MGU Mathematics Series — Arithmetic & Pre-Algebra Foundations
Part I: Foundations of Arithmetic
Numbers · Number Line · Signed Operations · Number Sets
Chapter 1 of 14
MGU Mathematics Series  ·  Arithmetic & Pre-Algebra Foundations  ·  Library Textbook Edition

Chapter 1:
Numbers and
Operations

The number line, positive and negative numbers, number sets, identity numbers, opposites, absolute value, and all four signed operations — the foundation every later chapter requires

■ Math Level: Preparatory / Pre-Algebra ■

■ 14 Sections  •  Part I of 5 ■

Opens Arithmetic & Pre-Algebra Foundations  ·  Prepares students for Chapter 2 through Chapter 14 and Algebra I

Chapter 1 — Front Matter

Purpose and Overview

Every mathematical journey begins with numbers. Before a student can solve an equation, read a graph, calculate a percentage, or understand a scientific formula, that student must have a clear and confident understanding of what numbers are, how they are organized, and how they behave under the basic operations of arithmetic.

This chapter establishes that foundation. It introduces the number line as a visual model, explains why negative numbers exist and how they are used, organizes numbers into sets, and walks through the four basic operations applied to signed numbers. The chapter closes by connecting signed number arithmetic to the kind of thinking that algebra requires.

No prior course is assumed. Students who feel uncertain about negative numbers, or who have operated on instinct rather than understanding, will find this chapter a reliable starting point. Students who already feel confident with the basics will find that this chapter deepens and organizes what they already know.

Opening Question

Imagine the temperature outside is 4 degrees and drops overnight by 9 degrees. What is the temperature in the morning? If your first instinct is to subtract, you are already thinking arithmetically. If you are not sure what to do when the answer would be less than zero, this chapter is exactly where you need to begin.

Why This Chapter Matters

Signed numbers appear constantly in real life. Temperatures drop below zero. Bank accounts can carry negative balances. Elevations fall below sea level. In physics, forces act in opposite directions. In algebra, variables can represent negative quantities just as easily as positive ones. Students who skip over signed number work often struggle later, not because algebra is too hard, but because the operations feel unpredictable. This chapter removes that unpredictability.

14Sections
11Objectives
19Key Terms
12+Worked Examples

Learning Objectives

  1. 1Describe what arithmetic studies and how it connects to algebra.
  2. 2Place numbers on a number line and interpret their positions.
  3. 3Identify positive numbers, negative numbers, and zero and explain the role of each.
  4. 4Classify numbers as members of the natural numbers, whole numbers, integers, or rational numbers.
  5. 5Identify and apply the additive and multiplicative identity numbers.
  6. 6Find the opposite of any number.
  7. 7Calculate the absolute value of any number.
  8. 8Add, subtract, multiply, and divide signed numbers correctly.
  9. 9Recognize and correct common sign mistakes.
  10. 10Apply signed number operations to real-world problems involving temperature, money, elevation, and direction.
  11. 11Connect signed number operations to the foundations of algebraic reasoning.
Chapter 1 — Contents

Sections at a Glance

Sections 1.1 – 1.4
Arithmetic, the Number Line, and Number Sets
What Arithmetic Studies · The Number Line · Positive and Negative Numbers · Number Sets
Section 1.1

What Arithmetic Studies

Arithmetic is the branch of mathematics that deals with numbers and the four basic operations: addition, subtraction, multiplication, and division. These operations are the tools that make it possible to count, measure, compare, combine, separate, scale, and describe quantitative relationships.

Many students think of arithmetic as the computation they learned in early schooling: columns of addition problems, multiplication tables, long division. Those calculations are part of arithmetic. But arithmetic is also the reasoning behind those calculations. It answers questions such as: Why do two negatives multiplied together give a positive? What does it mean to divide by a fraction? When does a problem have no answer?

This textbook treats arithmetic as both skill and understanding. Computation matters, but so does knowing why the computation works.

OperationWhat It DoesResult Name
AdditionCombines quantitiesSum
SubtractionRemoves or compares quantitiesDifference
MultiplicationScales or groups quantitiesProduct
DivisionSeparates into equal parts or finds a rateQuotient
Inverse Operations

Addition and subtraction are inverses of each other. Multiplication and division are inverses of each other. This inverse relationship is one of the most important ideas in algebra: to undo an operation, apply its inverse. This is the core of equation solving.

MGU Library Connection
  • The MGU Mathematics Dictionary contains entries for all four arithmetic operations with extended definitions and historical notes
  • Algebra I, Chapter 2: The algebraic properties (commutative, associative, distributive) that make these operations reliable
Practice 1.1Concept Check
  1. Name the four operations of arithmetic and the result name for each.
  2. Explain in your own words what it means for two operations to be inverses. Give an example using numbers.
  3. A recipe calls for 3 cups of flour per batch. You need to make 5 batches. Which operation gives you the total flour needed? Which operation would tell you how many batches you could make from 12 cups?
Section 1.2

Numbers and the Number Line

Numbers can be represented as points on a line. This is called the number line, and it is one of the most useful tools in all of mathematics. A number line is a straight line with a point marked as zero in the center. Numbers to the right of zero are positive. Numbers to the left of zero are negative. The line extends infinitely in both directions, shown by arrows at each end.

