Chapter 2 — Front Matter

Purpose and Overview

Algebra is built on numbers. Before a student can solve equations, evaluate functions, graph relationships, or work with polynomials, they need a clear and confident understanding of the number system that underlies all of those activities. Chapter 2 provides that foundation.

In Arithmetic and Pre-Algebra Foundations, you worked with whole numbers, fractions, decimals, percents, negative numbers, and the basic rules of arithmetic. You learned how to add, subtract, multiply, and divide. You learned order of operations. You learned what a signed number means. This chapter does not simply repeat that work. It organizes it. It asks you to look at the number system as a whole, understand how different kinds of numbers relate to one another, and connect the arithmetic properties you already know to the algebraic reasoning you are about to begin.

Properties such as the commutative property and the distributive property are not vocabulary words to memorize and forget. They are rules that allow you to rewrite expressions without changing their meaning. They are the reason you can rearrange terms, simplify expressions, and solve equations step by step without making errors. Understanding why a step is legal is just as important as knowing how to perform it.

By the end of this chapter, you will understand the structure of the real number system, feel confident working with signed numbers in any context, recognize and apply the major properties of operations, and use those properties to simplify both numerical and algebraic expressions. This chapter is the bridge between arithmetic and formal algebra. Every chapter that follows depends on the ideas developed here.

Opening Question

A student is simplifying the expression −3(x − 4) + 2x. She writes: −3(x − 4) + 2x = −3x − 12 + 2x = −x − 12. Another student writes: −3(x − 4) + 2x = −3x + 12 + 2x = −x + 12. Both students followed a procedure. Only one is correct. The difference between them is not a matter of being fast or slow, smart or careless — it is a precise understanding of signed numbers and the distributive property. By the time you complete Chapter 2, you will know exactly which answer is correct, why it is correct, and how to explain the error in the other solution.

14Sections

11Objectives

19Key Terms

12+Worked Examples

Learning Objectives

1Identify and classify real numbers as natural, whole, integer, rational, or irrational.
2Place real numbers on a number line and compare their values using inequality symbols.
3Compute absolute value and interpret it as distance from zero.
4Add, subtract, multiply, and divide real numbers, including all signed number cases.
5Find and use the opposite and reciprocal of a real number.
6State and apply the commutative and associative properties of addition and multiplication.
7Apply the distributive property, including distributing negative factors over subtraction.
8Identify identity elements and inverse elements for addition and multiplication.
9Simplify numerical and algebraic expressions using number properties.
10Apply real number operations to measurement and real-world models.
11Connect the properties of real numbers to the logic of equation solving.

Chapter 2 — Contents

Sections at a Glance

2.1The Real Number SystemNested structure of number families; rational vs. irrational; closure; number system map
2.2Natural Numbers, Whole Numbers, Integers, and Rational NumbersDefining each family; counting, zero, negatives, fractions, terminating and repeating decimals; classifying examples
2.3Irrational Numbers and Real NumbersNon-terminating non-repeating decimals; π and square roots; perfect vs. non-perfect squares; the complete real line
2.4Number Lines and Ordering Real NumbersPlotting real numbers; left-right order; comparing with < and >; common ordering mistakes with negatives
2.5Absolute Value and DistanceDefinition as distance from zero; formal piecewise definition; distance between two numbers; evaluating expressions with absolute value
2.6Adding and Subtracting Real NumbersSame-sign and different-sign addition; rewriting subtraction as addition of the opposite; signed examples
2.7Multiplying and Dividing Real NumbersSign rules for two factors; same/different sign outcomes; products of multiple signed numbers; division; undefined division by zero
2.8Opposites and ReciprocalsAdditive inverses; multiplicative inverses; sum to zero; product to one; zero has no reciprocal
2.9Properties of Addition and MultiplicationCommutative property; associative property; why order and grouping matter (or don't); non-examples with subtraction and division
2.10The Distributive Property RevisitedFull restatement; distributing negatives; every term must be reached; connecting back to the opening question
2.11Identity and Inverse PropertiesAdditive identity (0); multiplicative identity (1); additive inverse; multiplicative inverse; preview of equation solving
2.12Simplifying Numerical and Algebraic ExpressionsOrder of operations with signed numbers; combining like terms; distributing then combining; multi-variable expressions
2.13Real Numbers in Measurement and ModelsSigned numbers in context (temperature, elevation, finance); formulas with real numbers; equivalent formula forms
2.14Preparing for Equation SolvingHow each property maps to an equation-solving step; preview of Chapter 3; why arithmetic precision determines algebraic reliability

Section 2.1

The Real Number System

Mathematics would not get very far if it only used one kind of number. Over time, mathematicians extended the number system to handle problems that simpler numbers could not solve. The result is the real number system: a collection of numbers organized into layers, each layer containing and extending the previous one.

Think of the real number system as a set of nested categories. The largest category is the real numbers, and smaller categories live inside it. Understanding this structure helps you know what kind of number you are working with and what rules apply.

Category	Description	Examples
Natural Numbers	Counting numbers; positive integers only	`1, 2, 3, 4, 5, ...`
Whole Numbers	Natural numbers plus zero	`0, 1, 2, 3, 4, ...`
Integers	Whole numbers and their negatives	`..., -2, -1, 0, 1, 2, ...`
Rational Numbers	Expressible as a/b where a, b are integers, b ≠ 0	`-3/4, 0.25, 0.333..., 7`
Irrational Numbers	Non-terminating, non-repeating decimals; cannot be written as a/b	`√2, π, √5`
Real Numbers	All rational and irrational numbers; the entire number line	Every number listed above

Structure — Containment

Every natural number is also whole, integer, rational, and real. Every integer is rational and real. But not every rational number is an integer, and not every real number is rational. The categories nest inward: Natural ⊂ Whole ⊂ Integer ⊂ Rational ⊂ Real.

