terms, such as x² + 5x + 6.
quadratic equation: A quadratic equation in one variable can be written as ax² + bx + c = 0 with a not zero. The variable is squared, and there are at most two real solutions. Its graph, y = ax² + bx + c, is a parabola.
Zero Product Property: If a product equals zero, then at least one of the factors equals zero. From (x − 2)(x + 5) = 0, either x − 2 = 0 or x + 5 = 0, so x = 2 or x = −5.

5.7 Functions

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

relation: A relation is any pairing of inputs with outputs. It may be a set of ordered pairs, a table, a graph, or a rule. One input is allowed to pair with more than one output.
domain: The domain is the set of allowed inputs, the x-values. In an application, the domain is also limited by what makes sense, such as a length that cannot be negative.
function: A function is a relation in which each input has exactly one output. Two points may share a y-value. They may not share an x-value unless they also share that same y-value.
mapping: A mapping diagram draws inputs in one region and outputs in another, with an arrow for each pair. The relation is a function when every input has exactly one arrow leaving it.
vertical line test: If any vertical line crosses a graph more than once, that x-value has more than one output, so the graph is not a function. If no vertical line crosses more than once, the graph is a function.

5.8 Graphing Functions

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

intercepts of a line: The x-intercept is where the graph crosses the x-axis, so y = 0. The y-intercept is where it crosses the y-axis, so x = 0. A line that is neither horizontal nor vertical has one of each.
slope: Slope is the steepness of a line: the change in y divided by the change in x, often called rise over run. A positive slope rises from left to right, a negative slope falls, zero slope is horizontal, and the slope of a vertical line is undefined.
slope-intercept form: Slope-intercept form is y = mx + b. The coefficient m is the slope, and b is the y-intercept, the point (0, b) where the line crosses the y-axis.

5.9 Systems of Linear Equations in Two Variables

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

system of linear equations: A system of linear equations is two or more linear equations considered together. In two variables, you are looking for the point or points that lie on every line.
solutions of a system of equations: A solution of a system is an ordered pair that makes every equation true at the same time. On the graph it is a point where the lines intersect.
contradictions: A contradiction is an equation that is never true, such as 0 = 5 after you simplify. In a system, a contradiction means the lines are parallel and distinct, so there is no solution.
identities: An identity is true for every allowed value of the variable, such as 2x = 2x, or 0 = 0 after simplifying. In a system, an identity means the two equations describe the same line.
coincident lines: Coincident lines are the same line written in two different ways. They meet at every point, so a system of coincident lines has infinitely many solutions.
consistent system of linear equations: A consistent system has at least one solution. One intersection point makes it consistent and independent. Coincident lines make it consistent and dependent, with infinitely many solutions.
inconsistent system of linear equations: An inconsistent system has no solution. The graphs are parallel lines that never meet.

5.10 Systems of Linear Inequalities in Two Variables

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

system of linear inequalities: A system of linear inequalities is two or more linear inequalities required at the same time. Graph each boundary, shade each solution, and keep only the overlap. A boundary is included only for the inequality that uses ≤ or ≥.

5.11 Linear Programming

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

linear programming: Linear programming finds the greatest or least value of a linear objective inside a region shaped by linear constraints. On a bounded region with straight edges, that best value occurs at a corner point, so you test the objective at each corner.
objective function: The objective function is the quantity you want to maximize or minimize, written as a linear expression such as P = 3x + 5y. Substitute the coordinates of each corner of the feasible region and compare the results.
constraint: A constraint is a limit written as a linear equation or inequality, such as a limit on time, money, or materials. Together, the constraints form the feasible region, the set of points you are allowed to use.

Chapter 6

Money management

Read chapter 6 in the book

Summary

Percent becomes a price, a tax, or a discount, and then interest. Simple interest is always charged on the original principal, I = Prt. Compound interest pays interest on earlier interest, A = P(1 + r)^t. The rest of the chapter follows that growth into savings, credit cards, loans, and mortgages, where early payments are mostly interest.

Key terms

Each term has a plain-language definition written for this guide. The book’s own key-terms page mostly names the word and sends you back to the section.

Open this chapter’s key terms in the book

6.1 Understanding Percent

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Percent: Percent means “per hundred.” 35% is 35 out of 100, which is the fraction 35/100 and the decimal 0.35. To change a decimal to a percent, move the decimal point two places right.
Fractional form: The fractional form of a percent is that percent written as a fraction of 100, then reduced when possible. 35% = 35/100 = 7/20.
Decimal form: The decimal form of a percent is the percent divided by 100. 8% = 0.08, and 150% = 1.5. Multiply by the decimal form when you take a percent of a number.
Total: The total is the whole amount that a percent is taken from. In “30% of 80,” the total is 80, and the part is 0.30 × 80 = 24.
Base: In a percent problem, the base is the whole that the percent refers to, the number after “of” in “percent of what.” Finding the base means the part and the percent are known and the whole is not.
Percent of the total: The percent of the total is the part divided by the whole, then written as a percent. If 18 students out of 40 play a sport, the percent is 18/40 = 0.45 = 45%.
Part: The part is the portion of the whole named by the percent. Part = percent, as a decimal, times the total. If any two of part, percent, and total are known, the third can be found.
Amount: The amount is the quantity you get after applying the percent: the part itself, or the result of a percent increase or decrease. An amount of tax, tip, or discount is the percent times the starting figure.

6.2 Discounts, Markups, and Sales Tax

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Discount: A discount is an amount taken off an original price. Discount = discount rate × original price, and the sale price is original price minus that discount.
Cost: Cost is what the seller paid for an item before any markup. It is the starting amount for a retail price, not the price the customer sees.
Markup: Markup is the amount added to the cost to get a selling price. Markup = markup rate × cost, and selling price = cost + markup. The rate may instead be stated as a percent of the selling price, so read which base the problem uses.
Retail price: The retail price is the price charged to the customer. It equals cost plus markup, and a later discount or sales tax is applied to that retail price unless the problem says otherwise.

6.3 Simple Interest

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Interest: Interest is the fee paid for borrowing money, or the money earned for lending it. It depends on how much money is involved, the interest rate, and how long the money is kept.
Principal: The principal is the starting amount of money, before interest is added. In a loan it is the amount borrowed. In a savings account it is the amount deposited.
Annual percentage rate: The annual percentage rate, or APR, is the stated interest rate for one year. In the simple-interest formula it is the decimal r. An APR of 6% means r = 0.06.
Simple interest: Simple interest is interest charged only on the original principal, not on interest already earned. The interest is I = Prt, and the future value is A = P(1 + rt), with t in years and r as a decimal.
Term: The term is the length of time the money is borrowed or invested. In I = Prt, the term is t, measured in years. Three months is t = 3/12 = 0.25.
Due: The amount due is what must be paid at the end of the term: the principal plus the interest earned or charged. For simple interest, the amount due is P + Prt.
Origination date: The origination date is the day the loan or note begins, the day interest starts to accumulate. The term is counted from that date to the due date.
Payoff amount: The payoff amount is what it costs to settle the debt on a chosen day, including the principal still owed and the interest up to that day. An early payoff uses a shorter time than the full term.
Future value: Future value is what a present amount grows into after interest. For simple interest, the future value is A = P(1 + rt).
Partial payment: A partial payment pays only part of what is owed. The payment first covers the interest due so far, and any remainder reduces the principal. Later interest is then computed on the smaller principal.
Present value: Present value is the amount that must be set aside now to grow into a chosen future amount. For simple interest, P = A / (1 + rt).

6.4 Compound Interest

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Compound interest: Compound interest pays interest on the principal and also on interest already earned. If interest is compounded n times a year, A = P(1 + r/n)^(nt). More compounding periods produce a slightly larger future value at the same annual rate.
Effective annual yield: The effective annual yield is the actual percent gained in one year after compounding is taken into account. It is (1 + r/n)^n − 1. A nominal rate of 6% compounded monthly earns a little more than 6% over the year.

6.5 Making a Personal Budget

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Budget: A budget is a plan that assigns expected income to categories of spending and saving. The amounts in the plan have to add up to no more than the income.
Necessary expenses: Necessary expenses are the costs required to live and keep working, such as housing, basic food, utilities, and required transportation. Wants, such as entertainment, are not in this group.
Fixed expenses: A fixed expense is the same amount each period, such as rent or a set loan payment. You can plan the exact dollar amount because it does not change with use.
Variable expenses: A variable expense changes from period to period, such as groceries, gasoline, or the electric bill. A budget estimates it from recent months rather than treating it as a fixed bill.
50-30-20 budget philosophy: The 50-30-20 plan sends about 50% of after-tax income to needs, 30% to wants, and 20% to savings and extra debt payments. It is a guideline for splitting a paycheck, not a rule that every household can meet exactly.