←——|——|——|——|——|——|——|——|——|——|——→ -5 -4 -3 -2 -1 0 1 2 3 4 5

The number line communicates several things at once. Order: numbers increase from left to right. Direction: moving right means increasing, moving left means decreasing. Distance: the space between two points represents how far apart those numbers are. Symmetry: for every positive number, there is a corresponding negative number at the same distance from zero on the other side.

Example 1.2.1
Comparing numbers using the number line
  • ProblemWhich is greater, −3 or −1?
  • ThinkBoth numbers are negative. On the number line, −1 is to the right of −3. Numbers farther to the right are always greater.
  • Solve−1 > −3
  • CheckThink of temperature. −1 degree is warmer than −3 degrees. Warmer means higher, which means greater. ✓
Answer: −1 is greater than −3.
Student Warning

A common error is to assume that larger-looking negative numbers are greater. The number −10 is not greater than −2 just because 10 is greater than 2. On the number line, −10 is far to the left of −2, which means it is less. Always locate numbers on the line before comparing.

Practice 1.2Number Line and Comparison
  1. Which is greater: −7 or −2?
  2. Which is greater: −1 or −100?
  3. List in order from least to greatest: −4, 1, −6, 0, 3
  4. Place on a number line: −3, 1.5, −0.5, 2
  5. True or false: −8 > −3. Explain.
  6. What number is halfway between −4 and 2?
Section 1.3

Positive Numbers, Negative Numbers, and Zero

CategoryDefinitionReal-World Examples
Positive numbersAny number greater than zero; written without sign or with a + sign72°F temperature; $200 balance; 1,400 ft elevation; 5 steps forward
Negative numbersAny number less than zero; written with a − sign before the number−15°F temperature; −$50 overdraft; −282 ft below sea level; 5 steps backward
ZeroNeither positive nor negative; the reference point and center of the number lineFreezing point; empty account; sea level; a starting position before movement

The concept of negative numbers was not universally accepted in mathematics for a long time. For centuries, many mathematicians refused to treat negative numbers as real quantities. Today, negative numbers are understood to be as legitimate and necessary as positive ones. Their existence allows subtraction to always have an answer and makes algebra far more powerful.

Zero has important special properties that appear throughout this chapter and throughout the course. Adding zero to any number does not change the number. Multiplying any number by zero always gives zero. Dividing by zero is not defined, a topic addressed in Chapter 3.

Practice 1.3Signed Numbers in Context
  1. Write a signed number for each situation: (a) 12 feet above sea level, (b) a loss of $35, (c) 8 degrees below zero, (d) 20 steps forward.
  2. Give two real-world examples of your own where negative numbers are the appropriate representation.
  3. Explain why zero is neither positive nor negative. Use the number line in your explanation.
  4. A diver is at the surface (0 feet). She descends 40 feet, then descends another 15 feet. Write the final depth as a signed number.
Section 1.4

Number Sets

Mathematics organizes numbers into sets. A number set is a collection of numbers that share a particular property. Understanding these sets helps students know which numbers are available for a given situation and how one type of number relates to another.

Set NameMembersExamples
Natural NumbersThe counting numbers; begin at 1 and continue without end1, 2, 3, 4, 5, …
Whole NumbersThe natural numbers plus zero0, 1, 2, 3, 4, …
IntegersThe whole numbers and their negatives; extend infinitely in both directions…, −3, −2, −1, 0, 1, 2, 3, …
Rational NumbersAny number expressible as a fraction a/b where a and b are integers and b ≠ 03/4, −5/8, 0.5, 0.333…, all integers
Natural Numbers └─ are inside Whole Numbers └─ which are inside Integers └─ which are inside Rational Numbers Every natural number is also a whole number, an integer, and a rational number. Not every integer is a natural number. Not every rational number is an integer.
Example 1.4.1
Classifying numbers by set membership
  • ProblemClassify each number. Identify every set it belongs to: natural, whole, integer, rational.
  • Numbers7  |  0  |  −4  |  3/5
  • Solve 7 → Natural ✓ Whole ✓ Integer ✓ Rational ✓ 0 → Natural ✗ Whole ✓ Integer ✓ Rational ✓ −4 → Natural ✗ Whole ✗ Integer ✓ Rational ✓ 3/5 → Natural ✗ Whole ✗ Integer ✗ Rational ✓
  • CheckEach classification follows from the definitions. The integer −4 equals −4/1, confirming it is rational. ✓
Answer: See table above. Every integer is rational. Not every rational is an integer.
Note on Irrational Numbers

Some numbers cannot be written as fractions of integers. These are called irrational numbers and include numbers like π and √2. Irrational numbers are not covered in depth in this preparatory course, but students should be aware they exist. The MGU Library Algebra I textbook introduces the real number system in full.