Closure

A set is closed under an operation if performing that operation on members of the set always produces another member of the same set. The integers are closed under addition (integer + integer = integer) but not under division (3 ÷ 4 is not an integer). The real numbers are closed under addition, subtraction, multiplication, and division — except division by zero.

Practice 2.1Concept Check

Draw the nested oval diagram of the real number system from memory. Label all five regions and place one example number in each.
Give an example of a number that is an integer but not a natural number.
Give an example of a number that is rational but not an integer.
Is the set of natural numbers closed under subtraction? Explain with an example.

Section 2.2

Natural Numbers, Whole Numbers, Integers, and Rational Numbers

Natural numbers are the numbers used for counting: 1, 2, 3, 4, 5, and so on. They do not include zero, fractions, or negatives.

Whole numbers include all natural numbers and also zero. The number zero represents the absence of quantity and acts as the additive identity (Section 2.11).

Integers extend whole numbers into the negatives: . . . , −4, −3, −2, −1, 0, 1, 2, 3, 4, . . . Negative integers describe temperatures below zero, debts, depths below sea level, and positions behind a starting point.

Rational numbers are all numbers expressible as a fraction a/b where a and b are integers and b ≠ 0. This includes all integers (since n = n/1), all proper fractions, all mixed numbers, all terminating decimals, and all repeating decimals.

Example 2.2.1

Classifying numbers across the rational families

ProblemClassify each number by listing every applicable category: {natural, whole, integer, rational}.

12 −7 0 3/5 −2/3 0.4
ThinkStart with the most restrictive (natural) and work outward. A number belongs to a category if it meets that category's definition.

Solve

Number	Natural	Whole	Integer	Rational
12	✓	✓	✓	✓
−7			✓	✓
0		✓	✓	✓
3/5				✓
−2/3				✓
0.4				✓

CheckVerify each row. The number 12 is a counting number (→ natural), non-negative (→ whole), an integer, and equals 12/1 (→ rational). The number −7 is not a counting number or non-negative, but it is an integer and equals −7/1.

Answer: Every natural number is whole, integer, and rational. Every whole number is integer and rational. Every integer is rational. Fractions and non-integer decimals are only rational.

Practice 2.2Classification Practice

8
−3
0
5/6
−7/2
0.121212…

List every category from {natural, whole, integer, rational} that applies to each number.

Section 2.3

Irrational Numbers and Real Numbers

Not every number can be expressed as a fraction. Some numbers, when written as decimals, go on forever without repeating any pattern. These are irrational numbers. The most famous example is π ≈ 3.14159265…, whose digits never end and never repeat. Another class of irrationals comes from square roots of non-perfect squares: √2 ≈ 1.41421356… continues forever without settling into a repeating block.

Not every square root is irrational. The square roots of perfect squares are rational: √4 = 2, √9 = 3, √25 = 5. But √3, √5, √7, and √10 are all irrational.

The real numbers are the union of the rational and irrational numbers. Together they fill the entire number line without gaps. Every point on the number line corresponds to exactly one real number, and every real number corresponds to exactly one point.

Example 2.3.1

Identifying rational and irrational numbers

ProblemClassify each as rational or irrational: √9 √11 0.666… π −√16 √5
ThinkCan this number be expressed exactly as a ratio of two integers? If yes, rational. If it is a non-terminating, non-repeating decimal or the square root of a non-perfect square, it is irrational.
Solve √9 = 3 → rational (equals a whole number) √11 → irrational (11 is not a perfect square) 0.666... = 2/3 → rational (repeating decimal) π → irrational −√16 = −4 → rational (equals a negative integer) √5 → irrational
CheckThe key test is whether the exact value can be written as a/b. Calculator approximations do not determine rationality.

Answer: √9, 0.666…, and −√16 are rational. √11, π, and √5 are irrational.

Practice 2.3Rational or Irrational?

√36
√7
−√49
π/2
0.313131…
√100
√6
1.41421…

Section 2.4

Number Lines and Ordering Real Numbers

A number line is a straight line with a designated point for zero, positive numbers extending to the right, and negative numbers extending to the left. Every real number corresponds to exactly one location on this line.

The number line gives a visual representation of order. For any two real numbers a and b:

Order on the Number Line

a < b a is to the left of b

a > b a is to the right of b

a = b a and b occupy the same location

The further left a number is, the smaller it is. Negative numbers are always less than zero and less than any positive number.

Common Mistake — Negative Numbers

Do not confuse the size of the digits with the size of the number. The number −100 has large digits, but −100 < −1 because −100 is further left on the number line. Think in context: owing $100 is a worse financial position than owing $1.

Example 2.4.1

Ordering real numbers

ProblemPlace in order from least to greatest: −3 1.5 −1/2 0 √5 −2.8
ThinkConvert to decimal approximations, then order on a mental number line.
Solve −3 = −3.000 −2.8 = −2.800 −1/2 = −0.500 0 = 0.000 1.5 = 1.500 √5 ≈ 2.236 Order: −3, −2.8, −1/2, 0, 1.5, √5
CheckEach number is larger than the one before. Moving left to right on a number line, we encounter them in this order.