6.6 Methods of Savings

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Savings account: A savings account holds money at a bank or credit union and pays interest. It is meant for money you want kept safe and available, and the interest rate is usually lower than the return on riskier investments.
1099 form: A 1099 form is a tax record of certain income that was not paid as regular wages. Banks use a 1099-INT to report interest you earned, which is taxable income even if you leave it in the account.
Certificate of deposit: A certificate of deposit, or CD, locks a deposit for a stated term in exchange for a stated interest rate. Taking the money out early usually costs a penalty.
Money market account: A money market account is a deposit account that often pays a higher rate than a regular savings account and may offer limited checks or transfers. The balance may need to stay above a minimum.
Return on investment: Return on investment compares what you gained or lost with what you put in. As a percent, it is (ending value − starting amount) / starting amount. A higher return usually comes with a higher chance of loss.
Ordinary annuity: An ordinary annuity is a series of equal payments made at the end of each equal time period, such as a monthly deposit into a retirement account. Interest compounds between payments, so the first payment earns interest for less time than if it had been deposited at the start.

6.7 Investments

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Bonds: A bond is a loan you make to a government or a company. You pay a price now, and the issuer pays interest and later returns the face value on the maturity date, if it does not default.
Maturity date: The maturity date is the day a bond or other debt ends and the face amount is scheduled to be paid back. Interest obligations for that bond stop at maturity.
Stocks: Stock is a share of ownership in a company. The price can rise or fall, and the company may or may not pay a dividend. Unlike a bond, stock does not promise to return a fixed amount on a fixed day.
Dividend: A dividend is a portion of a company’s earnings paid to shareholders, usually as cash per share. The dividend yield is the yearly dividend divided by the current price of the share.
Mutual fund: A mutual fund pools money from many investors and buys a collection of stocks, bonds, or other assets. One share of the fund spreads your money across that collection, and the fund charges expenses.
Prospectus: A prospectus is the legal document that describes an investment offered to the public. It states the goal, the fees, the risks, and how the fund or security works, so the buyer is not relying only on an advertisement.
Issue price: The issue price is the price of a security when it is first sold, as in an initial public offering of stock. After that sale, the market price can move above or below the issue price.
Shares: A share is one unit of ownership in a company or a fund. Owning more shares means owning a larger fraction, and dividends are paid per share.
Stock table: A stock table is a listing of market information for a security: the ticker symbol, prices, the change from the previous close, the volume traded, and sometimes the dividend and yield. The columns are abbreviations, so each heading has to be read carefully.
Individual retirement account: An individual retirement account, or IRA, is a personal account with tax advantages for retirement savings. Traditional IRA contributions may be tax-deductible, and withdrawals in retirement are generally taxed.
Roth IRA: A Roth IRA is funded with money that has already been taxed. Qualified withdrawals in retirement, including the earnings, are tax-free. Income limits can restrict who may contribute.
401(k): A 401(k) is a retirement plan offered through an employer. Contributions come out of pay, often before income tax, and many employers add a match. Withdrawals before retirement age usually bring a penalty as well as tax.

6.8 The Basics of Loans

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Fixed interest rate: A fixed interest rate stays the same for the term of the loan. The scheduled payment does not change because of a change in market rates.
Variable interest rate: A variable interest rate can go up or down with a published index. The payment or the cost of the loan can change even though you borrowed the same principal.
Installment loan: An installment loan is paid back in scheduled payments over a set term. Each payment covers interest and also reduces the principal, as with a car loan or a mortgage.
Loan amortization: Amortization is the schedule that pays off a loan by regular payments. Early payments are mostly interest, because interest is computed on a large balance. Later payments retire more principal.
Revolving credit: Revolving credit lets you borrow again, up to a limit, as you pay the balance down. A credit card is revolving credit. There is no fixed number of payments unless you stop borrowing and pay the balance off.
Amortization table: An amortization table lists every payment, and for each one it shows the interest portion, the principal portion, and the balance left afterward. The interest portion is the periodic rate times the previous balance.
Cost of finance: The cost of finance is the extra amount you pay beyond the money you received: interest and required fees. On an installment loan it is the total of the payments minus the amount financed.

6.9 Understanding Student Loans

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

FAFSA: The FAFSA is the Free Application for Federal Student Aid. It collects household financial information and is used to decide federal grants, work-study, and federal student loans.
College funding gap: The funding gap is the part of the college bill that is still unpaid after grants, scholarships, family contributions, and work income. Loans are often used to fill that gap.
Subsidized loan: On a subsidized federal loan, the government pays the interest while you are enrolled at least half time and during certain other periods. Eligibility depends on financial need.
Unsubsidized loan: On an unsubsidized federal loan, interest starts accruing as soon as the loan is disbursed, including while you are in school. You do not need to show financial need, but unpaid interest is added to the balance if you do not pay it.
Parent loan for undergraduate students: A Parent PLUS loan is a federal loan that a parent of a dependent undergraduate can take for the student’s education costs. The parent, not the student, is the borrower, and interest begins right away.
Private student loan: A private student loan comes from a bank, a credit union, or another private lender rather than from the federal government. Rates and repayment rules are set by the lender, and federal protections may not apply.
School-channel loan: A school-channel loan is a private loan arranged through the college. The school certifies enrollment and the cost of attendance before the lender releases the money.
Direct-to-consumer loan: A direct-to-consumer loan is a private education loan the student or family gets straight from a lender, without the school packaging it. The school may still need to confirm enrollment.
Standard repayment plan: The standard federal plan spreads the loan over 10 years of fixed monthly payments. It usually costs less interest than a longer plan, because the balance is paid off sooner.
Federal consolidation: Federal consolidation combines federal student loans into one new federal loan. The new rate is a weighted average of the old rates, and the repayment term may be longer. It does not lower the underlying rates the way a true refinance might.
Refinancing: Refinancing replaces one or more loans with a new loan, usually from a private lender, at a new rate and term. A lower rate can cut the cost, but moving a federal loan into a private refinance gives up federal repayment options.
Private consolidation: Private consolidation combines loans under a private lender into one payment. It is a form of refinancing, and federal loans included in it leave the federal program.
Graduated repayment plan: A graduated plan starts with lower payments that rise every two years. The early payments may cover little principal, so the total interest paid is higher than on the standard plan.
Extended repayment plan: An extended plan lengthens the term, up to 25 years for eligible balances, which lowers the monthly payment and increases the total interest.
Discretionary income: Discretionary income, for these plans, is the part of your income above a set multiple of the poverty guideline for your family size. Income-driven payments are a percentage of that amount, and they can be as low as zero.
Pay as you earn (PAYE) repayment plan: PAYE sets the monthly payment at about 10% of discretionary income and extends the term, typically to 20 years. Any remaining eligible balance at the end of the term may be forgiven, and forgiven amounts can be taxed.
Revised pay as you earn (REPAYE) repayment plan: REPAYE also uses about 10% of discretionary income, but it does not cap the payment at the standard 10-year amount the way PAYE can. The repayment period is 20 years for undergraduate loans and 25 years if graduate loans are included.
Income-based (IBR) repayment plan: Income-based repayment sets the payment at 10% or 15% of discretionary income, depending on when you borrowed, and never above the standard 10-year payment. The remaining balance may be forgiven after 20 or 25 years.
Income-contingent (ICR) repayment plan: Income-contingent repayment sets the payment from your income, family size, and loan balance, using 20% of discretionary income or a 12-year adjusted payment, whichever is less. Forgiveness is considered after 25 years.
Delinquent: A loan is delinquent when a payment is past due but the loan has not yet reached default. Late fees can apply, and the missed payment can be reported to credit agencies.
Default: Default is a serious failure to repay, reached after a loan stays delinquent for the period the program specifies, often 270 days for a federal student loan. Default can bring collection, garnishment, and the loss of federal aid eligibility.
Rehabilitation: Rehabilitation is a way out of default. You make a required number of on-time agreed payments, and the loan can return to good standing. The default may then be removed from your credit report.

6.10 Credit Cards

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Reward program: A reward program gives points, miles, or cash back on purchases. The reward is worth something only after you subtract the annual fee and the interest you pay if you carry a balance.
Annual fee: An annual fee is a yearly charge just for holding the card, separate from interest. A card with a fee can still be cheaper if its rewards or lower rate save more than the fee costs.
Credit limit: The credit limit is the most you are allowed to owe on the card at one time. Going over it can bring a fee and a penalty rate. The unused part of the limit is limit minus current balance.
Bank-issued credit card: A bank-issued card is offered by a bank or card network, and it can be used at any merchant that accepts that network. Approval and the interest rate depend on your credit.
Store-issued credit card: A store-issued card is tied to one retailer or retail group. It often carries a higher interest rate and may be usable only at that store, sometimes in exchange for a discount.
Travel and entertainment cards: Travel and entertainment cards are aimed at travel spending and often charge an annual fee in exchange for rewards or perks. Some of them must be paid in full each month.
Charge cards: A charge card expects you to pay the full balance every month. It may have no preset spending limit, but it is not free credit: the whole amount is due, and a late payment brings a penalty.
Billing period: The billing period is the stretch of days, often about a month, covered by one statement. Purchases in that window, plus any older balance and fees, make up the statement.
Balance: The balance is the amount you currently owe. A new purchase raises it, and a payment or a credit lowers it. Interest, if any, is computed from the balance the card’s method uses.
Minimum payment: The minimum payment is the smallest amount the issuer will accept for the billing period. Paying only the minimum keeps the account current, but most of a large balance remains and continues to collect interest.
Average daily balance: The average daily balance adds up the balance at the end of each day in the billing period and divides by the number of days. Many cards multiply that average by the daily rate and by the number of days to get the interest charge.