Practice 1.4Number Sets
  1. Is 0 a natural number? Explain.
  2. Is every integer a rational number? Why?
  3. Classify −9: name all sets it belongs to.
  4. Classify 1/2: name all sets it belongs to.
  5. Name a whole number that is not a natural number.
  6. Name a rational number that is not an integer.
Sections 1.5 – 1.7
Identity Numbers, Opposites, and Absolute Value
Additive and Multiplicative Identity · Opposites · Absolute Value as Distance
Section 1.5

Identity Numbers

Every operation in arithmetic has an identity number: a number that leaves any other number unchanged when that operation is applied to it. Understanding these identities is not merely a vocabulary exercise. They are tools that appear constantly in algebra whenever a term must be removed without changing a value, or a fraction must be renamed without changing its worth.

The Two Identity Numbers

Additive Identity: Zero — Adding zero to any number leaves it unchanged: a + 0 = a

Multiplicative Identity: One — Multiplying any number by one leaves it unchanged: a × 1 = a

Zero does not change a number under addition. In algebra, this is used to isolate variables: when the goal is to eliminate a term, adding its opposite produces zero, which disappears because adding zero is the same as doing nothing. One does not scale a number under multiplication. In algebra, multiplying by one in clever forms, such as 5/5 or x/x, is how fractions are renamed and how equations are manipulated without changing their values.

Student Warning

Zero is not the multiplicative identity. Multiplying any number by zero always gives zero, not the original number. The multiplicative identity is specifically one. Do not confuse the two identities.

Practice 1.5Identity Numbers
  1. What is 15 + 0?
  2. What is −7 × 1?
  3. What is 0 + (−3/4)?
  4. What is 100 × 0?
  5. Which identity is used when a student rewrites 6/6 as 1 to simplify a fraction?
  6. Explain why multiplying by 0 does NOT produce the original number.
Section 1.6

Opposites of Numbers

Every number on the number line has an opposite: a number at the same distance from zero, but on the other side. The opposite of a number a is written −a. To find the opposite of any number, change its sign. The opposite of 5 is −5. The opposite of −3 is 3. The opposite of 0 is 0, because zero is its own opposite.

−5 0 5 ←———————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————————→ <——— 5 units ———|——— 5 units ———>
Additive Inverse Property

A number and its opposite always sum to zero: a + (−a) = 0

For example: 7 + (−7) = 0     −4 + 4 = 0     (3/5) + (−3/5) = 0

The opposite of a number is also called its additive inverse, because adding a number and its opposite always produces the additive identity, which is zero.

Example 1.6.1
Finding the opposite of a number
  • ProblemFind the opposite of each: 12, −8, −1/4, 0.
  • ThinkTo find the opposite, change the sign.
  • Solve Opposite of 12 → −12 Opposite of −8 → 8 Opposite of −1/4 → 1/4 Opposite of 0 → 0
  • CheckEach pair sums to zero: 12 + (−12) = 0, −8 + 8 = 0. ✓
Answer: −12, 8, 1/4, and 0.
Notation Note

The symbol − serves two purposes. Between two numbers it means subtract. Before a single number it means the opposite of, or negative. The expression −(−3) means the opposite of −3, which equals 3. Context determines meaning.

Practice 1.6Opposites
  1. Opposite of 25
  2. Opposite of −11
  3. Opposite of 0
  4. Opposite of −1/3
  5. Evaluate: −(−7)
  6. Evaluate: −(−(−4))
  7. What value of x satisfies x + 9 = 0?
  8. True or false: The opposite of a negative number is always positive.
Section 1.7

Absolute Value

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

Definition of Absolute Value

If a ≥ 0, then |a| = a.    (The number is already non-negative; return it unchanged.)

If a < 0, then |a| = −a.    (The number is negative; its distance is its opposite, which is positive.)

|6| = 6      |−6| = 6      |0| = 0

Example 1.7.1
Evaluating absolute value expressions
  • ProblemEvaluate: (a) |−9|   (b) |4|   (c) −|7|   (d) |−3| + |5|
  • ThinkFor (c): the absolute value of 7 is 7, then the negative sign outside is applied after. The result is −7, not 7.
  • Solve (a) |−9| = 9 (b) |4| = 4 (c) −|7| = −7 (negative sign is outside the bars) (d) |−3| + |5| = 3 + 5 = 8
  • CheckEach result is consistent with the definition of distance. ✓
Answer: (a) 9, (b) 4, (c) −7, (d) 8.
Student Warning

Absolute value does not always make a result positive. If a negative sign appears outside the absolute value bars, it is applied after the absolute value is found. The expression −|7| equals −7, not 7. The bars and the negative sign are separate operations.

