Take out is add the inverse
Multiplying by a negative adds the inverse, again and again.
An easier way to take out
To work out (−4) × 2 we took out 2 green tokens, 4 times. Each time we needed zero pairs first. Taking away a number is the same as adding its inverse.
Instead of taking 2 greens out, what could we put in?
What this lesson covers
The idea
Since removing a number is the same as adding its inverse, multiplying by a negative integer can be done by adding the inverse of the multiplicand as many times as the multiplier's magnitude.
An easier way to take out
To work out (−4) × 2 we took out 2 green tokens, 4 times. Each time we needed zero pairs first. Taking away a number is the same as adding its inverse.
Instead of taking 2 greens out, what could we put in?
- 2 green tokens
- 2 red tokens
- 4 red tokens
Two ways to the same bag
Do each product both ways: take out (with zero pairs), and put in the inverse tokens. Then compare the two bags.
Add the inverse, again and again
Taking out 2 greens is the same as putting in 2 reds. So (−4) × 2 means adding the inverse of 2, which is −2, 4 times.
The number of times is the size of the multiplier: 4 and 3. The sign of the multiplier says: use the inverse of the multiplicand.
Removing a number is the same as adding its inverse. So multiplying by a negative integer can be done by adding the inverse of the multiplicand as many times as the magnitude of the multiplier.
- Product | Add the inverse | Value
- (−4) × 2 | (−2) + (−2) + (−2) + (−2) | −8
- (−3) × (−4) | 4 + 4 + 4 | 12
Notes
Multiplying by a negative integer can be done by adding the inverse of the multiplicand as many times as the magnitude of the multiplier: (−4) × 2 = (−2) + (−2) + (−2) + (−2).
Check yourself
Find (−6) × 4 by adding the inverse of 4, six times.
Answer: -24
(−4) + (−4) + (−4) + (−4) + (−4) + (−4) = −24.
Find (−3) × (−7) by adding the inverse of −7, three times.
Answer: 21
7 + 7 + 7 = 21.
How do we work out (−3) × 7 by adding?
How do we work out (−5) × (−2) by adding?
- Add 7 three times. That is 3 × 7. A negative multiplier adds the inverse of 7, which is −7.
- Add (−7) three times — correct. Yes! The inverse of the multiplicand 7 is −7, added 3 times (the size of the multiplier).
- Add (−7) seven times. The number of times is the size of the multiplier, 3, not the multiplicand.
- Add 2 five times — correct. Yes! The inverse of −2 is 2, added 5 times.
- Add (−2) five times. That is 5 × (−2). The inverse of the multiplicand −2 is 2, not −2.
- Add (−5) two times. The multiplicand is −2, so we add its inverse, 2, and we do it 5 times.