Chapter 1: Integers & Number Sense

Pre-Algebra • Chapter 1 of 10 • Updated February 2026

Numbers are the foundation of all mathematics. Before we can solve equations or work with variables, we need a solid understanding of the integers — the counting numbers, their negatives, and zero. This chapter builds that foundation: we explore the different families of numbers, how to place them on a number line, how to compare them, and how to perform every arithmetic operation with both positive and negative integers. We finish with the order of operations, a set of rules that ensures every expression has exactly one correct value.

1.1 Families of Numbers

Mathematicians organize numbers into sets based on their properties. Understanding these sets helps you know which numbers are available in a given context and which operations always produce results within the same set.

Natural Numbers $\mathbb{N}$

The counting numbers: $1, 2, 3, 4, 5, \ldots$
Used for counting objects. Does not include zero.

Whole Numbers $\mathbb{W}$

Natural numbers plus zero: $0, 1, 2, 3, 4, \ldots$
Useful when "none" is a valid quantity.

Integers $\mathbb{Z}$

Whole numbers and their negatives: $\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots$
Every natural number has an opposite here.

Rational Numbers $\mathbb{Q}$

Any number expressible as $\frac{p}{q}$ where $p, q \in \mathbb{Z}$, $q \neq 0$.
Includes all integers, fractions, and terminating or repeating decimals.

Definition: Integer

An integer is any number in the set $\mathbb{Z} = \{\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots\}$. The positive integers are $1, 2, 3, \ldots$ and the negative integers are $-1, -2, -3, \ldots$. Zero is an integer that is neither positive nor negative.

Notice the nesting structure: every natural number is a whole number, every whole number is an integer, and every integer is a rational number. We write this as $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q}$.

1.2 The Number Line

The number line is a visual tool that gives every real number a unique location in space. Negative numbers appear to the left of zero; positive numbers appear to the right. The further a number is from zero, the greater its distance from the origin.

Interactive number line: drag the slider to place an integer n (blue dot). Its opposite −n (red dot) appears automatically. The dashed segment shows absolute value |n| = distance from zero.

Number line showing integers from −8 to 8. The blue dot at −5 and red dot at +5 are opposites: same distance (5 units) from zero.

Key observations from the number line:

Numbers increase as you move to the right and decrease as you move to the left.
Every positive number $n$ has an opposite $-n$ located the same distance from zero on the other side.
$-(-n) = n$ — the opposite of an opposite returns the original number.
Zero is its own opposite: $-0 = 0$.

1.3 Absolute Value

Definition: Absolute Value

The absolute value of a number $a$, written $|a|$, is its distance from zero on the number line. Distance is always non-negative, so: $$|a| = \begin{cases} a & \text{if } a \geq 0 \\ -a & \text{if } a < 0 \end{cases}$$

Absolute value strips away the sign and keeps the magnitude. For example, $|7| = 7$ and $|-7| = 7$ because both $7$ and $-7$ are seven units from zero. Note that $|0| = 0$.

Example 1.1 — Evaluating Absolute Value Expressions

Simplify each expression.

(a) $|-15|$

Solution: $-15$ is $15$ units from zero, so $|-15| = 15$.

(b) $-|{-8}|$

Solution: First find $|-8| = 8$, then negate: $-|{-8}| = -8$.

(c) $|3 - 9|$

Solution: Evaluate inside first: $3 - 9 = -6$, then $|-6| = 6$.

(d) $|{-4}| + |{-11}|$

Solution: $4 + 11 = 15$.

1.4 Comparing and Ordering Integers

On the number line, a number to the left is always less than a number to the right. We use the inequality symbols $<$ (less than) and $>$ (greater than) to compare integers.

Ordering Rule

For any two integers $a$ and $b$:

$a < b$ means $a$ lies to the left of $b$ on the number line.
$a > b$ means $a$ lies to the right of $b$ on the number line.
Every negative integer is less than zero, and zero is less than every positive integer.
Among two negative integers, the one with the larger absolute value is actually smaller: $-10 < -3$ because $|-10| > |-3|$.

Example 1.2 — Ordering Integers

Order the following integers from least to greatest: $-7,\ 3,\ -1,\ 0,\ -12,\ 5$.

Strategy: Locate each on a number line or use the rule that more-negative means smaller.