Practice 1.7Absolute Value
  1. |−15|
  2. |8|
  3. |0|
  4. −|4|
  5. −|−10|
  6. |−7| + |3|
  7. |−6| − |2|
  8. |−8| × |−2|
  9. Which is greater: |−20| or |15|?
  10. Explain: can |x| ever be negative?
Sections 1.8 – 1.11
The Four Signed Number Operations
Adding · Subtracting · Multiplying · Dividing Signed Numbers
Section 1.8

Adding Signed Numbers

Adding signed numbers requires thinking about direction as well as size. The number line is the best tool for building this understanding. There are two distinct cases: adding numbers with the same sign, and adding numbers with different signs.

Rules for Adding Signed Numbers

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

   Example: (−4) + (−3): |−4| + |−3| = 4 + 3 = 7 → result is −7

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

   Example: (−9) + 4: |−9| − |4| = 9 − 4 = 5 → the larger absolute value belongs to −9 → result is −5

Example 1.8.1
Adding numbers with different signs
  • ProblemAdd −9 + 4.
  • ThinkSigns differ. Subtract the smaller absolute value from the larger: 9 − 4 = 5. The number with the greater absolute value is −9, so the result is negative.
  • Solve−9 + 4 = −5
  • CheckOn the number line, start at −9 and move 4 units to the right. You land at −5. ✓
Answer: −5
Example 1.8.2
Adding numbers with the same sign
  • ProblemAdd (−6) + (−11).
  • ThinkBoth signs are negative. Add the absolute values: 6 + 11 = 17. Keep the negative sign.
  • Solve(−6) + (−11) = −17
  • CheckBoth numbers are debts. Combining two debts gives a larger debt. Negative 17 makes sense. ✓
Answer: −17
SignsRuleExampleResult
Both positiveAdd, result is positive3 + 58
Both negativeAdd absolute values, result is negative(−3) + (−5)−8
Different signsSubtract absolute values, keep sign of larger−7 + 2−5
The Tug-of-War Model

When adding a positive and a negative number, think of it as a tug of war. Whichever side has greater absolute value wins, and the result carries that side's sign. A draw, where both sides have equal absolute value, results in zero.

Practice 1.8Adding Signed Numbers
  1. (−4) + (−9)
  2. 7 + (−13)
  3. −6 + 6
  4. −15 + 8
  5. (−3) + (−3) + (−3)
  6. 12 + (−5) + (−4)
  7. −7 + 20 + (−6)
  8. −4 + (−8) + 15
Section 1.9

Subtracting Signed Numbers

Subtraction of signed numbers can feel confusing because the subtraction symbol and the negative sign look identical. Understanding the principle in this section will remove that confusion entirely. The key insight is this: subtracting a number gives the same result as adding its opposite.

Subtraction as Adding the Opposite

a − b = a + (−b)

Rewrite the subtraction as addition of the opposite, then apply the addition rules from Section 1.8.

Example 1.9.1
Subtracting a positive number
  • ProblemCompute 8 − 12.
  • ThinkRewrite as adding the opposite. The opposite of 12 is −12.
  • Solve8 − 12 = 8 + (−12) = −4
  • CheckDifferent signs. |−12| − |8| = 4. Sign belongs to −12. Result is −4. ✓
Answer: −4
Example 1.9.2
Subtracting a negative number
  • ProblemCompute 5 − (−3).
  • ThinkRewrite as adding the opposite. The opposite of −3 is +3.
  • Solve5 − (−3) = 5 + 3 = 8
  • CheckStarting at 5 and moving 3 units to the right gives 8. ✓
Answer: 8
Example 1.9.3
Subtracting a negative from a negative
  • ProblemCompute −4 − (−7).
  • ThinkRewrite as adding the opposite of −7, which is +7.
  • Solve−4 − (−7) = −4 + 7 = 3
  • Check|7| − |−4| = 3. Sign belongs to 7. Result is 3. ✓
Answer: 3
Student Warning

Subtracting a negative number produces a positive result. This surprises many students. If the reasoning is not clear, always use the rule: rewrite as adding the opposite. The expression 5 − (−3) becomes 5 + 3 because the opposite of −3 is +3. The double negative cancels, and the result is addition.

Practice 1.9Subtracting Signed Numbers
  1. 9 − 15
  2. −3 − 8
  3. 6 − (−4)
  4. −5 − (−12)
  5. 0 − (−7)
  6. −10 − (−10)
  7. −3 − (−3) − 3
  8. −2 − 9 − (−4)
Section 1.10

Multiplying Signed Numbers

Multiplication of signed numbers follows a consistent set of sign rules. These rules can be understood through patterns, not just memorized as arbitrary laws. The pattern argument appears below.