Answer: −3 < −2.8 < −1/2 < 0 < 1.5 < √5

Practice 2.4Number Lines and Ordering

Plot and label on a number line: −3, −1.5, 0, 3/4, 2, √5
Write in order from least to greatest: −4, 2.5, −1/3, 0, −3.7, 1
Insert <, >, or =: (a) −5 ___ −2 (b) |−8| ___ |8| (c) −3/4 ___ −0.8 (d) √4 ___ √5

Section 2.5

Absolute Value and Distance

The absolute value of a number is its distance from zero on the number line. Distance is always non-negative, so absolute value is always non-negative. The absolute value of a is written |a|.

Definition of Absolute Value

|a| = a if a ≥ 0 |a| = −a if a < 0

If a number is positive or zero, its absolute value is itself. If a number is negative, its absolute value is its opposite (which is positive).

|7| = 7 |−7| = 7 |0| = 0

The distance between a and b on the number line is |a − b|, which always equals |b − a|.

Example 2.5.1

Computing absolute value expressions

ProblemEvaluate: (a) |−15| (b) |3 − 8| (c) −|6| (d) |−4 + 9|
ThinkComplete any arithmetic inside the bars first, then take the non-negative version. Note: a negative sign outside the bars is applied after the absolute value is evaluated.
Solve (a) |−15| = 15 (b) |3 − 8| = |−5| = 5 (c) −|6| = −(6) = −6 (The negative sign is outside the bars; it applies after.) (d) |−4 + 9| = |5| = 5
CheckIn (c), the absolute value of 6 is 6, and then the negative outside makes it −6. The bars are not a magic "make positive" button applied to the whole expression.

Answer: (a) 15 (b) 5 (c) −6 (d) 5

Common Mistake — Negative Signs Outside the Bars

−|−7| is NOT 7. Evaluate the absolute value first: |−7| = 7. Then apply the negative outside: −7. Always follow the order of operations around absolute value bars.

Practice 2.5Absolute Value

|9|
|−14|
−|−6|
|3 − 10|
|−5 + 2|
|4| + |−7|
|4 + (−7)|
|−3 − (−9)|

Section 2.6

Adding and Subtracting Real Numbers

In arithmetic, you learned to add and subtract positive numbers. In algebra, you need to be equally confident with negative numbers. Signed number arithmetic is the foundation of expression simplification, equation solving, and graphing.

Adding Real Numbers

Sign Rules for Addition

Same signs: Add the absolute values. Keep the common sign.

5 + 8 = 13 (−5) + (−8) = −13

Different signs: Subtract the smaller absolute value from the larger. Keep the sign of the number with the larger absolute value.

(−9) + 4 = −5 9 + (−4) = 5

Subtracting Real Numbers

Subtraction as Addition of the Opposite

a − b = a + (−b)

Every subtraction problem can be rewritten as an addition problem. This is the most important rule for signed subtraction.

7 − 12 = 7 + (−12) = −5

−4 − (−9) = −4 + 9 = 5

−3 − 8 = −3 + (−8) = −11

Example 2.6.1

Adding and subtracting signed numbers

ProblemEvaluate: (a) (−6) + (−10) (b) 14 + (−20) (c) −5 − (−13) (d) −8 − 7
ThinkFor subtractions, rewrite as addition of the opposite first. Then apply the same-sign or different-sign rule for addition.
Solve (a) (−6) + (−10): same signs → −(6 + 10) = −16 (b) 14 + (−20): different signs, |−20| > |14| → −(20 − 14) = −6 (c) −5 − (−13) = −5 + 13: different signs, |13| > |5| → +(13 − 5) = 8 (d) −8 − 7 = −8 + (−7): same signs → −(8 + 7) = −15
CheckUse a mental number line. Starting at −8 and moving 7 units left (adding −7) lands at −15. ✓

Answer: (a) −16 (b) −6 (c) 8 (d) −15

Practice 2.6Adding and Subtracting Signed Numbers

(−12) + (−8)
15 + (−23)
−9 − (−4)
−6 − 13
18 + (−18)
−7 − (−7)
0 − (−5)
−14 + 9 − 3

Section 2.7

Multiplying and Dividing Real Numbers

Sign Rules for Multiplication and Division

positive × positive = positive negative × negative = positive

positive × negative = negative negative × positive = negative

Same signs → positive result Different signs → negative result

These rules apply equally to division.

Multiple Factors

When multiplying three or more signed numbers, count the negative factors. An even number of negative factors gives a positive product. An odd number of negative factors gives a negative product.

Example 2.7.1

Multiplying and dividing signed numbers

ProblemEvaluate: (a) (−8)(5) (b) (−6)(−9) (c) −36 / −4 (d) (−1)(−2)(−3)(−4)
ThinkApply the sign rule first, then multiply or divide the absolute values. For (d), count the negatives.
Solve (a) (−8)(5): different signs → negative → −40 (b) (−6)(−9): same signs → positive → 54 (c) −36/−4: same signs → positive → 9 (d) (−1)(−2)(−3)(−4): 4 negative factors (even) → positive (−1)(−2) = 2, (2)(−3) = −6, (−6)(−4) = 24

Answer: (a) −40 (b) 54 (c) 9 (d) 24

Division by Zero

Division by zero is undefined. No real number equals a/0 because no number times zero equals a nonzero number. You will encounter this as an excluded value when studying rational functions in Algebra II. For now, never divide by zero.

Practice 2.7Multiplying and Dividing

(−7)(8)
(−5)(−9)
−42/7
−30/−5
(−2)(−3)(−4)
(−1)&sup5;
(−2)&sup4;
−2&sup4;

Section 2.8

Opposites and Reciprocals

Opposites (Additive Inverses)

The opposite of a real number a is −a. Opposites are the same distance from zero but on opposite sides of the number line. Their sum is always zero:

Opposite (Additive Inverse)

a + (−a) = 0

Opposite of 7 is −7 Opposite of −5 is 5 Opposite of 0 is 0

This property is used in equation solving: adding the opposite of a quantity to both sides isolates the variable.