6.11 Buying or Leasing a Car

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Title and registration fees: Title and registration fees are government charges to record you as the owner and to put the car on the road legally. They are part of the out-the-door cost, not part of the vehicle’s advertised price.
Destination fee: The destination fee is the manufacturer’s charge for shipping the car to the dealer. It is usually not optional, and it is added on top of the sticker price of the vehicle.
Documentation fee: The documentation fee is the dealer’s charge for preparing the sales and title paperwork. It is set by the dealer, and it belongs in the total you compare between offers.
Dealer preparation fee: A dealer preparation fee is a charge for getting the car ready for delivery. Ask what work it covers, because some of that preparation is already included in the price of a new car.
Extended warranty: An extended warranty, or service contract, is optional coverage you buy for repairs after the manufacturer’s warranty ends. Its price should be judged against the repairs it actually covers and the ones it excludes.
Down payment: A down payment is cash paid at the start of a purchase or lease. On a loan it reduces the amount financed. A larger down payment means less principal and less interest.
Acquisition fee: An acquisition fee is an upfront charge from a leasing company for setting up the lease. It is part of the cost of leasing even though it is not a monthly payment.
Security deposit: A security deposit on a lease is money held against damage or missed payments. It should be returned at the end if the contract’s conditions are met.
Disposition fees: A disposition fee is charged at the end of a lease for the lessor to take the car back and sell it. You may avoid it if you buy the car or, under some contracts, lease another one from the same company.
Liability insurance: Liability insurance pays for injury and property damage you cause to other people. It does not pay to repair your own car. States require a minimum amount.
Collision insurance: Collision insurance pays to repair your car after a crash with another vehicle or object, regardless of who is at fault, minus your deductible. It is required by many lenders while the car is financed.
Comprehensive insurance: Comprehensive insurance pays for non-collision losses such as theft, hail, flood, or a broken windshield, minus the deductible. It is the coverage people mean when they separate “theft and weather” from crash damage.
Uninsured or underinsured motorist insurance: This coverage pays your injuries, and sometimes your car, when the at-fault driver has no insurance or too little to cover the loss. It fills the gap that the other driver’s liability insurance leaves.
Medical payment insurance: Medical payments coverage pays medical bills for you and your passengers after a crash, without first deciding who was at fault. It is separate from health insurance and from liability coverage.
Personal injury insurance: Personal injury protection covers medical costs and, in many states, some lost wages and other expenses for you and your passengers, no matter who caused the crash. It is required in no-fault states.
Gap insurance: Gap insurance covers the difference when a car is totaled and the insurance payout is less than the remaining loan or lease balance. That gap is common early in a loan, when the balance is still high and the car has already depreciated.
Rental reimbursement insurance: Rental reimbursement pays, up to a daily and total limit, for a rental car while your insured car is being repaired after a covered claim. It does not pay for a rental you want for a vacation.

6.12 Renting and Homeownership

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Lease: A lease is a contract that lets you use property for a stated period in exchange for rent. It names the rent, the length, the deposit, and who is responsible for repairs.
Landlord: The landlord is the owner, or the owner’s agent, who rents the property out. The landlord must deliver a habitable place and follow the lease and housing law; the tenant must pay rent and care for the property.
Mortgage: A mortgage is a loan used to buy real estate, with the property itself pledged as security. The payment typically covers principal, interest, and often taxes and insurance. Missing payments can lead to foreclosure.
Escrow account: An escrow account, in a mortgage, holds the portion of your payment that covers property taxes and homeowners insurance. The loan servicer pays those bills out of the account when they come due.
Assessed value: Assessed value is the value a local government assigns to a property for tax purposes. Property tax is a rate applied to that assessed value, which may be lower than the price the home would sell for.

6.13 Income Tax

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Gross income: Gross income is all income you must report, before adjustments and deductions: wages, interest, some benefits, and other taxable receipts. It is the starting figure on a tax return, not the amount you take home.
Adjusted gross income: Adjusted gross income, or AGI, is gross income minus specific adjustments such as certain retirement contributions and student-loan interest. Several deductions and credits are computed from AGI.
Exemption: An exemption was a set amount subtracted for each person the return supported. The 2017 tax law suspended the personal exemption, so current returns reduce income through the standard or itemized deduction and through credits instead.
Taxable income: Taxable income is the amount the tax brackets are actually applied to. It is AGI minus the deduction you take, either the standard deduction or itemized deductions.
Deduction: A deduction lowers the income that is taxed. A $1,000 deduction does not cut your tax by $1,000; it cuts your taxable income by $1,000, and the tax savings is that amount times your marginal rate.
Tax credit: A tax credit cuts the tax itself, dollar for dollar. A $1,000 credit reduces the tax due by $1,000. A refundable credit can be paid out even if it is larger than the tax you owe; a nonrefundable credit cannot.
Earned income credit: The earned income tax credit is a refundable credit for workers with low to moderate earnings. The amount depends on income and on the number of qualifying children, and it can produce a refund.
American opportunity credit: The American Opportunity credit helps pay for the first four years of undergraduate study. Up to 40% of the allowable credit is refundable, and the student must be enrolled at least half time in a degree program.
Lifetime learning credit: The lifetime learning credit helps pay for undergraduate, graduate, or job-skill courses, with no limit on the number of years. It is nonrefundable, so it can reduce your tax to zero but not produce a refund by itself.
Child tax credit: The child tax credit reduces tax for each qualifying child under the age limit who meets the relationship, residency, and support tests. A portion of it can be refundable as the additional child tax credit.
Child and dependent care tax credit: This credit offsets part of the cost of care for a child under 13, or for a disabled dependent, so that you can work or look for work. The percentage depends on your income, and there is a cap on the expenses that count.
Premium tax credit: The premium tax credit lowers the cost of health insurance bought through the federal or state marketplace. It can be paid in advance to the insurer, and the return later reconciles that advance with the credit you actually qualify for.

Chapter 7

Probability

Read chapter 7 in the book

Summary

Probability is a number from 0 to 1. The chapter computes it by counting equally likely outcomes, uses the complement rule, and separates events that cannot happen together from events that do not affect each other. Odds compare favorable cases with unfavorable ones. Expected value is the sum of each payoff times its probability.

Key terms

Each term has a plain-language definition written for this guide. The book’s own key-terms page mostly names the word and sends you back to the section.

Open this chapter’s key terms in the book

7.1 The Multiplication Rule for Counting

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

combinatorics: Combinatorics counts the ways an event can happen without listing every one. The basic tools are the multiplication rule, permutations, and combinations, and the choice among them depends on whether order matters and whether repeats are allowed.
Multiplication Rule for Counting (Fundamental Counting Principle): If one choice can be made in m ways and a second, independent choice can be made in n ways, the two choices together can be made in m × n ways. With more choices, multiply all of the numbers of options. A meal of 3 entrees and 4 drinks can be chosen in 3 × 4 = 12 ways.

7.2 Permutations

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permutation: A permutation is an arrangement in which order matters. The number of permutations of n distinct objects taken r at a time is P(n, r) = n! / (n − r)!. Arranging 5 books in 3 spots on a shelf is P(5, 3) = 5 × 4 × 3.
factorial: The factorial of a whole number n, written n!, is the product of all whole numbers from n down to 1. For example, 4! = 4 × 3 × 2 × 1 = 24. By agreement, 0! = 1, so formulas that divide by 0! still work.

7.3 Combinations

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combination: A combination is a selection in which order does not matter. The number of ways to choose r objects from n distinct objects is C(n, r) = n! / (r!(n − r)!). Choosing 3 people from 5 is C(5, 3) = 10, because the order of the three names does not create a new group.

7.4 Tree Diagrams, Tables, and Outcomes

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experiment: An experiment is an activity with an observable result, such as rolling a die or drawing a name. The result you record is the outcome.
replication: A replication is one more trial of the same experiment under the same conditions. Repeating a coin flip 50 times is 50 replications, and the collection of results is what an empirical probability is built from.
sample space: The sample space is the set of all possible outcomes of an experiment. For one fair die it is {1, 2, 3, 4, 5, 6}. A tree or a table is a way to list that set when there is more than one stage.
independent/dependent: Two stages are independent when the result of the first does not change the probabilities of the second, as in flipping a coin twice. They are dependent when the first result changes what is left, as in drawing a card and not putting it back.