Answer: $-12,\ -7,\ -1,\ 0,\ 3,\ 5$

Check: moving left to right on the number line, each number is larger than the one before it. ✓

1.5 Adding and Subtracting Integers

Adding Integers

Rules for Addition of Integers

Same signs: Add the absolute values; keep the common sign.
$7 + 5 = 12$ $(-7) + (-5) = -12$
Different signs: Subtract the smaller absolute value from the larger; keep the sign of the number with the larger absolute value.
$(-7) + 5 = -2$ $7 + (-5) = 2$

Subtracting Integers

Key Principle: Subtraction = Add the Opposite

To subtract an integer, add its opposite (additive inverse): $$a - b = a + (-b)$$ This converts every subtraction problem into an addition problem.

Example 1.3 — Adding and Subtracting Integers

(a) $(-13) + (-9)$

Same signs: $13 + 9 = 22$, keep negative. Answer: $-22$

(b) $(-8) + 15$

Different signs: $15 - 8 = 7$, keep the sign of $15$ (positive). Answer: $7$

(c) $6 - (-4)$

Rewrite: $6 + 4 = 10$. Answer: $10$

(d) $-3 - 8$

Rewrite: $-3 + (-8) = -11$. Answer: $-11$

(e) $-5 - (-12)$

Rewrite: $-5 + 12 = 7$. Answer: $7$

1.6 Multiplying and Dividing Integers

Multiplication and division with integers follow a simple sign rule that you can remember with one sentence: same signs give positive; different signs give negative.

Operation	Signs of Factors/Dividend&Divisor	Sign of Result	Example
Multiplication	$(+) \times (+)$	Positive $(+)$	$4 \times 3 = 12$
Multiplication	$(-) \times (-)$	Positive $(+)$	$(-4) \times (-3) = 12$
Multiplication	$(+) \times (-)$	Negative $(-)$	$4 \times (-3) = -12$
Multiplication	$(-) \times (+)$	Negative $(-)$	$(-4) \times 3 = -12$
Division	$(+) \div (+)$	Positive $(+)$	$20 \div 4 = 5$
Division	$(-) \div (-)$	Positive $(+)$	$(-20) \div (-4) = 5$
Division	$(+) \div (-)$	Negative $(-)$	$20 \div (-4) = -5$
Division	$(-) \div (+)$	Negative $(-)$	$(-20) \div 4 = -5$

Quick Sign Count for Products

When multiplying or dividing a chain of integers, count the number of negative factors. Even count → positive result. Odd count → negative result.
Example: $(-2)(-3)(-5) = -30$ (three negatives → odd → negative).

Example 1.4 — Multiplying and Dividing Integers

(a) $(-7) \times (-6)$

Same signs (both negative) → positive. $7 \times 6 = 42$. Answer: $42$

(b) $(-9) \times 4$

Different signs → negative. $9 \times 4 = 36$. Answer: $-36$

(c) $(-48) \div (-8)$

Same signs → positive. $48 \div 8 = 6$. Answer: $6$

(d) $72 \div (-9)$

Different signs → negative. $72 \div 9 = 8$. Answer: $-8$

(e) $(-2) \times (-3) \times (-4) \times (-1)$

Four negative factors → even count → positive. $2 \times 3 \times 4 \times 1 = 24$. Answer: $24$

1.7 Order of Operations (PEMDAS)

When an expression contains multiple operations, we need an agreed-upon order so that everyone arrives at the same answer. The acronym PEMDAS summarizes this order.

PEMDAS — Order of Operations

Parentheses (and all grouping symbols: brackets $[\,]$, braces $\{\,\}$, fraction bars, absolute values)
Exponents (powers and roots)
Multiplication and Division — left to right, equal priority
Addition and Subtraction — left to right, equal priority

Note: Multiplication and Division are at the same level; likewise Addition and Subtraction. When two operations share a level, work left to right.

Example 1.5 — Applying PEMDAS

Evaluate: $3 + 4 \times 2^2 - (10 - 3) \div 7$

Step 1 (Parentheses): $(10 - 3) = 7$

Expression becomes: $3 + 4 \times 2^2 - 7 \div 7$

Step 2 (Exponents): $2^2 = 4$

Expression becomes: $3 + 4 \times 4 - 7 \div 7$

Step 3 (Multiplication/Division, left to right): $4 \times 4 = 16$, then $7 \div 7 = 1$

Expression becomes: $3 + 16 - 1$

Step 4 (Addition/Subtraction, left to right): $3 + 16 = 19$, then $19 - 1 = 18$

Answer: $18$

Example 1.6 — PEMDAS with Integers

Evaluate: $-2 \times [3 + (-5)^2] \div (-11)$

Step 1 (Innermost grouping — brackets): Evaluate inside the brackets first.