Reciprocals (Multiplicative Inverses)

The reciprocal of a nonzero real number a is 1/a. For a fraction a/b, the reciprocal is b/a. Their product is always one:

Reciprocal (Multiplicative Inverse)

a · (1/a) = 1 (a ≠ 0)

Reciprocal of 4 is 1/4 Reciprocal of −2/3 is −3/2 Reciprocal of 1 is 1

This property is used in equation solving: multiplying both sides by the reciprocal of a coefficient isolates the variable.

Zero Has No Reciprocal

There is no number you can multiply by zero to get one, because anything times zero is zero. The expression 1/0 is undefined. This is not a detail to ignore — it becomes critical when you study domain restrictions in rational equations.

Example 2.8.1

Finding opposites and reciprocals

ProblemFind the opposite and the reciprocal of each: 9 −4/5 −3 7/2
Solve

Number Opposite Reciprocal

9 −9 1/9

−4/5 4/5 −5/4

−3 3 −1/3

7/2 −7/2 2/7
CheckVerify: −4/5 + 4/5 = 0 ✓ (−4/5)(·(−5/4)) = 20/20 = 1 ✓

Number	Opposite	Reciprocal
9	−9	1/9
−4/5	4/5	−5/4
−3	3	−1/3
7/2	−7/2	2/7

Answer: See table above. The opposite changes sign; the reciprocal flips the fraction (and keeps the sign).

Practice 2.8Opposites and Reciprocals

For each number, give (a) its opposite and (b) its reciprocal, if it exists.

10
−6
3/8
−5/4
1
−1
0
−3/7

Section 2.9

Properties of Addition and Multiplication

Arithmetic has structure. The way numbers combine under addition and multiplication follows rules that are always true, regardless of which real numbers are involved. These rules — called properties of operations — are what allow algebra to work. When you rearrange terms, regroup a calculation, or simplify an expression, you are applying one of these properties.

Learn each property by understanding what it allows you to do, not just by memorizing its name.

The Commutative Property

Commutative Property

a + b = b + a a · b = b · a

The order of addition or multiplication does not change the result. You can move the numbers around. This does NOT apply to subtraction or division.

3 + 7 = 7 + 3 = 10 4 · 6 = 6 · 4 = 24

3x + 5 = 5 + 3x (same expression, different appearance)

Commutativity Does Not Apply to Subtraction or Division

8 − 3 ≠ 3 − 8 and 12 ÷ 4 ≠ 4 ÷ 12. Do not apply the commutative property to subtraction or division expressions.

The Associative Property

Associative Property

(a + b) + c = a + (b + c)

(a · b) · c = a · (b · c)

The grouping of addition or multiplication does not change the result. You can re-associate (regroup) freely. This does NOT apply to subtraction or division.

(2 + 5) + 3 = 2 + (5 + 3) = 10

(3 · 4) · 5 = 3 · (4 · 5) = 60

Example 2.9.1

Identifying properties

ProblemIdentify the property in each statement.

(a) 5 + (x + 3) = (5 + x) + 3 (b) 7 · x = x · 7 (c) (ab)c = a(bc) (d) x + y = y + x
ThinkLook at whether the grouping changes or the order changes.
Solve (a) Grouping of addition changes → Associative Property of Addition (b) Order of multiplication reverses → Commutative Property of Multiplication (c) Grouping of multiplication changes → Associative Property of Multiplication (d) Order of addition reverses → Commutative Property of Addition

Answer: (a) Associative/Addition (b) Commutative/Multiplication (c) Associative/Multiplication (d) Commutative/Addition

Practice 2.9Identifying Properties

Name the property illustrated in each equation.

(x + 4) + 6 = x + (4 + 6)
3 · y = y · 3
(2 · 5) · 3 = 2 · (5 · 3)
a + b + c = c + a + b
4(xy) = (4x)y
mn = nm

Section 2.10

The Distributive Property Revisited

The distributive property was introduced in Chapter 1. In the context of signed numbers, it deserves a full revisit. Incorrect distribution — especially with negative signs — is one of the most common sources of error in algebra.

The Distributive Property

a(b + c) = ab + ac

a(b − c) = ab − ac

Multiply the factor by every term inside the parentheses. With more than two terms: a(b + c + d) = ab + ac + ad

The most error-prone case involves a negative number outside the parentheses. Consider −3(x − 4). The negative distributes to every term:

−3(x − 4) = (−3)(x) + (−3)(−4) = −3x + 12

Notice: (−3)(−4) = +12. This answers the opening question of the chapter. The correct simplification is:

−3(x − 4) + 2x = −3x + 12 + 2x = −x + 12

Example 2.10.1

Distributing with negative signs

ProblemSimplify each expression.

(a) 4(3x + 2) (b) −5(2y − 7) (c) −(a + 6) (d) 3(x + 4) − 2(x − 1)
ThinkMultiply every term inside by the factor outside. For negatives: a negative times a negative is positive.
Solve (a) 4(3x + 2) = 12x + 8 (b) −5(2y − 7) = (−5)(2y) + (−5)(−7) = −10y + 35 (c) −(a + 6) = (−1)(a) + (−1)(6) = −a − 6 (d) 3(x + 4) − 2(x − 1) = 3x + 12 − 2x + 2 [note: −2(−1) = +2] = x + 14
CheckFor (d), let x = 0: original gives 3(4) − 2(−1) = 12 + 2 = 14; simplified gives 0 + 14 = 14. ✓

Answer: (a) 12x + 8 (b) −10y + 35 (c) −a − 6 (d) x + 14

Distributing a Negative — Every Term Changes Sign

A negative sign in front of parentheses distributes to every term inside, not just the first. −(3x − 5) = −3x + 5, NOT −3x − 5. Write out −1 × each term if you are uncertain.