7.5 Basic Concepts of Probability

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event: An event is a set of outcomes you are interested in, a subset of the sample space. “Rolling an even number” is the event {2, 4, 6}.
probability: The probability of an event is a number from 0 to 1 that measures how likely the event is. 0 means it cannot happen, 1 means it must happen, and the probabilities of all the outcomes in the sample space add to 1.
theoretical probability: Theoretical probability is computed from equally likely outcomes, or from a model, without running the experiment. If every face of a fair die is equally likely, P(even) = 3/6 = 1/2.
empirical probability: Empirical probability is the number of times the event happened divided by the number of trials you actually ran. If a tack lands point up 28 times in 100 drops, the empirical probability is 0.28.
subjective probability: A subjective probability is a personal estimate of how likely something is, based on judgment rather than a complete list of equally likely outcomes or a long record of trials. “I think there is a 70% chance of rain” is subjective.

7.7 What Are the Odds?

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odds (for/against): Odds compare the ways an event can happen with the ways it can fail, instead of comparing the event with the whole sample space. Odds in favor are favorable to unfavorable; odds against reverse that ratio. If P(A) = 3/5, the odds in favor are 3 to 2.

7.8 The Addition Rule for Probability

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mutually exclusive: Events are mutually exclusive when they cannot happen at the same time, so their intersection is empty. For mutually exclusive events, P(A or B) = P(A) + P(B). If they can overlap, subtract the probability of the overlap so it is not counted twice.

7.9 Conditional Probability and the Multiplication Rule

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conditional probability: A conditional probability, P(A | B), is the probability of A given that B has already happened. It is P(A and B) / P(B), provided P(B) is not zero. The multiplication rule then says P(A and B) = P(B) · P(A | B). If the events are independent, P(A | B) = P(A).

7.10 The Binomial Distribution

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binomial experiment: A binomial experiment has a fixed number of independent trials, and each trial has exactly two outcomes, often called success and failure, with the same probability of success every time. Counting the number of successes gives a binomial random variable.
probability density function (PDF): For a discrete variable, the probability density function gives the probability of each single value. For a binomial count, P(exactly k successes) = C(n, k) p^k (1 − p)^(n − k).
cumulative distribution function (CDF): The cumulative distribution function gives the probability of a value less than or equal to a cutoff. For a binomial count it is the sum of the individual probabilities from 0 successes up through that cutoff.

7.11 Expected Value

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expected value: The expected value is the long-run average of a random quantity. Multiply each possible value by its probability and add those products. A game that pays $10 with probability 0.2 and $0 otherwise has expected value 10(0.2) + 0(0.8) = $2.

Chapter 8

Statistics

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Summary

This is the book’s short chapter on describing data, before a full statistics course. You summarize a list with the mean, median, mode, and range, and you see why one extreme value pulls the mean more than the median. Graphs and the shape of a distribution, including a long right tail, tell you which summary to trust.

Key terms

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8.1 Gathering and Organizing Data

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sample: A sample is the part of a population you actually observe. A good sample is chosen so that it can stand in for the larger group; a convenience sample often cannot.
units: The units are the people or objects you measure, one at a time. If you record the height of 40 students, each student is a unit.
data: Data are the recorded values. A single number or category is a datum; the whole collection is the data set you organize and summarize.
population: The population is the entire group you want to know about. A number that describes the population is a parameter; the matching number computed from a sample is a statistic.
simple random sample: A simple random sample gives every group of the chosen size the same chance to be selected. Drawing names from a mixed hat, or using a random-number method, is the idea.
systematic random sample: A systematic sample chooses a starting point at random and then takes every kth unit, such as every 10th name on a list. It is simple to carry out, but a hidden pattern in the list can bias it.
stratified sample: A stratified sample divides the population into groups, called strata, that are alike in some important way, and then draws a random sample from each group. This guarantees that each group is represented.
cluster sample: A cluster sample divides the population into groups that each resemble the whole, chooses some of those groups at random, and measures every unit inside the chosen groups. It saves travel when the groups are locations.
quantitative data: Quantitative data are numbers that measure or count something, such as height or the number of siblings. You can compute a mean from them. Categories such as eye color are not quantitative.
categorical data: Categorical data place each unit into a group, such as yes or no, or a favorite color. You summarize them with counts and percents, not with a meaningful average of the labels.
categorical frequency Distribution: A categorical frequency distribution is a table that lists each category and how many observations fall in it. Adding a relative-frequency column turns those counts into fractions or percents of the total.
binned frequency distribution: A binned frequency distribution groups numerical data into intervals, called bins, and counts how many values fall in each interval. The bins should cover the data without overlapping.

8.2 Visualizing Data

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proportion: Here a proportion is the relative frequency of a category: the count in that category divided by the total number of observations. A category with 15 of 60 responses has proportion 15/60 = 0.25.
bar chart: A bar chart shows a bar for each category, with the length of the bar equal to the count or the percent. The bars are separated, because the categories are not a continuous scale.
pie chart: A pie chart shows each category as a slice of a circle. The angle, or the area, of a slice matches that category’s proportion of the whole. Too many slices make it hard to compare.
stem-and-leaf plot: A stem-and-leaf plot splits each number into a stem and a final digit, the leaf. The plot lists every value and, at the same time, shows the shape of the distribution. 42 appears as stem 4 and leaf 2.
distribution (of quantitative data): The distribution of a quantitative variable is the pattern of its values: where they cluster, how far they spread, and whether a tail stretches to one side. A graph is a picture of that pattern.
histogram: A histogram displays a binned frequency distribution. The bars touch, because the bins meet, and the height of each bar is the count or the relative frequency in that interval.
bar chart for labeled data: A bar chart for labeled data uses categories that already have names, rather than numerical bins. Compare the bar heights to compare the categories, and do not read the gaps between bars as numerical distance.

8.3 Mean, Median and Mode

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mode: The mode is the value that appears most often. A data set can have more than one mode, and a data set in which every value appears once has no useful mode.
bimodal: A distribution is bimodal when it has two values, or two separated peaks, that occur most often. The two peaks can mean the data mix two different groups.
median: The median is the middle value after the data are ordered. If the count is even, average the two middle values. The median does not move much when one extreme value is added.
mean: The mean is the arithmetic average: add the values and divide by how many there are. One unusually large or small value pulls the mean toward itself, which is why a skewed set can have a mean far from its median.

8.4 Range and Standard Deviation

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range: The range is the largest value minus the smallest value. It is easy to compute, but one extreme value changes it completely, and it says nothing about the values in between.
standard deviation: The standard deviation measures typical distance from the mean. For a sample, subtract the mean from each value, square those differences, divide by one less than the count, and take the square root. A larger standard deviation means a more spread-out set.

8.5 Percentiles

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percentile: A percentile reports the position of a value in an ordered list. The value at the 80th percentile is at or above about 80% of the data. It is a position, not a percent score by itself.
quartile: Quartiles cut an ordered data set into four roughly equal parts. The first quartile is the 25th percentile, the second is the median, and the third is the 75th percentile. The interquartile range is Q3 − Q1.
quintile: Quintiles cut an ordered data set into five roughly equal parts, at the 20th, 40th, 60th, and 80th percentiles. Each fifth of the data is one quintile group.
quantile: A quantile is the general name for a cut point that leaves a chosen fraction of the data below it. Percentiles, quartiles, and quintiles are quantiles with particular fractions.

8.6 The Normal Distribution

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normal distribution: A normal distribution is a symmetric, bell-shaped pattern centered at its mean. The mean, median, and mode coincide, and the spread is described by the standard deviation. Many measurements are modeled this way, though real data are only approximately normal.
68-95-99.7 Rule: In a normal distribution, about 68% of the values lie within 1 standard deviation of the mean, about 95% within 2, and about 99.7% within 3. A value more than 3 standard deviations away is rare under this model.
standardized score (zz-score): A z-score says how many standard deviations a value is from the mean: z = (value − mean) / standard deviation. A positive z is above the mean, a negative z is below it, and the book’s label writes the letter as zz.

