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Foundations · Numbers

Negative numbers

Integers on the number line, adding and subtracting with signs, and the two rules for multiplying and dividing negatives.

First, the rules

Left of zero

Temperatures below freezing, floors below ground, money you owe: negative numbers sit to the left of zero on the number line, and every rule here is a move along that line.

★ Ordering
Example 1
Further left is smaller: −7 < −2 < 0 < 3. The "bigger-looking" negative is the smaller number.
★ Adding
Example 1
+ b moves right by b; + (−b) moves left by b. −3 + 5 = 2, 4 + (−6) = −2.
★ Subtracting
Example 2
− b moves left by b; − (−b) moves right by b (two minuses make a plus). 2 − 5 = −3, 3 − (−4) = 7.
★ Multiplying and dividing
Example 3
Same signs → positive, different signs → negative. (−3)(−4) = 12, (−3)(4) = −12, −12 ÷ 3 = −4, −12 ÷ (−3) = 4.
Many factors
Example 3
Count the minus signs: an even number gives +, an odd number gives −. (−1)(−2)(−3) = −6.
Powers
(−2)³ = −8, (−2)⁴ = 16, −2⁴ = −16 — without the bracket the minus is applied last.

See it

Arrows on the line

Set a and b and watch the arrow: addition goes right, subtraction goes left, and subtracting a negative turns the arrow round. Multiplication is repeated jumps.

Step by step

Worked examples

1

Temperatures

ordering · adding

At 6 am it is −4 °C. By noon it has risen 9 degrees; overnight it falls 12. Order the three temperatures.

Ask yourself: Rising is moving right on the line, falling is moving left.

Show the solution
  1. Noon: −4 + 9 = 5 °C.
  2. Night: 5 − 12 = −7 °C.
  3. Smallest to largest: −7 < −4 < 5.

Answer −7 °C < −4 °C < 5 °C

2

Minus a minus

subtracting negatives

Work out (a) 3 − (−8), (b) −5 − (−2), (c) −6 − 7.

Ask yourself: Rewrite every "− (−)" as "+" before you move.

Show the solution
  1. (a) 3 + 8 = 11.
  2. (b) −5 + 2 = −3 (start at −5, move right 2).
  3. (c) −6 − 7 = −13 (start at −6, move left 7).

Answer 11 · −3 · −13

3

Signs in a product

multiplying · dividing

Evaluate (a) (−4) × (−6), (b) 36 ÷ (−9), (c) (−2) × 5 × (−3) × (−1).

Ask yourself: Count the minus signs first, then multiply the sizes.

Show the solution
  1. (a) Two minuses → positive: 24.
  2. (b) One minus → negative: −4.
  3. (c) Three minuses → negative; sizes 2 × 5 × 3 × 1 = 30: −30.

Answer 24 · −4 · −30

Practice: integers

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Adding, subtracting (including minus a minus), multiplying, dividing and ordering negative numbers. Your best score is kept in this browser only.

About this lesson

Negative numbers for school mathematics: where integers sit on the number line, why subtracting a negative means adding, the sign rules for multiplication and division, and how to order and compare negative values. A number line you can move, worked examples and practice questions with explanations. Everything on this page runs in your browser; nothing you type is sent or stored. Spotted an error? Tell us.