Practice 2.10The Distributive Property

3(x + 5)
6(2n − 4)
−2(x + 3)
−4(3a − 1)
−(5x − 2)
7(m + n)
3(2x + 1) + 4x
5(n − 3) − 2(n + 1)
2(4a − 3) + 3(a + 5)
−3(x − 2) + 4(2x + 1)

Section 2.11

Identity and Inverse Properties

Identity Properties

Additive Identity: a + 0 = a The number 0 is the identity for addition.

Multiplicative Identity: a · 1 = a The number 1 is the identity for multiplication.

Adding 0 leaves a number unchanged. Multiplying by 1 leaves a number unchanged. Both identities will be used to verify equation-solving steps.

Inverse Properties

Additive Inverse: a + (−a) = 0 Every real number has an additive inverse (its opposite).

Multiplicative Inverse: a · (1/a) = 1 (a ≠ 0) Every nonzero number has a multiplicative inverse (its reciprocal).

Additive inverses combine with addition to give 0 (the identity). Multiplicative inverses combine with multiplication to give 1 (the identity). Zero has no multiplicative inverse.

Why These Properties Matter for Equations

When you solve x + 5 = 12, you add the additive inverse of 5 to both sides (−5) to isolate x. When you solve 3x = 15, you multiply by the multiplicative inverse of 3 (which is 1/3) to isolate x. These properties are the justification for every isolation step in Chapter 3.

Example 2.11.1

Naming the property used

ProblemState the property that justifies each step.

(a) x + 0 = x (b) 1 · (x + 3) = x + 3 (c) 7 + (−7) = 0 (d) 1/4 · 4 = 1
Solve (a) Additive Identity Property (adding 0 leaves x unchanged) (b) Multiplicative Identity Property (multiplying by 1 leaves (x + 3) unchanged) (c) Additive Inverse Property (7 and −7 are opposites; their sum is 0) (d) Multiplicative Inverse Property (1/4 and 4 are reciprocals; their product is 1)

Answer: (a) Additive Identity (b) Multiplicative Identity (c) Additive Inverse (d) Multiplicative Inverse

Practice 2.11Identity and Inverse Properties

Name the property that justifies each equation.

m + 0 = m
n · 1 = n
7 + (−7) = 0
(2/3)(3/2) = 1
1 · x = x
x + (−x) = 0

Section 2.12

Simplifying Numerical and Algebraic Expressions

Simplifying an expression means rewriting it in a more compact form without changing its value. The number properties you have studied give the justification for every step.

Example 2.12.1

Simplifying a numerical expression with signed numbers

ProblemSimplify: −4 + 3 · (−2)² − 5
ThinkFollow the order of operations: exponents first, then multiplication, then addition and subtraction left to right.
Solve −4 + 3 · (−2)² − 5 = −4 + 3 · 4 − 5 (evaluate exponent: (−2)² = 4) = −4 + 12 − 5 (multiply: 3 · 4 = 12) = 8 − 5 (add left to right: −4 + 12 = 8) = 3
CheckWork backward: 3 + 5 = 8, 8 − 12 = −4. Starting value matches. ✓

Answer: 3

(−2)² ≠ −4

(−2)² = (−2)(−2) = 4. Compare with −2² = −(2²) = −4. The placement of the negative sign relative to the exponent changes the result entirely. Always use parentheses when substituting negative values.

Example 2.12.2

Simplifying an algebraic expression with distribution

ProblemSimplify: 5x − 3(2x − 4) + x
ThinkApply the distributive property to −3(2x − 4) first. Then group and combine like terms.
Solve 5x − 3(2x − 4) + x = 5x − 6x + 12 + x (distribute −3) = (5x − 6x + x) + 12 (group x-terms) = 0x + 12 (combine: 5 − 6 + 1 = 0) = 12
CheckLet x = 2: original gives 10 − 3(0) + 2 = 12. Simplified gives 12. ✓

Answer: 12 (the x-terms cancel; the result is a constant)

Example 2.12.3

Simplifying with multiple variable types

ProblemSimplify: 4a + 3b − 2a + 7 − b + 1
ThinkGroup like terms: a-terms together, b-terms together, constants together.
Solve 4a + 3b − 2a + 7 − b + 1 = (4a − 2a) + (3b − b) + (7 + 1) = 2a + 2b + 8

Answer: 2a + 2b + 8

Practice 2.12Simplifying Expressions

5x + 3x − 2x
4a − 7 + 2a + 3
−3(2y + 5)
2(4x − 1) + 3x
−(5m − 6) + m
7 − 4(n − 2)
3(x + 2) − 2(x + 2)
5k − (3k − 8)
6 + 2(a − 5) − a
−2 + 4(x − 3) − x

Section 2.13

Real Numbers in Measurement and Models

Real numbers describe the measurable world. Length, area, volume, time, temperature, speed, weight, cost, and population are all quantities represented by real numbers. Many real-world quantities require negative numbers: temperatures below zero, depths below sea level, debts, and decreases. Algebra connects arithmetic with real-world measurement through formulas, equations, and models.