8.8 Scatter Plots, Correlation, and Regression Lines

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response variable (dependent variable): The response variable is the outcome you are trying to explain or predict. It is plotted on the vertical axis and is also called the dependent variable.
explanatory variable (independent variable): The explanatory variable is the one you think helps explain the response. It is plotted on the horizontal axis. Calling it independent does not by itself prove that it causes the response.
scatter plot: A scatter plot graphs each observation as a point, with the explanatory variable as x and the response as y. The cloud of points shows the direction, the form, and the strength of the relationship.
positive linear relationship: A positive linear relationship means the points rise together in a straight-line pattern. As the explanatory variable increases, the response tends to increase.
negative linear relationship: A negative linear relationship means the points fall in a straight-line pattern. As the explanatory variable increases, the response tends to decrease.
correlation coefficient: The correlation coefficient, r, measures the strength and direction of a straight-line relationship. It is between −1 and 1. Values near 1 or −1 are strong linear relationships; a value near 0 means little linear relationship. Correlation is not causation.
regression line (least-squares line, line of best fit): The least-squares regression line is the straight line that makes the sum of the squared vertical distances from the points as small as possible. Its equation predicts a typical response for a chosen x, and it should not be trusted far outside the data.

Chapter 9

Metric measurement

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Summary

The metric system converts by powers of ten inside one kind of measure. Kilo- means 1,000, centi- means 0.01, and milli- means 0.001. Kilometers become meters, and meters become centimeters, without leaving length. Mass, capacity, and length are different families and are not converted into each other by a prefix.

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9.1 The Metric System

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metric system: The metric system is a base-10 system of measurement. Each step to the next prefix multiplies or divides by a power of 10, so conversions move the decimal point instead of using mixed numbers such as feet and inches.
meter (m): The meter is the base unit of length. A meter is a little longer than a yard. The prefixes millimeter, centimeter, and kilometer mean one-thousandth, one-hundredth, and one thousand meters.
gram (g): The gram is the base unit of mass. A paper clip is about a gram, and a kilogram is 1,000 grams. Everyday weight in metric units is usually stated in kilograms or grams.
liter (L): The liter is the everyday metric unit of liquid volume. One liter is 1,000 milliliters, and a milliliter matches a cubic centimeter.
square meter (m2): A square meter is the area of a square that is 1 meter on each side. Area conversions use the square of the length factor: 1 square meter is 10,000 square centimeters, because 100 × 100 = 10,000.
cubic meter (m3): A cubic meter is the volume of a cube that is 1 meter on each edge. Volume conversions use the cube of the length factor. One cubic meter is 1,000 liters.
degrees Celsius (°C): Degrees Celsius measure temperature in the metric system. Water freezes at 0°C and boils at 100°C under standard conditions. The size of one Celsius degree is the same as one kelvin.
conversion factor: A conversion factor is a ratio equal to 1 that changes the unit and keeps the quantity. To change 2.5 meters to centimeters, multiply by 100 cm / 1 m. The unit you want to cancel goes on the opposite side of the fraction.

9.2 Measuring Area

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area: Area measures the amount of surface a flat figure covers. It is found from length measurements, and the unit is a square unit. Doubling every length of a figure multiplies its area by 4.
square units: Square units are the units of area, such as square centimeters or square meters. A length unit has to be squared because area covers a two-dimensional region. You cannot add square meters to meters.

9.3 Measuring Volume

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volume: Volume measures the space inside a three-dimensional object, or the capacity of a container. It is computed from length measurements and expressed in cubic units, or in liters for fluids.
cubic units: Cubic units are the units of volume, such as cubic centimeters. Because volume is three-dimensional, a length conversion factor is cubed. A cube 10 cm on a side holds 1,000 cubic centimeters.

9.4 Measuring Weight

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mass: Mass measures the amount of matter in an object. In the metric system the gram and the kilogram measure mass. Weight is the gravitational pull on that mass, so the same mass weighs less on the Moon, but everyday speech often uses “weight” for the kilogram reading on a scale.

9.5 Measuring Temperature

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temperature: Temperature measures how hot or cold something is. Celsius is the metric scale used in this chapter, with 0°C at the freezing point of water and 100°C at its boiling point. Fahrenheit converts by F = (9/5)C + 32.

Chapter 10

Geometry

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Summary

Plane figures are measured by perimeter and area, and solids by surface area and volume. A triangle’s interior angles add to 180°, and a right triangle obeys a² + b² = c². Circle circumference is 2πr and area is πr². When similar figures scale by a factor k, lengths scale by k and areas by k².

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10.1 Points, Lines, and Planes

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line segment: A line segment is the straight path connecting two endpoints, including those endpoints. Unlike a line, it has a finite length, which you can measure.
plane: A plane is a flat surface that extends without end in every direction along that surface. Three points that are not on the same line determine one plane. A tabletop is a model of a small piece of a plane.
union: The union of two geometric figures is the set of all points that belong to either figure or to both. The union of two segments that share an endpoint is the longer path made by both.
intersection: The intersection of two figures is the set of points they share. Two distinct lines in a plane intersect in one point, or not at all if they are parallel. Two planes intersect in a line, or not at all.
parallel: Parallel lines lie in the same plane and never meet, so the distance between them stays constant. Parallel planes never meet. A transversal crossing parallel lines creates congruent corresponding angles.
perpendicular: Perpendicular lines, segments, or planes meet at a right angle, 90°. The slopes of two perpendicular lines that are not vertical are negative reciprocals of each other.

10.2 Angles

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vertex: The vertex of an angle is the point where the two sides meet. An angle is named by that point, or by three letters with the vertex letter in the middle.
right angle: A right angle measures exactly 90°. The small square drawn in the corner of a diagram marks a right angle.
acute angle: An acute angle measures more than 0° and less than 90°. It is sharper than a right angle.
obtuse angle: An obtuse angle measures more than 90° and less than 180°. It is wider than a right angle and narrower than a straight angle.
straight angle: A straight angle measures 180°. Its sides form a straight line.
complementary: Two angles are complementary when their measures add to 90°. If one measures 35°, its complement measures 55°.
supplementary: Two angles are supplementary when their measures add to 180°. Adjacent angles that form a straight line are supplementary.

10.3 Triangles

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acute: An acute triangle has three acute angles. Every angle is less than 90°, and the three angles still add to 180°.
obtuse: An obtuse triangle has one obtuse angle. It can have only one, because two angles of 90° or more would already add to more than 180°.
isosceles: An isosceles triangle has at least two sides of equal length. The angles opposite those equal sides are also equal.
equilateral: An equilateral triangle has all three sides equal. All three angles are equal too, so each measures 60°. Every equilateral triangle is also isosceles.
hypotenuse: The hypotenuse is the side opposite the right angle in a right triangle. It is the longest side. If the legs are a and b, then a² + b² = c², where c is the hypotenuse.
congruence: Congruent figures have the same size and the same shape. Corresponding sides and corresponding angles are equal. Triangles can be shown congruent by SSS, SAS, ASA, or AAS, and by HL for right triangles.
similarity: Similar figures have the same shape but not necessarily the same size. Corresponding angles are equal, and corresponding sides are proportional. The ratio of any pair of matching sides is the scale factor.
scaling factor: The scaling factor is the ratio of a length on one figure to the matching length on a similar figure. If the factor is 3, every length is tripled, the area is multiplied by 9, and the volume, for similar solids, by 27.

10.4 Polygons, Perimeter, and Circumference

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perimeter: The perimeter is the distance around a polygon. Add the side lengths. For a rectangle it is 2(length + width).
polygon: A polygon is a closed flat figure made of straight sides that meet only at their endpoints. Triangles, quadrilaterals, and pentagons are polygons. A circle is not, because it has no straight sides.
pentagon: A pentagon is a polygon with five sides. In a regular pentagon all sides are equal and all interior angles are equal.
hexagon: A hexagon is a polygon with six sides. A regular hexagon splits into six equilateral triangles.
heptagon: A heptagon is a polygon with seven sides. The sum of its interior angles is (7 − 2) × 180° = 900°.
octagon: An octagon is a polygon with eight sides. A stop sign is a regular octagon. The sum of its interior angles is (8 − 2) × 180° = 1,080°.
quadrilateral: A quadrilateral is a polygon with four sides. The sum of its interior angles is 360°. Rectangles, squares, parallelograms, and trapezoids are special quadrilaterals.
trapezoid: A trapezoid has exactly one pair of parallel sides in the definition used by this course. Those parallel sides are the bases, and the height is the perpendicular distance between them.
parallelogram: A parallelogram is a quadrilateral with both pairs of opposite sides parallel. Opposite sides are equal, opposite angles are equal, and consecutive angles are supplementary.
circumference: The circumference is the distance around a circle. If the radius is r and the diameter is d, then C = 2πr = πd.

10.5 Tessellations

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tessellation: A tessellation is a covering of the plane by copies of one or more shapes, with no gaps and no overlaps. A regular tessellation uses one kind of regular polygon. Equilateral triangles, squares, and regular hexagons each tile the plane.
translation: A translation slides a figure along a straight path. Every point moves the same distance in the same direction, so the size and the orientation stay the same.
reflection: A reflection flips a figure across a line, the mirror line. Each point and its image are the same distance from that line, on opposite sides. Orientation reverses.
rotation: A rotation turns a figure around a fixed point by a stated angle. A 180° rotation sends each point to the opposite side of the center, at the same distance.
glide reflection: A glide reflection is a translation followed by a reflection across a line parallel to the slide. Footprints in a straight walk form a glide reflection.