Sign Rules for Multiplication

Same signs → Positive product

   Positive × Positive = Positive     Negative × Negative = Positive

Different signs → Negative product

   Positive × Negative = Negative     Negative × Positive = Negative

The pattern below explains why negative times negative equals positive. Each step decreases the first factor by 1. The product increases by 4 each time. Following the pattern into negative territory, (−1) × (−4) must equal 4:

3 × (−4) = −12 2 × (−4) = −8 1 × (−4) = −4 0 × (−4) = 0 −1 × (−4) = +4 ← pattern continues; negative × negative = positive −2 × (−4) = +8
Example 1.10.1
Multiplying signed numbers
  • ProblemCompute: (a) 6 × (−5)   (b) (−7) × (−3)   (c) (−2) × 9
  • ThinkApply sign rules, then multiply absolute values.
  • Solve (a) Different signs → negative. 6 × 5 = 30 → 6 × (−5) = −30 (b) Same signs → positive. 7 × 3 = 21 → (−7) × (−3) = 21 (c) Different signs → negative. 2 × 9 = 18 → (−2) × 9 = −18
Answer: (a) −30, (b) 21, (c) −18
Multiple Signed Factors

When multiplying three or more signed numbers, count the negative factors. An even count of negatives produces a positive product. An odd count produces a negative product. Then multiply the absolute values. Example: (−2) × (−3) × (−4) has three negatives (odd count) → result is −24.

Practice 1.10Multiplying Signed Numbers
  1. (−5) × 6
  2. (−8) × (−4)
  3. 3 × (−9)
  4. (−1) × 14
  5. (−2) × (−3) × 5
  6. (−4) × (−1) × (−6)
  7. (−3)²
  8. (−1) × (−1) × (−1) × (−1)
Section 1.11

Dividing Signed Numbers

Division follows the same sign rules as multiplication. Because division is the inverse of multiplication, their sign behaviors must match. If (−3) × 4 = −12, then it must be true that (−12) ÷ 4 = −3 and (−12) ÷ (−3) = 4.

Sign Rules for Division

Same signs → Positive quotient

Different signs → Negative quotient

Division by zero is undefined. There is no number that answers the question "how many times does zero go into 8?"

Example 1.11.1
Dividing signed numbers
  • ProblemCompute: (a) (−18) ÷ 3   (b) (−20) ÷ (−4)   (c) 30 ÷ (−6)
  • ThinkApply sign rules, then divide absolute values.
  • Solve (a) Different signs → negative. 18 ÷ 3 = 6 → (−18) ÷ 3 = −6 (b) Same signs → positive. 20 ÷ 4 = 5 → (−20) ÷ (−4) = 5 (c) Different signs → negative. 30 ÷ 6 = 5 → 30 ÷ (−6) = −5
Answer: (a) −6, (b) 5, (c) −5

Division is closely connected to fractions. Any division problem can be written as a fraction, and the sign of the fraction depends on the signs of its numerator and denominator. A negative sign in a fraction can be attached to the numerator, to the denominator, or placed in front of the entire fraction. All three are equivalent forms:

−12/4 = −3 12/(−4) = −3 −(12/4) = −3 ← all three are equivalent
Practice 1.11Dividing Signed Numbers
  1. (−24) ÷ 4
  2. (−36) ÷ (−9)
  3. 45 ÷ (−5)
  4. −72 / 8
  5. −50 / (−10)
  6. 0 / (−7)
  7. (−100) ÷ (−4) ÷ 5
  8. Explain why division by zero is undefined.
Sections 1.12 – 1.14
Common Mistakes, Real Applications, and Algebra Preparation
Sign Mistakes · Signed Numbers in Context · Bridge to Algebra I
Section 1.12

Common Sign Mistakes

Learning to recognize common sign errors is as important as learning the rules themselves. Students who know what to watch for can catch mistakes during checking rather than after receiving a graded assessment.

Mistake 1 — Confusing Subtraction with Negative
−4 − 6 = −2   (treating the second minus as canceling the first)

Correct: −4 − 6 = −4 + (−6) = −10. Two negatives added together produce a larger negative, not a smaller one.

Mistake 2 — Subtracting a Negative Stays Negative
5 − (−8) = −3   (reasoning that two negatives give a negative)

Correct: 5 − (−8) = 5 + 8 = 13. Subtracting a negative means adding the opposite, which increases the value.

Mistake 3 — Losing Track of Signs in Multi-Step Problems
(−3) × (−4) × (−2) = 24   (forgetting that three negatives give a negative)

Correct: Three negative factors, odd count, so the product is negative: (−3) × (−4) × (−2) = −24.

Mistake 4 — Misapplying Absolute Value
−|7| = 7   (assuming absolute value always makes everything positive)

Correct: −|7| = −(7) = −7. The negative sign is outside the absolute value bars and is applied after.

Mistake 5 — Adding Instead of Subtracting in Mixed-Sign Addition
8 + (−3) = 11   (adding absolute values instead of subtracting)

Correct: Different signs mean subtract the smaller absolute value from the larger: 8 − 3 = 5. Keep the positive sign. Answer is 5.

Self-Check Habit

When working with signed numbers, always ask: does this answer make sense given the direction of the numbers? If you started at a negative number and added another negative, the result should be more negative, not less. If you subtracted a negative, the result should be larger, not smaller.