Example 2.13.1

Using a formula with real numbers — perimeter

ProblemA rectangular room has a length of 14.5 feet and a width of 10.25 feet. Find the perimeter.
ThinkUse P = 2(l + w). The distributive property guarantees this is equivalent to P = 2l + 2w.
Solve P = 2(l + w) P = 2(14.5 + 10.25) P = 2(24.75) P = 49.5 feet

Answer: The perimeter is 49.5 feet.

Example 2.13.2

Net change with signed numbers — temperature

ProblemA city's temperature at 6 AM was −8°F. By noon it rose 19 degrees. By evening it fell 12 degrees from the noon reading. What was the evening temperature?
RepresentNoon temperature = −8 + 19. Evening temperature = (noon) − 12.
Solve Noon: −8 + 19 = 11°F Evening: 11 − 12 = −1°F
CheckA rise of 19 from −8 gives 11. A fall of 12 from 11 gives −1. The temperature ended just below zero. Reasonable. ✓

Answer: The evening temperature was −1°F.

Practice 2.13Real-World Applications

A diver starts at sea level and descends 48 feet. She then descends another 27 feet. Write and evaluate an expression for her current elevation.
A student's bank account had a balance of −$48. They deposited $125, then spent $67. What was the final balance?
The perimeter of a triangle is P = a + b + c. If the sides are 3x, 2x − 1, and x + 5, write and simplify an expression for the perimeter.
A business reports losses of $4,500 per month for 3 consecutive months. Use a signed number multiplication to represent the total financial change.
A stock's value changed by −$3, +$7, −$2, and −$4 on four consecutive days. What was the total net change? What was the average daily change?

Section 2.14

Preparing for Equation Solving

Every technique in Chapter 3 and beyond depends on the arithmetic and properties studied in this chapter. The table below connects each property directly to a step you will use in equation solving.

Property	How It Is Used in Equations	Example
Additive Inverse	Add the opposite to both sides to remove a constant term	`x + 7 = 15: add −7 to both sides`
Multiplicative Inverse	Multiply by the reciprocal to remove a coefficient	`3x = 15: multiply both sides by 1/3`
Distributive Property	Expand parentheses before isolating the variable	`2(x + 3) = 10: distribute first`
Commutative Property	Rearrange terms freely to group like terms	`5 + 3x = 3x + 5`
Associative Property	Regroup terms to combine them efficiently	`(3x + 7) + (−7) = 3x + (7 + (−7)) = 3x`

Students sometimes approach Chapter 3 as if it is completely new material. It is not. Chapter 3 is Chapter 2 applied to balanced statements. The variable is just an unknown real number. Your job in solving an equation is to use real number operations and properties systematically to figure out which real number the variable represents.

A note on precision: students who understand that adding −9 to both sides preserves equality because of the additive inverse property will make fewer errors than students who only remember "move the number to the other side." Meaning drives reliability. Keep these properties in mind as you work through Chapter 3.

Practice 2.14Readiness Check

Simplify −4x + 9x − 3. State which property allows you to combine the x-terms.
Simplify 2(x − 5) + 3. State the property used in the first step.
A student solves x + 8 = 20 by "subtracting 8 from both sides." Which property of real numbers justifies this step?
Evaluate −3x + 7 for x = −4. Show every substitution step with parentheses.
Two students simplify −2(x − 3) + 5x. Student A gets 3x + 6. Student B gets 3x − 6. Who is correct? Identify the error in the incorrect solution.

Chapter 2 — Back Matter

Chapter Summary

Sec 2.1The real number system is organized in nested layers: Natural ⊂ Whole ⊂ Integer ⊂ Rational ⊂ Real. Rational and irrational numbers together make up all real numbers. Every real number lives on the number line.
Sec 2.2Natural numbers count. Whole numbers include zero. Integers include negatives. Rational numbers include all numbers expressible as a/b (integers, fractions, terminating decimals, and repeating decimals).
Sec 2.3Irrational numbers cannot be expressed as a/b; their decimals are non-terminating and non-repeating. Examples: π, √2, √5. Square roots of perfect squares are rational; all others are irrational.
Sec 2.4On the number line, smaller numbers are to the left. Negative numbers are always less than zero. The number with larger digits is not necessarily larger — context and sign determine order.
Sec 2.5Absolute value is distance from zero; always non-negative. |a| = a when a ≥ 0; |a| = −a when a < 0. A negative sign outside the bars applies after the absolute value is evaluated.
Sec 2.6Same-sign addition: add absolute values, keep the sign. Different-sign addition: subtract smaller from larger, keep the sign of the larger. Subtraction is addition of the opposite: a − b = a + (−b).
Sec 2.7Same signs multiply/divide to a positive result; different signs to a negative result. An even number of negative factors gives a positive product; an odd number gives a negative product. Division by zero is undefined.
Sec 2.8The opposite of a is −a; opposites sum to zero. The reciprocal of a is 1/a; reciprocals multiply to one. Zero has no reciprocal. Both operations are tools for equation solving.
Sec 2.9Commutative property: order does not matter for addition or multiplication. Associative property: grouping does not matter for addition or multiplication. Neither applies to subtraction or division.
Sec 2.10The distributive property: a(b + c) = ab + ac. Every term inside must be multiplied. Distributing a negative changes every sign inside the parentheses. This was the key to the opening question.
Sec 2.11Additive identity: 0. Multiplicative identity: 1. Additive inverse: −a (sum with a equals 0). Multiplicative inverse: 1/a (product with a equals 1). These justify every isolation step in equation solving.
Sec 2.12To simplify: use order of operations for numerical expressions; distribute and combine like terms for algebraic expressions. (−2)² = 4 but −2² = −4. Always use parentheses when substituting negative values.
Sec 2.13Real numbers model temperature, elevation, finance, and measurement. Formulas such as P = 2(l + w) use the distributive property. Signed arithmetic describes real changes in context.
Sec 2.14Every equation-solving step is justified by a real number property. Additive and multiplicative inverses isolate variables. The distributive property expands parentheses. Commutativity and associativity allow free rearrangement.

