Operations on integers

Q: What are integers?

All whole numbers, positive and negative, including zero: \dots -2, -1, 0, 1, 2 \dots

Q: Why is a negative times a negative a positive?

Multiplying by -1 reverses direction on the number line. Doing it twice brings you back to the original direction.

Q: How do you subtract integers?

Change the subtraction to addition of the opposite number: 5 - 8 = 5 + (-8) = -3.

Integers are the whole numbers and their negatives: $\dots, - 3, - 2, - 1, 0, 1, 2, 3, \dots$ . Working with them comes down to a few sign rules, which are easiest to see on a number line.

Formulas

Adding

same signs: add and keep the sign different signs: subtract and keep the sign of the bigger number

Subtracting means adding the opposite

a - (- b) = a + b a - b = a + (- b)

Multiplying and dividing

(+) (+) = +, (-) (-) = +, (+) (-) = -, (-) (+) = -

Worked examples

Example 1. Find

(- 7) + (- 5)

Same signs: add $7 + 5 = 12$ and keep the negative sign.

Answer:

- 12

Example 2. Find

8 + (- 13)

Different signs: $13 - 8 = 5$ , and 13 is the bigger number, which is negative.

Answer:

- 5

Example 3. Find

6 - (- 4)

Subtracting a negative is adding a positive: $6 + 4 = 10$ .

Answer: 10

Example 4. Find

(- 9) \times (- 3)

and

(- 48) \div 6

Two negatives multiply to a positive: $9 \times 3 = 27$ .
A negative divided by a positive is negative: $48 \div 6 = 8$ , so $- 8$ .

Answer: 27 and

- 8

Example 5. The temperature is

- 3^{\circ}

C and falls by 5 degrees. What is it now?

$- 3 - 5 = - 8$ .

Answer:

- 8^{\circC}

Common mistakes

Treating $a - (- b)$ as $a - b$ . Two minus signs together make a plus.
Subtracting the wrong way round when the signs are different: always subtract the smaller size from the larger.
Applying the "two negatives make a positive" rule to addition. $(- 2) + (- 3) = - 5$ , not $5$ .

Practice questions with answers

1. Which of the following is NOT an integer?

A−12
B0
C0.5
D47

Show answer

Answer: C) 0.5

Integers are whole numbers and their negatives, including zero. 0.5 is a fraction (one half), so it is not an integer.

2. What is (-4) + (-6)?

A-10
B10
C-2
D2

Show answer

Answer: A) -10

Both are negative, so add their values and keep the negative sign: 4 + 6 = 10, giving -10.

3. What is 7 - (-2)?

Show answer

Answer: A) 9

Subtracting a negative is adding: 7 - (-2) = 7 + 2 = 9.

4. What is

(- 3) \times 5

A-15
B15
C-8
D8

Show answer

Answer: A) -15

A negative times a positive is negative:

3 \times 5 = 15

, so

(- 3) \times 5 = - 15

5. Evaluate: (-12) ÷ 4

Show answer

Answer: A) -3

A negative divided by a positive is negative. 12 ÷ 4 = 3, so (-12) ÷ 4 = -3.

6. A team's points in a treasure hunt change round by round: −15, then +22, then −9, then +5. The team starts at 0. What are its points at the end, and is that above or below the start?

A−3 points, below the start
B51 points, above the start
C0 points, level with the start
D3 points, above the start

Show answer

Answer: D) 3 points, above the start

Add in order: 0 + (−15) = −15, then −15 + 22 = 7, then 7 + (−9) = −2, then −2 + 5 = 3. The team finishes on 3 points, just above where it started. Adding every round as a positive gives 51, and losing track of one sign is what turns 3 into −3.

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Frequently asked questions

What are integers?

All whole numbers, positive and negative, including zero: $\dots - 2, - 1, 0, 1, 2 \dots$

Why is a negative times a negative a positive?

Multiplying by $- 1$ reverses direction on the number line. Doing it twice brings you back to the original direction.

How do you subtract integers?

Change the subtraction to addition of the opposite number: $5 - 8 = 5 + (- 8) = - 3$ .

Is zero positive or negative?

Neither. Zero is an integer that is not positive and not negative.

Operations on integers

Formulas

Worked examples

Common mistakes

Practice questions with answers

Practice by class

Frequently asked questions

Related topics