Practice 1.12Error Analysis

Each problem contains a student error. Identify the mistake and write the correct answer.

  1. Student writes: −6 − (−2) = −8. Identify and correct the error.
  2. Student writes: (−3) × (−4) = −12. Identify and correct the error.
  3. Student writes: |−5| = −5. Identify and correct the error.
  4. Student writes: −4 + (−5) = 1. Identify and correct the error.
  5. Student writes: (−2) × (−3) × (−1) = 6. Identify and correct the error.
  6. Student writes: 8 − 11 = 3. Identify and correct the error.
  7. Student classifies −4 as a whole number. Identify and correct the error.
  8. Student writes: −|−9| = 9. Identify and correct the error.
Section 1.13

Using Signed Numbers in Real Situations

Signed numbers are especially powerful in situations where direction, comparison, change, or opposition matters. This section applies the problem-solving method: Read → Represent → Solve → Check → Explain.

Example 1.13.1
Temperature change
  • ReadThe morning temperature in a mountain town is −6°C. By afternoon, the temperature has risen by 14 degrees. What is the afternoon temperature?
  • Represent−6 + 14
  • SolveDifferent signs. |14| − |−6| = 8. Sign belongs to 14. Result: 8
  • CheckRising from below zero by more than 6 degrees should produce a positive temperature. 8°C is positive and reasonable. ✓
  • ExplainThe afternoon temperature is 8°C.
Answer: 8°C
Example 1.13.2
Bank account balance
  • ReadA student's bank account has a balance of $45. The student spends $60 on supplies.
  • Represent45 − 60 = 45 + (−60)
  • Solve|−60| − |45| = 15. Sign belongs to −60. Result: −15
  • CheckSpending more than is available creates an overdraft. A negative balance is correct. ✓
  • ExplainThe new balance is −$15, meaning the account is $15 overdrawn.
Answer: −$15
Example 1.13.3
Change in elevation
  • ReadA hiker begins at an elevation of 240 feet below sea level, recorded as −240 feet. The hiker descends an additional 85 feet.
  • Represent−240 + (−85)
  • SolveSame signs (both negative). 240 + 85 = 325. Result: −325
  • ExplainThe new elevation is −325 feet, or 325 feet below sea level.
Answer: −325 feet
Example 1.13.4
Net displacement
  • ReadA robot moves 15 meters to the right, then 22 meters to the left. What is the final position relative to the starting point?
  • Represent+15 + (−22)
  • Solve|−22| − |15| = 7. Sign belongs to −22. Result: −7
  • ExplainThe robot is 7 meters to the left of its starting position.
Answer: −7 meters (7 meters left of start)
Practice 1.13Real-World Applications
  1. Temperature: At midnight the temperature is −11°C. By noon it has increased by 18 degrees. What is the noon temperature?
  2. Temperature: The high on Monday was 7°C. The high on Tuesday was −3°C. By how much did the temperature change? Was it an increase or decrease?
  3. Finance: A student starts the week with a balance of $32, deposits $50, then spends $95. What is the ending balance? What does a negative balance mean?
  4. Finance: A business records monthly changes: +$4,200, −$1,800, −$3,100, +$900. What is the overall change over four months?
  5. Elevation: A submarine is at −420 feet. It descends 75 more feet. What is the new elevation?
  6. Elevation: A mountain peak is at 14,340 feet. A nearby valley is at −180 feet. What is the difference in elevation?
  7. Direction: A diver starts at −15 feet, descends 5 feet, ascends 8 feet, then descends 3 more feet. What is the final depth?
  8. Pattern: The temperature at a mountaintop is −8°F. For every 1,000 feet you descend, the temperature rises 3°F. What is the temperature 3,000 feet below the summit?
Section 1.14

Preparing for Algebra with Signed Operations

This chapter has covered the number line, number sets, opposites, absolute value, and all four signed number operations. None of this was accidental preparation. Every idea in this chapter will reappear in algebra, and recognizing the connections now will make algebra far more accessible.

Variables work the same way signed numbers do. In algebra, a variable is a symbol representing a number. That number might be positive, negative, zero, or fractional. The operations on variables follow exactly the same sign rules as the operations on signed numbers. When a student writes x + (−x) = 0, that is the additive inverse property applied to a variable. When a student writes −(−x) = x, that is the double negative rule.

Equations use inverse operations. Solving an equation involves using inverse operations to isolate a variable. The inverse of adding a positive number is adding the corresponding negative. The inverse of multiplying by a negative is dividing by that negative. These inverses are signed number operations.

Expressions may include negative terms. Algebraic expressions include terms such as −5x, 3 − 7y, and −2a + 4b − 1. Reading and evaluating these expressions correctly requires fluency with signed number operations. For example, evaluating −5x + 3 when x = −2:

−5(−2) + 3 = 10 + 3 = 13 ← negative × negative = positive, then add constant

Graphs use signed coordinates. The coordinate plane, introduced in Chapter 12, uses signed numbers for both the horizontal and vertical positions of every point. Without comfort with negative numbers, the second, third, and fourth quadrants of the plane are inaccessible.