Properties Reference

Property	Addition Form	Multiplication Form
Commutative	a + b = b + a	ab = ba
Associative	(a + b) + c = a + (b + c)	(ab)c = a(bc)
Identity	a + 0 = a	a · 1 = a
Inverse	a + (−a) = 0	a · (1/a) = 1, a ≠ 0
Distributive	a(b + c) = ab + ac and a(b − c) = ab − ac

Key Terms Review

absolute value

The distance of a number from zero on the number line; always non-negative. Written |a|.

additive identity

The number 0; adding 0 to any number leaves it unchanged: a + 0 = a.

additive inverse

The opposite of a number; the number that sums with it to give 0. The additive inverse of a is −a.

associative property

Grouping does not affect the result of addition or multiplication: (a + b) + c = a + (b + c).

closure

A set is closed under an operation if performing that operation on members always produces a member of the same set.

commutative property

Order does not affect the result of addition or multiplication: a + b = b + a and ab = ba.

distributive property

a(b + c) = ab + ac. Multiplication distributes over addition and subtraction. Every term inside must receive the factor.

integer

A whole number or its negative: …, −2, −1, 0, 1, 2, …

irrational number

A real number that cannot be written as a/b; its decimal expansion is non-terminating and non-repeating. Examples: √2, π.

multiplicative identity

The number 1; multiplying any number by 1 leaves it unchanged: a · 1 = a.

multiplicative inverse

The reciprocal of a nonzero number a; equals 1/a. The product of a number and its reciprocal is 1.

natural number

A positive counting number: 1, 2, 3, 4, …

number line

A line representing all real numbers, with zero at the origin, positives to the right, negatives to the left.

opposite

The additive inverse of a number; the same distance from zero on the opposite side. The opposite of a is −a.

rational number

A number expressible as a/b where a and b are integers and b ≠ 0. Includes all integers, fractions, and repeating or terminating decimals.

real number

Any number on the number line; the union of all rational and irrational numbers.

reciprocal

The multiplicative inverse; for a nonzero number a, the reciprocal is 1/a. For a fraction a/b, the reciprocal is b/a.

signed number

A number with an explicit positive or negative sign, indicating both magnitude and direction.

whole number

A non-negative integer: 0, 1, 2, 3, …

Concept Review Questions

What is the difference between an integer and a rational number? Give an example of a rational number that is not an integer.
Why is every integer also a rational number?
Describe how to tell whether a number is irrational based on its decimal expansion.
What does absolute value measure? Why is absolute value never negative?
Explain the rule for adding two negative numbers. Why does adding two negatives always give a negative result?
Why does subtracting a negative number give a result larger than the original? Illustrate with a number line description.
How do you determine the sign of a product when more than two signed numbers are multiplied together?
What is the additive inverse of a number, and why is it useful in algebra?
What is the multiplicative inverse of a number, and why is it useful in algebra?
Explain why the commutative property applies to addition but not subtraction. Give a numerical example for each.
What does the distributive property allow you to do with an expression like 4(x + 7)?
Why must you distribute to every term inside parentheses? What goes wrong if you distribute to only the first term?
What is the additive identity, and what role does it play in simplifying expressions?
Why can zero never have a multiplicative inverse?
Give an example of a real-world quantity that requires negative numbers to represent it correctly.

Skill Practice

Section AClassifying Real Numbers

List all categories from {natural, whole, integer, rational, irrational, real} that apply.

8
−3
0
5/6
−7/2
√36
√7
−√25
0.121212…
π

Section BOrdering and Absolute Value

Write in order from least to greatest: −4, 2.5, −1/3, 0, −3.7, 1
Insert <, >, or =: (a) −5 ___ −2 (b) |−8| ___ |8| (c) −3/4 ___ −0.8 (d) √4 ___ √5
Evaluate: (a) |−9| (b) −|−6| (c) |3 − 10| (d) |−5 + 2| (e) |4 + (−7)|

Section CSigned Number Operations

(−12) + (−8)
15 + (−23)
−9 − (−4)
−6 − 13
(−7)(8)
(−5)(−9)
−42 / 7
−30 / −5
(−2)(−3)(−4)
(−1)&sup5;

Section DProperties of Operations

Identify the property illustrated in each equation.

(x + 4) + 6 = x + (4 + 6)
3 · y = y · 3
5(a − b) = 5a − 5b
m + 0 = m
n · 1 = n
7 + (−7) = 0
(2/3)(3/2) = 1
(2 · 5) · 3 = 2 · (5 · 3)

Section ESimplifying Expressions

5x + 3x − 2x
4a − 7 + 2a + 3
−3(2y + 5)
2(4x − 1) + 3x
−(5m − 6) + m
7 − 4(n − 2)
3(x + 2) − 2(x + 2)
5k − (3k − 8)
(1/2)(8x − 4) + 3x
6 + 2(a − 5) − a

Word Problems

Use the problem-solving method: Read → Represent → Solve → Check → Explain.