10.6 Area

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triangle: The area of a triangle is (1/2) × base × height. The height must be perpendicular to the base you chose, and it may fall outside the triangle if the triangle is obtuse.
square: The area of a square with side s is s². Because all sides are equal and all angles are right angles, one side determines the area.
rectangle: The area of a rectangle is length × width. The two dimensions must be perpendicular, and they must be in the same unit before you multiply.
rhombus: A rhombus is a quadrilateral with all sides equal. Its area is (1/2) × d₁ × d₂, where d₁ and d₂ are the lengths of the diagonals, which cross at right angles.
apothem: The apothem of a regular polygon is the perpendicular distance from the center to a side. The area of a regular polygon is (1/2) × apothem × perimeter.
radius: The radius is the distance from the center of a circle to a point on the circle. It is half the diameter. The area of the circle is πr².
circle: A circle is the set of all points in a plane at a fixed distance, the radius, from a center. Its circumference is 2πr and its area is πr².

10.7 Volume and Surface Area

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surface area: Surface area is the total area of all the outer faces of a solid, including the bases. It is measured in square units. For a box, add the areas of the six rectangular faces.
volume: Volume is the space inside a solid, measured in cubic units. For a prism or a cylinder with parallel bases, volume = area of the base × height, with the height perpendicular to the base.
right prism: A right prism has two parallel bases that are congruent polygons, and the sides meet those bases at right angles. Its volume is the area of one base times the height.
right cylinder: A right cylinder has two parallel circular bases, and its side rises perpendicular to those bases. Its volume is πr²h, and its lateral surface, unrolled, is a rectangle of width 2πr and height h.

10.8 Right Triangle Trigonometry

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right triangle: A right triangle has one 90° angle. The other two angles are acute and add to 90°. The side opposite the right angle is the hypotenuse; the other two sides are the legs.
sine: For an acute angle in a right triangle, the sine is the length of the opposite side divided by the length of the hypotenuse. If sin(A) = 0.5, the opposite side is half as long as the hypotenuse.
cosine: For an acute angle in a right triangle, the cosine is the length of the adjacent side divided by the length of the hypotenuse. The adjacent side is the leg that touches the angle and is not the hypotenuse.
tangent: For an acute angle in a right triangle, the tangent is the opposite side divided by the adjacent side. Tangent does not use the hypotenuse. It also equals sine divided by cosine.

Chapter 11

Voting and apportionment

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Summary

The same ballots can crown different winners under different rules. Plurality takes the most first-place votes and does not require a majority. A majority is more than half. Borda count awards points down the ranking, and a Condorcet winner beats every rival head to head. Apportionment, including Hamilton’s method, divides seats by quota and can produce paradoxes such as the Alabama paradox.

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11.1 Voting Methods

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majority: A majority is more than half of the votes cast. With 40 votes, a majority is at least 21. A candidate can finish first and still lack a majority when the rest of the votes are split.
plurality: A plurality is the largest share of the votes, even if it is not more than half. In plurality voting, the candidate with the most first-place votes wins.
runoff election: A runoff is a second election held when no candidate meets the winning requirement, usually a majority. Only the top finishers from the first round appear on the runoff ballot.
runoff voting system: A runoff system uses a first round and, if needed, a second round between the leading candidates. It tries to produce a winner supported by a majority of the people who vote in the final round.
two-round system: The two-round system is a runoff method with at most two rounds. If someone wins a majority in the first round, that person wins. Otherwise the top two candidates face each other in the second round.
Hare Method: The Hare method eliminates the candidate with the fewest first-place votes, transfers each of those ballots to the next choice still in the race, and repeats until someone has a majority. It is a form of instant runoff.
preference ranking: A preference ranking lists the candidates in a voter’s order, from most wanted to least wanted. Voting methods such as Hare, Borda, and pairwise comparison need the full ranking, not only a first choice.
ranked ballot: A ranked ballot is the ballot on which a voter writes that order of preference. Tallying it depends on the method: first places only, points by rank, or one-on-one comparisons.
ranked-choice voting (RCV): Ranked-choice voting lets voters rank the candidates, then eliminates last-place candidates and transfers ballots until a winner has a majority of the remaining ballots. Voters do not have to return for a separate runoff.
instant runoff voting (IRV): Instant runoff voting is ranked-choice voting that imitates a series of runoffs in one count. The weakest candidate is removed, ballots move to the next listed choice, and the process repeats.
Borda count method: The Borda count gives points for every rank, not only for first place. With n candidates, a first-place vote is often worth n points, a second-place vote n − 1, and so on down to 1. The highest total wins.
Borda score: A candidate’s Borda score is the total of the points that candidate receives from every ballot. A candidate who is rarely first but often second can outscore a candidate with more first-place votes.
Compromise candidate: A compromise candidate is not many voters’ first choice but is acceptable to a wide group. Point-based methods such as Borda tend to reward this kind of candidate more than plurality does.
divisive candidate: A divisive candidate is strongly liked by some voters and strongly disliked by others. That candidate can win a plurality of first places while losing methods that look at the rest of the rankings.
pairwise comparison method: The pairwise comparison method matches every candidate against every other candidate, one pair at a time, and gives a point to the one preferred by more voters. The candidate with the most pairwise points wins.
Condorcet voting methods: Condorcet methods decide elections from the one-on-one comparisons. If one candidate beats every other candidate head to head, that candidate is the Condorcet winner. Some sets of ballots have no such candidate.
Condorcet candidate: The Condorcet candidate is the one who would beat each of the other candidates in a two-person contest. A method that fails to elect that person, when one exists, violates the Condorcet criterion.
approval voting system: In approval voting, a voter may approve of as many candidates as the voter wants, one unranked vote each. The candidate with the most approvals wins. There is no first, second, and third.
approval voting ballot: An approval ballot is a list of candidates on which the voter marks every acceptable candidate and leaves the others blank. Marking two names does not split or rank the vote.
spoiler: A spoiler is a candidate with little chance to win whose presence changes which of the leading candidates wins. In plurality voting, a spoiler can pull enough votes from a similar candidate to elect someone else.

11.2 Fairness in Voting Methods

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

majority criterion: The majority criterion says that if a candidate is ranked first by more than half of the voters, that candidate should win. Plurality satisfies this. Borda count can fail it.
tyranny of the majority: Tyranny of the majority is the worry that a group larger than half can always impose its first choice and ignore the preferences of the rest. Fairness criteria ask what a method owes to voters besides the majority.
Condorcet criterion: The Condorcet criterion says that a candidate who beats every rival head to head should win the election. Pairwise comparison satisfies it. Plurality, Hare, and Borda can fail it.
Condorcet method: A Condorcet method is any voting method that elects the Condorcet candidate whenever one exists. Pairwise comparison is such a method. The other standard methods in this chapter are not.
monotonicity criterion: The monotonicity criterion says that moving a candidate up on some ballots, and changing nothing else, should not cause that candidate to lose. Instant runoff can fail this, because a change in elimination order can undo a winner.
up-rank: To up-rank a candidate is to move that candidate higher on a ballot, ahead of someone who used to be preferred. A monotone method never punishes a candidate for being up-ranked.
down-rank: To down-rank a candidate is to move that candidate lower on a ballot. Under monotonicity, down-ranking a losing candidate should not turn that candidate into the winner.
independence of irrelevant alternatives criterion (IIA): Independence of irrelevant alternatives says that if A beats B, then removing some other candidate C, or adding one, should not reverse the result between A and B. Many common methods fail this test.
Arrow’s Impossibility Theorem: Arrow’s theorem says that no ranked voting method with three or more candidates can satisfy a short list of reasonable fairness conditions all at once, including independence of irrelevant alternatives and the requirement that it not be a dictatorship. Every method gives something up.
cardinal voting system: A cardinal method asks voters for a rating or a score, not only a rank order. Approval voting is a simple cardinal method: each candidate is scored yes or no. Range and score voting allow a wider scale.

