Integers

What you'll learn

• Understand negative numbers and where they appear in real life
• Place integers on the number line and compare them
• Add, subtract, multiply, and divide integers correctly
• Apply commutative, associative, and distributive properties

Key concepts

Negative Numbers and the Number Line

Integers = { …, −4, −3, −2, −1, 0, 1, 2, 3, 4, … }

Negative numbers arise whenever a quantity goes below zero.

Number line:

← −6 −5 −4 −3 −2 −1  0  1  2  3  4  5  6 →

• Moving right → increasing value
• Moving left → decreasing value
• Every positive integer has a corresponding negative integer (called its opposite): opposite of 5 is −5

Absolute value |n| = distance from 0 (always positive or zero)

• |7| = 7, |−7| = 7, |0| = 0

Ordering integers:

• On the number line: the further right, the greater the value
• −1 > −5 (because −1 is to the right of −5)
• 0 > −3

Worked Example: Arrange in ascending order: 3, −8, 0, −2, 6, −1 Answer: −8, −2, −1, 0, 3, 6

Addition of Integers

Rule 1 — Same sign: Add absolute values, keep the common sign.

• (+4) + (+3) = +7
• (−4) + (−3) = −7

Rule 2 — Different signs: Subtract the smaller absolute value from the larger, take the sign of the larger.

• (+7) + (−3) = +(7−3) = +4
• (−7) + (+3) = −(7−3) = −4

Number line method: Start at the first number, move right for positive addend, left for negative.

Worked Example: (−5) + 8 Start at −5, move 8 right → land on 3

Additive inverse (opposite): a + (−a) = 0

• 6 + (−6) = 0

Subtraction of Integers

Key rule: Subtracting an integer = adding its opposite.

a − b = a + (−b)

Worked Example: A submarine at −150 m dives a further 80 m. New depth? −150 − 80 = −150 + (−80) = −230 m

Multiplication of Integers

Sign rules for multiplication:

Memory tip: Same signs → Positive; Different signs → Negative

Product of multiple integers:

• Count the negative factors.
• Even number of negatives → Positive product
• Odd number of negatives → Negative product

Worked Example: (−2) × (−3) × (−4) × (+1) Three negatives (odd) → result is negative 2 × 3 × 4 × 1 = 24 → answer is −24

Division of Integers

Sign rules for division — same as multiplication:

Worked Example: The temperature drops 18°C equally over 6 hours. Change per hour? −18 ÷ 6 = −3°C per hour

Properties of Integer Operations

Commutative Property

Associative Property

Distributive Property

a × (b + c) = a×b + a×c

Works for integers with signs:

• (−3) × (4 + 5) = (−3)×4 + (−3)×5 = −12 + (−15) = −27
• Check: (−3) × 9 = −27 ✓

Worked Example (mental maths): Calculate (−7) × 99 = (−7) × (100 − 1) = (−7)×100 − (−7)×1 = −700 + 7 = −693

Identity and Zero Properties

Quick check

1. What is the value of |−15| + |−8|?
2. Compute: (−12) + 7 − (−5) + (−3)
3. Evaluate: (−6) × (−4) × (−2)
4. The temperature at midnight was −4°C. It rose by 9°C by noon and fell by 12°C by evening. What was the evening temperature?
5. Verify the distributive property: (−5) × (3 − 7)

Open the Practice tab for graded questions on Integers.