Temperature Change. The temperature at midnight was −14°F. By 10 AM it had risen 23 degrees. By 4 PM it had fallen 9 degrees from the 10 AM reading. What was the temperature at 4 PM? Write an expression for each step.
Bank Balance. A student's account had a balance of −$48. They deposited $125. Two days later they spent $67. What was the final balance?
Elevation. A hiker descends 320 meters from a peak at 1,840 meters above sea level, then descends another 515 meters. Write and evaluate an expression for their elevation.
Perimeter. The perimeter of a triangle is P = a + b + c. If the sides are 3x, 2x − 1, and x + 5, write and simplify an expression for the perimeter.
Business Loss. A business reports losses of $4,500 per month for 3 consecutive months. Use signed number multiplication to represent the total financial change. Explain what the sign of your answer means.
Average Change. A stock's value changed by −$3, +$7, −$2, and −$4 on four consecutive trading days. What was the total net change? What was the average change per day?

Error Analysis

Each problem shows a student's work containing an error. Identify the error and write the correct solution.

Problem 1

−2(x − 5) = −2x − 10

What went wrong? What is the correct result?

Problem 2

(−3)² = −9

What went wrong? Explain the correct evaluation of (−3)².

Problem 3

−|−7| = 7

What went wrong? What is the correct result?

Problem 4

5 − (3 − x) = 5 − 3 − x = 2 − x

What went wrong? What is the correct result?

Problem 5

To simplify 8 − 3 · 2: 8 − 3 · 2 = 5 · 2 = 10

Which rule was violated? What is the correct answer?

Problem 6

−12 / −3 = −4 (student says "negative divided by negative is negative")

What is the error in the student's reasoning? What is the correct result?

Chapter 2 Checkpoint

Complete without assistance — assesses all key skills and concepts from Chapter 2

Part 1 — Number Classification

List every set from {natural, whole, integer, rational, irrational, real} to which each number belongs: (a) −9/2 (b) √49 (c) √3 (d) 0
Give one example of a number that is rational but not an integer.
Place in order from least to greatest: √10, −2, 7/4, 0, −5/2

Part 2 — Operations with Signed Numbers

Evaluate: (−13) + 8 − (−5)
Evaluate: (−4)³
Evaluate: −(−3)² + 5 · (−2)
Evaluate: −24/−6 · (−2)
Find the opposite and reciprocal of −7/3.

Part 3 — Absolute Value

Evaluate: |−6| − |3|
Evaluate: |7 − 15| + |−4 + 10|
For what values of x is |x| = 8? (There may be more than one answer.)

Part 4 — Properties

Name the property that justifies each step:
(a) 3(x − 2) = 3x − 6
(b) x + 7 + y = x + y + 7
(c) 9 · (1/9) = 1
(d) (x + 3) + 0 = x + 3

Part 5 — Simplification

Simplify: 6x − 4(x − 3) + 2
Simplify: 3(2a + 5) − 4(a − 1)
Simplify: −5(y − 2) + 3y + 1

Part 6 — Applications

A diver starts at sea level (elevation = 0) and descends 48 feet, then descends an additional 27 feet. Write and evaluate an expression for her current elevation.
A store bought a product for $3.50 and sold it for $5.25. Write an expression using the distributive property for the total profit from selling n items. Simplify.
Reading and Writing. Explain in your own words why the commutative property applies to multiplication but not division. Include a numerical example for each case.

Bridge Note — Chapter 3

Chapter 3: Linear Equations in One Variable

Chapter 3 begins your formal study of linear equations. Every concept and skill in Chapter 2 is a tool you will use immediately. When you encounter the equation −4x + 9 = 1, you will need signed number arithmetic to simplify and check, the additive inverse to isolate the variable term, the multiplicative inverse to isolate x, the distributive property for equations like 2(x − 5) = 6, and the commutative and associative properties to rearrange terms freely.

Chapter 3 is not new material. It is Chapter 2 applied to balanced statements. The variable is simply an unknown real number, and your job is to use the properties of real numbers systematically to figure out which real number it is. Precision with signed arithmetic and properties in Chapter 2 is what makes Chapter 3 reliable.

MGU Library Connection

Appendix B: Real Number Properties Reference — one-page summary of all five properties with examples
Appendix C: Integer and Fraction Operations Review — additional signed number practice before Chapter 3
Arithmetic and Pre-Algebra Foundations, Chapters 8–10: Order of Operations, Signed Numbers, Properties of Operations
Algebra II, Chapter 4: Complex Numbers — extends the real number system to include square roots of negative numbers
MGU Dictionary entries: real number, rational number, irrational number, absolute value, distributive property, additive inverse

Chapter 2:Real Numbers andAlgebraic Properties

Purpose and Overview

Learning Objectives

Sections at a Glance

The Real Number System

Natural Numbers, Whole Numbers, Integers, and Rational Numbers

Irrational Numbers and Real Numbers

Number Lines and Ordering Real Numbers

Absolute Value and Distance

Adding and Subtracting Real Numbers

Adding Real Numbers

Subtracting Real Numbers

Multiplying and Dividing Real Numbers

Opposites and Reciprocals

Opposites (Additive Inverses)

Reciprocals (Multiplicative Inverses)

Properties of Addition and Multiplication

The Commutative Property

The Associative Property

The Distributive Property Revisited

Identity and Inverse Properties

Identity Properties

Inverse Properties

Simplifying Numerical and Algebraic Expressions

Real Numbers in Measurement and Models

Preparing for Equation Solving

Chapter Summary

Properties Reference

Key Terms Review

Concept Review Questions

Skill Practice

Word Problems

Error Analysis

Chapter 2 Checkpoint

Chapter 2:
Real Numbers and
Algebraic Properties