11.3 Standard Divisors, Standard Quotas, and the Apportionment Problem

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

apportion: To apportion is to divide a fixed number of identical seats among groups according to their populations. The result has to be whole seats, so the exact proportional shares usually cannot be given.
apportionment problem: The apportionment problem is how to hand out a fixed number of whole seats in proportion to population when the exact shares are not whole numbers. Different rounding rules give different fair-looking results.
proportional: A distribution is proportional when each group’s share of the seats matches its share of the population. Exact proportion would give state population ÷ standard divisor, which is the standard quota, often not an integer.
part-to-part ratio: A part-to-part ratio compares one group’s population with another group’s population, rather than with the whole. Apportionment instead uses each group’s part of the total, a part-to-whole comparison.
representative democracies: In a representative democracy, seats in a legislature stand for the people of districts or states. Apportionment decides how many of those seats each state receives.
standard divisor: The standard divisor is the total population divided by the number of seats. It is the number of people each seat would represent if the seats split the population evenly.
states: In an apportionment problem, the states are the groups that receive seats. They may be actual states, or any other parties sharing the seats, such as classes or offices.
seats: Seats are the identical items being divided: members of a legislature, or any other whole units that cannot be split. The house size is the number of seats.
house size: The house size is the total number of seats to be given out. It is fixed before the seats are assigned. Changing it can change not only the totals but, under some methods, which state gains a seat.
state population: A state’s population is the count used for that state in the apportionment. A state’s standard quota is its population divided by the standard divisor.
total population: The total population is the sum of the populations of all the states. The standard divisor is this total divided by the house size.
standard quota: The standard quota of a state is its population divided by the standard divisor. It is the exact fractional number of seats the state would get. The lower quota is that number rounded down, and the upper quota is that number rounded up.

11.4 Apportionment Methods

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

apportionment problem: An apportionment method is a rule for turning standard quotas into whole seats that add up to the house size. Hamilton’s method uses the quotas and gives the leftover seats to the largest fractional parts. Divisor methods, such as Jefferson, Adams, and Webster, adjust the divisor until the rounded quotas add up.
lower quota: The lower quota is the standard quota rounded down to the next whole number. It is the least number of seats a state receives under a method that respects the lower quota. Hamilton’s method never gives a state fewer seats than its lower quota.
upper quota: The upper quota is the standard quota rounded up to the next whole number. A method violates the upper quota when it gives a state more seats than that. Some divisor methods can break the quota.

11.5 Fairness in Apportionment Methods

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

apportionment paradox: An apportionment paradox is a reasonable-looking result that a method can still produce, such as a state losing a seat when the house grows, or a faster-growing state losing a seat to a slower-growing one.
Alabama paradox: The Alabama paradox happens when increasing the house size causes a state to lose a seat, with no change in any population. Hamilton’s method can do this. The name comes from Alabama’s loss of a seat in a historical calculation.
reapportionment: Reapportionment assigns the seats again after a new census, or after the house size changes. The populations are updated and the method is run once more, so a state’s seat count can rise or fall.
population growth rate: The population growth rate compares the change in a population with its starting population. In the population paradox, a state with the faster growth rate can still lose a seat to a state growing more slowly.
population paradox: The population paradox happens when one state gains population faster than another, yet loses a seat to that slower-growing state in a new apportionment. Hamilton’s method can produce it.
new-state paradox: The new-state paradox happens when a new state joins, the house grows by the number of seats the new state receives, and the seat counts of the older states still change among themselves. Adding one party should not reshuffle the others, but under Hamilton’s method it can.

Chapter 12

Graph theory

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Summary

A graph is vertices joined by edges. The degree of a vertex counts the edges that meet it, and the handshaking lemma says the degrees add up to twice the number of edges. A connected graph has an Euler circuit exactly when every degree is even, and an Euler path when zero or two degrees are odd. The Königsberg bridges fail that test. Trees and complete graphs are the other standard families.

Key terms

Each term has a plain-language definition written for this guide. The book’s own key-terms page mostly names the word and sends you back to the section.

Open this chapter’s key terms in the book

12.1 Graph Basics

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

vertex: A vertex is a dot in a graph, also called a node. Vertices stand for the objects in the problem, such as cities, people, or rooms. The plural is vertices.
edge: An edge is a connection between two vertices. It can stand for a road, a handshake, or any relationship the problem cares about. In a simple graph an edge has no direction.
loop: A loop is an edge that begins and ends at the same vertex. A loop contributes 2 to the degree of that vertex, because it touches the vertex twice.
graph (simple graph): A simple graph is a set of vertices and a set of edges, with at most one edge between any pair of vertices and no loops. If a pair may be joined by several edges, the drawing is a multigraph instead.
multigraph: A multigraph allows more than one edge between the same pair of vertices, and it may allow loops. Use one when the problem has several distinct connections between the same two objects.
adjacent (neighboring): Two vertices are adjacent when an edge joins them. They are neighbors. Two edges are adjacent when they share a vertex.
degree: The degree of a vertex is the number of edge ends that touch it. A loop counts twice. In any graph, the sum of all the degrees equals twice the number of edges, because each edge has two ends.

12.2 Graph Structures

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

complete: A complete graph has an edge between every pair of distinct vertices. The complete graph on n vertices has n(n − 1) / 2 edges, and every vertex has degree n − 1.
subgraph: A subgraph is a graph formed by keeping some of the vertices and some of the edges of a larger graph. Every edge you keep must still connect two vertices you kept.
cycle: A cycle is a path that starts and ends at the same vertex and repeats no other vertex and no edge. The smallest cycle in a simple graph has three vertices.
cyclic subgraph: A cyclic subgraph is a subgraph that contains at least one cycle. A graph with no cyclic subgraph is a forest of trees.
clique: A clique is a set of vertices in which every pair is joined by an edge, so the subgraph they form is complete. A triangle is a clique of three vertices.

12.3 Comparing Graphs

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

isomorphic: Two graphs are isomorphic when they are the same graph drawn differently. There is a one-to-one match of their vertices that preserves adjacency: two vertices are connected in one graph exactly when their matches are connected in the other.
isomorphism: An isomorphism is the matching that shows two graphs have the same structure. It renames the vertices and keeps every edge relationship. Equal degree lists are necessary for an isomorphism, but they are not enough by themselves.
planar: A graph is planar when it can be drawn in a plane with no edges crossing. A graph may look crossed in one drawing and still be planar if a different drawing separates the edges.
nonplanar: A nonplanar graph cannot be drawn without at least one pair of edges crossing. The complete graph on five vertices, and the complete bipartite graph that joins two groups of three, are the classic nonplanar examples.
complement: The complement of a graph has the same vertices, and it has an edge exactly where the original graph does not. If two vertices are adjacent in the graph, they are not adjacent in the complement, and the reverse is also true. Loops are not added.
complementary: Complementary graphs are a graph and its complement. Every possible edge between distinct vertices appears in exactly one of the two. The degrees of a vertex in the two graphs add to n − 1.

12.4 Navigating Graphs

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

walk (directed walk): A walk is a sequence of vertices and edges in which each edge connects the vertex before it to the vertex after it. Vertices and edges may be repeated. A directed walk follows the arrows if the edges have direction.
trail (directed trail): A trail is a walk that repeats no edge. Vertices may be repeated. A directed trail is the same idea along directed edges.
path (directed path): A path is a walk that repeats no vertex, and therefore repeats no edge. It is the most restrictive of the three. A directed path follows the direction of each edge.
closed: A walk is closed when it ends at the same vertex where it started. A circuit and a cycle are closed.
open: A walk is open when it ends at a different vertex from the one where it started. A path from one town to a different town is open.
closed walk: A closed walk begins and ends at the same vertex and may repeat vertices or edges along the way. Not every closed walk is a cycle.
circuit (closed trail): A circuit is a closed trail: it starts and ends at the same vertex and repeats no edge. A cycle is a circuit that also repeats no vertex except the start.
directed cycle (closed path): A directed cycle is a closed directed path. Following the arrows, it returns to the start and visits no other vertex twice.
coloring (graph coloring): A coloring assigns a color to every vertex so that no two adjacent vertices share a color. The goal is often to do this with as few colors as possible.
nn-coloring: An n-coloring is a proper coloring that uses at most n colors. The book writes the count as nn. A graph has a 2-coloring exactly when it has no odd cycle.
chromatic number: The chromatic number is the smallest number of colors that can color the graph properly. A complete graph on n vertices has chromatic number n, because every vertex is adjacent to every other.

12.5 Euler Circuits

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

connected: A graph is connected when there is a path between every pair of vertices. You can travel from any vertex to any other along edges.
component: A component is a largest connected piece of a graph. A connected graph has one component. Two vertices are in the same component when a path joins them.
disconnected: A graph is disconnected when at least two vertices have no path between them. It then has two or more components.
Euler circuit: An Euler circuit is a closed trail that uses every edge of the graph exactly once. The graph must be connected, and every vertex must have even degree.
Eulerian graph: An Eulerian graph has an Euler circuit. For a connected graph, that happens exactly when every vertex has even degree. The circuit can start at any vertex.
Chinese postman problem: The Chinese postman problem asks for the shortest closed route that travels every street at least once. If the graph is Eulerian, an Euler circuit is the answer. If not, some edges must be repeated.
Eulerization: Eulerization adds duplicate edges along existing routes until every vertex has even degree, using as little added length as possible. The new multigraph has an Euler circuit, which is an efficient postman route on the original streets.

