courses › Pre-Algebra

Negative numbers

level 9 course

Add, subtract, multiply, and divide with signs.

Best in your head · no pen, no calculator why?

Opens at level 9.

Builds on: Two-digit add & subtract (not open yet)

the lesson

The idea, the techniques and a tip for each skill, right here. The Learn page adds worked examples for every skill and untimed practice.

Read the lesson · about 3 minutes

The idea

A negative number sits below zero: −5 is five steps down. On a number line, subtracting 5 moves you 5 steps left, so subtracting −5 moves you 5 steps right. It is the same as adding 5.

For × and ÷, work the numbers as if both were positive, then fix the sign: same signs give a positive, different signs give a negative.

Absolute value, written with bars like |−7|, is a number's distance from zero. A distance is never negative, so |−7| is 7.

Techniques

Turn subtraction into addition

Any sum or difference with a negative number in it.

Rewrite subtracting as adding the opposite: − (−12) becomes + 12, and − 5 becomes + (−5).
Same signs: add the sizes and keep the sign.
Different signs: take the smaller size from the bigger. The answer gets the sign of the number farther from zero.
Equal sizes with different signs cancel, so the answer is 0.

worked example

Example: −8 − (−15)

Subtracting −15 is adding 15: −8 + 15.
Different signs: 15 − 8 = 7.
15 is farther from zero and positive, so the answer is positive.

Answer: 7

Sign rule for × and ÷

Multiplying or dividing two signed numbers.

Work out the size, ignoring the signs.
Same signs give a positive answer. Different signs give a negative one.

worked example

Example: (−56) ÷ 7

56 ÷ 7 = 8.
One negative and one positive: the signs differ, so the answer is negative.

Answer: −8

Absolute value is distance

You see bars, like |−9|.

Bars around a calculation: work it out inside first.
Then drop the sign. A distance from zero is never negative.
Bars on each number separately: take each absolute value first, then do the calculation, which can come out negative.

worked example

Example: |−3 − 8|

Inside first: −3 − 8 = −11.
Its distance from zero is 11.

Answer: 11

Tips by skill

TipAdd & subtract with signs: Change any subtraction to adding the opposite. Same signs add; different signs subtract, the sign farther from zero wins, and equal sizes give 0.
TipMultiply & divide with signs: Do the arithmetic without signs. Then same signs give a positive, different signs a negative.
TipAbsolute value: Bars around a calculation? Work inside first, then drop the sign. Bars on each number? Take each value first, then calculate.

Watch out for

Reading − (−12) as − 12. Two minus signs in a row make a plus.
Dropping the sign when dividing: 72 ÷ 8 is 9, but (−72) ÷ 8 is −9.
Using the wrong order for the bars. |−4 − 9| works inside first, giving 13. |−4| − |9| takes each value first: 4 − 9, which is −5.
Giving the answer the sign of the first number instead of the number farther from zero: 6 + (−15) is −9.

skills · practice stats

From rounds of this course only: box, review and test-out answers are left out. Once a skill has 40 tries, it compares your first 20 tries with your last 20.

Add & subtract with signs par 6 s not tried yet
worked example
−10 − (−17)

Answer: 7
1. Subtracting a negative is adding: −10 − (−17) = −10 + 17.
2. The signs differ: 17 − 10 = 7, and 17 is farther from zero, so the answer is 7.
Multiply & divide with signs par 5 s not tried yet
worked example
(−6) × 8

Answer: -48
1. 6 × 8 = 48.
2. The signs differ, so the result is negative: −48.
Absolute value par 7 s not tried yet
worked example
|−13| − |−12|

Answer: 1
1. Each number has its own bars, so take each absolute value first: |−13| = 13 and |−12| = 12.
2. Then 13 − 12 = 1.

rest ladder

Win 3 of your last 4 rounds and the course rests. A win is 90% right, within 2× the round's par. Pass the review when it comes back and the next rest is longer.

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