Practice 1.14Algebra Readiness Check
  1. Evaluate −5x + 3 when x = −2. Show each step.
  2. Evaluate 2t² − 3t + 1 when t = −2. Use parentheses at each substitution step.
  3. A student simplifies −3(x − 4) and gets −3x − 12. Is this correct? Explain using signed number rules.
  4. If x + (−9) = 0, what is x? Which section's rule explains this?
  5. Write the additive inverse of −7x. What is the sum of −7x and its additive inverse?
Chapter 1 — Back Matter

Chapter Summary

Key Terms Review

arithmetic
The branch of mathematics dealing with numbers and the four basic operations: addition, subtraction, multiplication, and division.
number line
A visual representation of numbers as points on a line, with zero at the center, positive numbers to the right, and negative numbers to the left.
positive number
Any number greater than zero. Written without a sign or with a + sign.
negative number
Any number less than zero. Written with a − sign placed directly before it.
zero
The number that is neither positive nor negative. The center of the number line and the additive identity.
integer
A whole number or the negative of a whole number: …, −3, −2, −1, 0, 1, 2, 3, …
whole number
A natural number or zero: 0, 1, 2, 3, 4, …
natural number
A counting number: 1, 2, 3, 4, … Does not include zero.
rational number
A number expressible as a/b where a and b are integers and b ≠ 0. Includes all integers, fractions, terminating decimals, and repeating decimals.
number set
A collection of numbers that share a defining property, such as the set of integers or the set of rational numbers.
additive identity
Zero. Adding zero to any number leaves it unchanged: a + 0 = a.
multiplicative identity
One. Multiplying any number by one leaves it unchanged: a × 1 = a.
opposite
The number at the same distance from zero on the other side of the number line. The opposite of a is −a.
additive inverse
Another name for the opposite of a number. A number and its additive inverse always sum to zero: a + (−a) = 0.
absolute value
The distance of a number from zero on the number line. Always non-negative. Notation: |a|.
signed number
A number carrying a positive or negative sign; used to indicate both magnitude and direction.
sum
The result of addition.
difference
The result of subtraction.
product
The result of multiplication.
quotient
The result of division.

Concept Review Questions

These questions check whether students understand the ideas of this chapter, not just the procedures.

  1. In your own words, explain what a negative number represents. Give two real-world examples.
  2. Why does the number −10 appear to the left of −2 on the number line, even though 10 is greater than 2?
  3. What is the difference between the absolute value of a number and the opposite of a number? Give an example where they are equal, and one where they differ.
  4. Explain why adding two negative numbers produces a more negative result, not a less negative result.
  5. Why is subtracting a negative number the same as adding a positive number? Use the rule for opposites in your explanation.
  6. A student says: "Negative times negative is positive, so two wrongs make a right in math." Is this reasoning mathematically sound? Explain what actually causes the negative-times-negative rule.
  7. What is the additive identity, and why does adding it never change a number?
  8. Why is every integer a rational number, but not every rational number is an integer? Give an example.
  9. Is −|x| always negative? Explain your answer.
  10. Describe a real-world situation that could be modeled by the expression −45 + 70. What would the result represent?

Computation Practice

Set AAbsolute Value
  1. |−8|
  2. |15|
  3. |−3/4|
  4. −|12|
  5. −|−6|
  6. |−20| + |9|
  7. |−3| × |−5|
Set BAdding Signed Numbers
  1. (−4) + (−9)
  2. 7 + (−13)
  3. −6 + 6
  4. −15 + 8
  5. (−3) + (−3) + (−3)
  6. 12 + (−5) + (−4)
  7. −7 + 20 + (−6)
Set CSubtracting Signed Numbers
  1. 9 − 15
  2. −3 − 8
  3. 6 − (−4)
  4. −5 − (−12)
  5. 0 − (−7)
  6. −10 − (−10)
  7. −3 − (−3) − 3
Set DMultiplying Signed Numbers
  1. (−5) × 6
  2. (−8) × (−4)
  3. 3 × (−9)
  4. (−1) × 14
  5. (−2) × (−3) × 5
  6. (−4) × (−1) × (−6)
  7. (−3)²
Set EDividing Signed Numbers
  1. (−24) ÷ 4
  2. (−36) ÷ (−9)
  3. 45 ÷ (−5)
  4. −72 / 8
  5. −50 / (−10)
  6. 0 / (−7)
  7. (−100) ÷ (−4) ÷ 5
Set FMixed Operations
  1. Find the opposite of: 9, −14, 2/3, 0
  2. Classify each number (natural/whole/integer/rational): −5, 0, 7/2, 3
  3. Evaluate: |−4| + (−4)
  4. Evaluate: −(−(−2))
  5. Evaluate: (−3) × (−2) × (−1) × (−1)
  6. Evaluate: −|−9| + |−3|

Answers may be placed in the answer key.