12.6 Euler Trails

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

algorithm: An algorithm is a finite list of exact steps that solves a problem. Fleury’s algorithm is an algorithm for building an Euler circuit or trail.
Fleury’s algorithm: Fleury’s algorithm builds an Euler trail by leaving a vertex along an unused edge, and it never uses a bridge of the remaining graph unless there is no other unused edge. That rule keeps you from getting stuck with unused edges somewhere else.
Euler trail: An Euler trail uses every edge exactly once but does not have to end where it started. A connected graph has one exactly when it has zero or two vertices of odd degree. The trail starts at one odd vertex and ends at the other.
bridge: A bridge is an edge whose removal increases the number of components. It is the only connection between two parts of the graph. Fleury’s algorithm saves a bridge of the remaining graph until no other choice is left.
local bridge: A local bridge is an edge that is the only short connection between its endpoints. Removing it does not have to disconnect the graph, but it forces a longer alternate route between those two vertices.

12.7 Hamilton Cycles

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Hamilton cycle, or Hamilton circuit: A Hamilton cycle visits every vertex exactly once and returns to the start. Unlike an Euler circuit, it does not have to use every edge. There is no simple degree test that decides whether one exists.
nn factorial: n factorial, written n!, is the product of the integers from 1 through n. The book writes the count as nn. There are (n − 1)! / 2 distinct Hamilton cycles on a complete graph of n vertices once direction and starting point are ignored.
weighted graph: A weighted graph assigns a number, the weight, to each edge. The weight may be a distance, a time, or a cost. The total weight of a route is the sum of the weights of its edges.
total weight: The total weight of a path or cycle is the sum of the weights of the edges it uses. In a traveling-salesperson problem, you want a Hamilton cycle whose total weight is as small as possible.

12.8 Hamilton Paths

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

Hamilton path: A Hamilton path visits every vertex exactly once and does not need to return to the start. A Hamilton cycle contains a Hamilton path, but a graph can have a Hamilton path and no Hamilton cycle.

12.9 Traveling Salesperson Problem

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

brute force algorithm: The brute force algorithm lists every Hamilton cycle, computes each total weight, and chooses the smallest. It finds the best route, but the number of cycles grows so fast that it is practical only for small graphs.
greedy algorithm: A greedy algorithm makes the locally best choice at each step and does not go back to reconsider it. It is fast, and for the traveling salesperson problem it can miss the shortest tour.
traveling salesperson problem (TSP): The traveling salesperson problem asks for the shortest Hamilton cycle in a weighted complete graph: a route that visits every city once, returns home, and has the least total distance or cost.
brute force method: The brute force method solves a small TSP by checking every possible tour and keeping the one with the least total weight. The answer is exact. The work becomes unrealistic as soon as the number of cities is more than modest.
nearest neighbor method: The nearest neighbor method starts at a vertex and repeatedly travels to the closest unvisited vertex, then returns home at the end. It builds a Hamilton cycle quickly. Different starting cities can give different totals, and none of them is guaranteed to be the shortest.

12.10 Trees

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

acyclic: An acyclic graph contains no cycle. Every path in it is the only simple route between its endpoints.
tree: A tree is a connected graph with no cycles. A tree with n vertices has exactly n − 1 edges. Between any two vertices there is exactly one path.
forest: A forest is a graph with no cycles. Its components are trees. A connected forest is a single tree.
path graph or linear graph: A path graph is a tree whose vertices lie in one chain: two vertices have degree 1 and the others have degree 2. It is also called a linear graph.
star tree: A star tree has one central vertex joined to every other vertex, and it has no other edges. The center has degree n − 1, and every other vertex is a leaf of degree 1.
root: The root is the vertex chosen as the starting point of a rooted tree. Every other vertex lies on exactly one path from the root, so the edges point away from the root in levels.
starlike tree: A starlike tree has exactly one vertex whose degree is greater than 2. Every branch leaving that vertex is a path. A star is the starlike tree in which those paths are single edges.
caterpillar tree: A caterpillar is a tree that becomes a path when all of its leaves are removed. The remaining path is the spine, and every other vertex is a leaf attached to that spine.
lobster tree: A lobster is a tree that becomes a caterpillar when all of its leaves are removed. The vertices one step in from the leaves form a caterpillar.
spanning tree: A spanning tree of a connected graph is a subgraph that is a tree and includes every vertex. It keeps the graph connected while using the smallest possible number of edges, which is one less than the number of vertices.
minimum spanning tree: A minimum spanning tree is a spanning tree whose total edge weight is as small as possible. Kruskal’s and Prim’s algorithms build one by adding the cheapest edge that does not create a cycle.

Chapter 13

Math and art

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Summary

The chapter connects counting and geometry to design. The Fibonacci sequence adds the two previous terms, and the ratio of neighbors approaches the golden ratio, about 1.618. Reflection, rotation, and translation are different symmetries, and a tessellation covers the plane with no gaps and no overlaps.

Key terms

Each term has a plain-language definition written for this guide. The book’s own key-terms page mostly names the word and sends you back to the section.

Open this chapter’s key terms in the book

13.1 Math and Art

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

golden ratio: The golden ratio is the number (1 + √5) / 2, about 1.618. A segment is divided in this ratio when the long part divided by the short part equals the whole divided by the long part. It appears in the growth of the Fibonacci sequence and in some proportions used in art.
ϕ: The symbol ϕ, phi, stands for the golden ratio, (1 + √5) / 2. It satisfies the equation ϕ = 1 + 1/ϕ, and its decimal begins 1.618033….
Fibonacci sequence: The Fibonacci sequence begins 1, 1, 2, 3, 5, 8, and each later term is the sum of the two terms before it. The ratio of one term to the previous term gets closer and closer to the golden ratio.
golden rectangle: A golden rectangle has a length and a width in the golden ratio. Cutting a square from it leaves a smaller rectangle that is also golden. Spirals are often drawn through those successive squares.

13.2 Math and the Environment

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

greenhouse gas: A greenhouse gas, such as carbon dioxide or methane, absorbs heat that the Earth radiates and sends some of it back toward the surface. More of these gases in the atmosphere raises the energy the climate system holds.
CO2 emissions: CO₂ emissions are the carbon dioxide released by an activity, often measured in kilograms or metric tons. Comparing activities requires the same unit and a clear statement of what is included, such as fuel burned or electricity used.
watt: A watt is a unit of power, the rate of energy use. One watt is one joule per second. A device rated 60 watts uses 60 joules of energy every second it runs.
kilowatt (kW): A kilowatt is 1,000 watts. Energy bills use kilowatt-hours: a kilowatt of power used for one hour. A 2 kW heater run for 3 hours uses 6 kilowatt-hours.

13.3 Math and Medicine

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

concentrations/dosages of drugs: A concentration says how much drug is in a given amount of mixture, such as milligrams per milliliter. A dosage is the amount given to a patient, often computed from body weight. Units have to match before you multiply or divide.
mathematical modeling: A mathematical model is an equation or a set of equations used to describe a real situation and predict what happens if an input changes. In medicine, a model might relate dose, time, and the amount of drug still in the body. A model leaves some real details out, so its prediction is only as good as its assumptions.

13.4 Math and Music

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

frequency: Frequency is how many cycles of a wave occur in one second. A higher frequency is a higher note. Doubling the frequency raises the pitch by one octave.
pitch: Pitch is how high or low a tone sounds. It is what we hear from the frequency of the vibration. The same note name returns every time the frequency doubles.
Hertz: A hertz, written Hz, is one cycle per second. The note A above middle C is tuned near 440 Hz, which means the sound wave completes 440 cycles each second.
decibel: A decibel is a unit of sound intensity on a logarithmic scale. An increase of 10 decibels multiplies the intensity by 10. Because the scale is logarithmic, you cannot compare loudness by subtracting decibel readings as if they were ordinary amounts.
half-step: A half-step is the smallest move between adjacent keys on a piano, counting both white and black keys, such as from C to C♯. Twelve half-steps make one octave.
whole step: A whole step is two half-steps, such as from C to D. Major scales are built from a set pattern of whole steps and half-steps.
flat: A flat lowers a note by one half-step. B♭ is one half-step below B.
sharp: A sharp raises a note by one half-step. F♯ is one half-step above F. On the piano, one black key is both the sharp of the key below it and the flat of the key above it.
octave: An octave is the interval between a frequency and double that frequency. The two notes have the same letter name, and the higher one vibrates twice as fast.

13.5 Math and Sports

Definitions written for this guide, in enough detail to study from. The link opens the section in the book.

data analytics: Data analytics in sports uses collected numbers, such as shots, distances, and results, to describe what happened and to estimate what is likely next. A rate has to be attached to the right opportunity, or a player with more playing time will look better only because of the extra chances.
seeding: Seeding ranks teams or players before a tournament so that the strongest, by the ranking rule, do not meet in the earliest rounds. A number-1 seed is the highest-ranked entry. Seeding does not change anyone’s skill; it changes the bracket.