Skill: integer-operations

Integer Operations: Add, Subtract, Multiply, Divide

Integers are the whole numbers plus their negatives. Adding and subtracting them is walking the number line — right for positive, left for negative. Multiplying and dividing follow one clean rule: same signs → positive, different signs → negative.

1 Understand

The rules of signed numbers, in plain language.

What it is

Integers are the whole numbers plus their negatives: …, −3, −2, −1, 0, 1, 2, 3, …. Adding and subtracting them is walking the number line — right for positive, left for negative. Multiplying and dividing follow one clean rule: same signs → positive, different signs → negative. Every signed-number calculation in algebra runs on these rules.

Why it matters

Bank balances, temperatures, elevations, and game scores all go negative. And every equation you will ever solve needs integer arithmetic to be automatic — sign errors are the #1 source of “I knew how but got it wrong.”

Where it is used

Temperatures below zero · bank overdrafts · football yardage lost and gained · elevation below sea level.

2 See It

Two pictures that carry the whole skill.

Addition is walking. Start at −3, walk 5 steps right, land on 2. The number line never lies about signs.

MULTIPLY / DIVIDE

SIGN

(+)(+)

(−)(−)

(+)(−)

−

(−)(+)

−

One rule for × and ÷. Same signs make positive; different signs make negative. Memorize the grid, not four separate facts.

3 Worked Examples

Follow each step. The pattern is always the same: picture the walk, or check the sign grid.

Example 1 Add: −7 + 4

Thinking: start at −7, walk 4 right.
Walking 4 right from −7: −7 → −6 → −5 → −4 → −3.
The bigger size wins the sign: 7 beats 4, so the answer is negative, and 7 − 4 = 3.

−7 + 4 = −3

Example 2 Subtract a negative: −5 − (−9)

Thinking: subtracting a negative is the same as adding its opposite: −5 + 9.
Rewrite first: −5 − (−9) = −5 + 9.
Walk it: start at −5, walk 9 right — 5 steps reach 0, then 4 more land on 4.

−5 − (−9) = 4

Example 3 Multiply: (−6)(−4)

Thinking: same signs → positive. 6 × 4 = 24.
Check the grid: (−)(−) gives +.
Two negatives “cancel” in multiplication.

(−6)(−4) = 24

Example 4 Divide: 56 ÷ (−7)

Thinking: different signs → negative. 56 ÷ 7 = 8.
Check: −8 × 7 = −56, and here the signs differ, so −8 is right.

56 ÷ (−7) = −8

4 Common Mistakes

These three errors show up on almost every integer quiz. Spot them now and they will never cost you points.

Mistake 1: −5 − 3 = 8 (dropping the negative)

Wrong

Treating it as 5 − 3 with a sign slapped on: 8, or “positive 8”.

Right

−8 — you are at −5 and walk 3 more left.

Rule: when both numbers pull the same direction, the answer keeps that sign and the values add.

Mistake 2: −4 − (−4) = −8

Wrong

“Two negatives make a bigger negative”: −8.

Right

−4 + 4 = 0. The walk goes left 4, then right 4 — back where you started.

Rule: minus-a-negative is plus-a-positive. Rewrite it first: −4 − (−4) → −4 + 4.

Mistake 3: (−3)(−3) = −9

Wrong

“There is a negative in there, so the answer is negative”: −9.

Right

9 — same signs → positive.

Rule: “Same → positive, different → negative.” Say it every time until it is reflex.

5 Quick Check

Try each one on paper first, then reveal the answer.

1. Compute −8 + 13.

Answer

5 — start at −8, walk 13 right: 8 steps reach 0, then 5 more. Positive wins because 13 > 8.

2. Compute (−5)(7).

Answer

−35 — different signs → negative; 5 × 7 = 35.

3. Compute −4 + (−9).

Answer

−13 — both negative, so walk left twice: 4 + 9 = 13, keep the negative sign.

Key Points to Remember

Adding is walking: right for positive, left for negative.
Subtracting a negative is the same as adding its opposite: a − (−b) = a + b.
For × and ÷: same signs → positive, different signs → negative.
When signs fight in addition, the bigger size wins; the answer is the difference of the sizes.
Work multi-step problems strictly left to right.

Practice integer problems Try the integer tool Explore the number line