$(-5)^2 = 25$ (exponent before addition)

$3 + 25 = 28$

Step 2 (Multiply/Divide left to right):

$-2 \times 28 = -56$

$-56 \div (-11)$ — Hmm, this does not divide evenly. Leave as a fraction: $\dfrac{-56}{-11} = \dfrac{56}{11}$.

Answer: $\dfrac{56}{11}$

Example 1.7 — Evaluating with Absolute Value

Evaluate: $|3 - 8| \times (-2) + 5^2 \div (-5)$

Step 1 (Grouping — absolute value): $|3 - 8| = |-5| = 5$

Step 2 (Exponent): $5^2 = 25$

Expression: $5 \times (-2) + 25 \div (-5)$

Step 3 (Multiply/Divide, left to right): $5 \times (-2) = -10$; $25 \div (-5) = -5$

Expression: $-10 + (-5) = -15$

Answer: $-15$

1.8 Real-World Applications

Integers appear naturally whenever quantities can be negative: temperatures below zero, depths below sea level, financial losses, or positions below a starting point.

Example 1.8 — Temperature Change

At 6 a.m., the temperature was $-14°F$. By noon, it had risen $23°F$. By 6 p.m., it dropped $11°F$ from the noon high. What was the temperature at 6 p.m.?

Noon temperature: $-14 + 23 = 9°F$

6 p.m. temperature: $9 - 11 = -2°F$

Answer: $-2°F$

Example 1.9 — Checking Account

Maya's checking account has a balance of $-\$45$ (overdrawn by $45). She deposits $\$120$ and then writes a check for $\$38$. What is her final balance?

$-45 + 120 - 38 = -45 + 120 - 38$

$= 75 - 38 = \$37$

Answer: Maya's balance is $\$37$.

Common Mistakes to Avoid

Double negatives: $-(-3) = +3$, not $-3$. Subtracting a negative is the same as adding a positive.
Exponent vs. multiplication: $(-3)^2 = (-3)(-3) = 9$, but $-3^2 = -(3^2) = -9$. Parentheses matter!
Left-to-right for same-level ops: $12 \div 4 \times 3 = 3 \times 3 = 9$, not $12 \div 12 = 1$.
Sign of zero: Zero has no sign. $|0| = 0$ and $-0 = 0$.

Practice Problems

Chapter 1 Practice

Absolute Value & Comparisons

Evaluate: $|{-23}| - |{-9}|$
Order from least to greatest: $-8, 2, -15, 0, 7, -3$
True or false: $-|{-6}| = |6|$. Explain.

Integer Operations

$(-17) + 9 + (-4)$
$-8 - (-13) - 5$
$(-6) \times 7 \times (-2)$
$(-144) \div (-12)$
$(-3)^4$

Order of Operations

$5 \times 3 - 2^3 + 4$
$[(-4) + 10] \div 3 \times (-2)$
$|{-7}| + (-3)^2 \div (-3) \times 2$

Word Problem

A submarine starts at a depth of $-200$ ft. It descends an additional $75$ ft, then rises $130$ ft. What is its final depth?

Answers: 1) 14 2) $-15, -8, -3, 0, 2, 7$ 3) False: $-|{-6}| = -6 \neq 6$ 4) $-12$ 5) $0$ 6) $84$ 7) $12$ 8) $81$ 9) $11$ 10) $-4$ 11) $1$ 12) $-145$ ft

Chapter Summary

Numbers are organized into sets: $\mathbb{N} \subset \mathbb{W} \subset \mathbb{Z} \subset \mathbb{Q}$.
The number line provides a geometric picture: larger numbers are to the right.
Absolute value $|a|$ measures distance from zero; it is always $\geq 0$.
Adding integers: same signs → add & keep sign; different signs → subtract & take sign of the larger magnitude.
Subtracting: add the opposite — $a - b = a + (-b)$.
Multiplying/Dividing: same signs → positive; different signs → negative.
PEMDAS: Parentheses, Exponents, Multiplication/Division (left to right), Addition/Subtraction (left to right).

Chapter 2: Fractions & Decimals →