Word Problems

For each problem: write the signed expression, solve, and explain the answer in words.

  1. At midnight the temperature in a city is −11°C. By noon it has increased by 18 degrees. What is the noon temperature?
  2. The high temperature on Monday was 7°C. The high on Tuesday was −3°C. By how much did the temperature change from Monday to Tuesday?
  3. A scientist records four daily low temperatures: −2°, −8°, 4°, and −6°. What is the sum of all four temperatures?
  4. A student starts the week with a bank balance of $32. During the week, the student deposits $50 and then spends $95. What is the ending balance? What does a negative balance mean in context?
  5. A business records the following monthly changes in account value: +$4,200, −$1,800, −$3,100, +$900. What is the overall change over the four months?
  6. A submarine is at an elevation of −420 feet. It descends 75 more feet. What is the new elevation?
  7. A mountain peak is at 14,340 feet above sea level. A nearby valley is at 180 feet below sea level. What is the difference in elevation between the peak and the valley?
  8. The temperature at a mountain summit is −8°F. For every 1,000 feet you descend, the temperature rises 3°F. What is the temperature at 3,000 feet below the summit?
  9. A diver begins at a depth of −15 feet. The diver descends 5 feet, then ascends 8 feet, then descends 3 more feet. What is the diver's final depth?

Error Analysis

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

Problem 1
−6 − (−2) = −8

What is the error? What is the correct answer?

Problem 2
(−3) × (−4) = −12

What is the error? What is the correct answer?

Problem 3
|−5| = −5

What is the error? What is the correct answer?

Problem 4
−4 + (−5) = 1

What is the error? What is the correct answer?

Problem 5
(−2) × (−3) × (−1) = 6

What is the error? What is the correct answer?

Problem 6
8 − 11 = 3

What is the error? What is the correct answer?

Problem 7
A student classifies −4 as a whole number.

What is the error? Which sets does −4 actually belong to?

Problem 8
−|−9| = 9

What is the error? What is the correct answer?

Chapter 1 Checkpoint

Chapter 1 Checkpoint
Complete without assistance — assesses all key skills and concepts from Chapter 1
Part 1 — Number Knowledge
  1. Plot the following numbers on a number line and list them in order from least to greatest: −3, 1, −5, 0, 4, −1.
  2. Classify each number. Identify all sets it belongs to (natural, whole, integer, rational): (a) 0, (b) −7, (c) 5/3, (d) 6.
  3. What is the opposite of each number? (a) 13, (b) −9, (c) 0.
  4. Evaluate: (a) |−14|, (b) −|6|, (c) |−1/2|.
Part 2 — Signed Number Computation
  1. Add: −8 + (−5)
  2. Add: −12 + 17
  3. Subtract: 3 − (−9)
  4. Subtract: −7 − 4
  5. Multiply: (−6) × (−8)
  6. Multiply: 5 × (−3)
  7. Divide: (−42) ÷ 7
  8. Divide: (−60) ÷ (−12)
Part 3 — Mixed Expressions
  1. Evaluate: (−4) + (−4) + (−4)
  2. Evaluate: −3 − (−8) + (−2)
  3. Evaluate: (−2) × (−5) × (−3)
Part 4 — Applications
  1. The temperature at sunrise was −9°F. The temperature at sunset was 6°F. What was the total temperature change?
  2. A business owes $2,400 in expenses and has earned $1,950 in revenue this month. Write and evaluate a signed expression for the business's net balance. What does the result mean?
  3. A submarine is at −380 feet. It ascends 145 feet. What is the new depth? Express as a signed number and interpret it.
Part 5 — Error Analysis
  1. A student claims that −(−4) = −4. Is this correct? Explain.
  2. A student evaluates (−3) × (−3) × (−3) and gets 27. Is this correct? Explain.

Answers may be placed in the answer key.

Looking Ahead
Chapter 2: Factors and Multiples

Chapter 1 gave you a foundation in signed numbers and the structure of the number system. In Chapter 2, the focus shifts from signed operations to the internal structure of whole numbers. You will explore divisibility, factors, multiples, prime numbers, and the tools used to simplify fractions and find common denominators. The greatest common factor and the least common multiple are the hidden machinery behind much of the fraction work that follows in Chapters 3 through 5 and the proportional reasoning that appears in Chapter 6.

MGU Library Connection
  • Number Sets Diagram — Reference Appendix of this textbook
  • Signed Number Reference Card — Appendix F: Symbols Reference
  • Arithmetic 74 Course, Unit 1: Numbers and Operations — the compressed ebook-course companion to this chapter
  • Algebra I, Chapter 2: Real Numbers and Algebraic Properties — extends this chapter's number set work to the full real number system
  • MGU Dictionary entries: integer, absolute value, additive inverse, rational number, number line