Take away is add the opposite

The idea

Subtracting an integer is the same as adding its additive inverse; the additive inverse of an integer a is written – a, so – (– a) = a.

Not enough tokens

A bag holds 7 green tokens. We want to take away 18 green tokens: 7 − 18.

What will be in the bag after 18 greens are taken away?

• Nothing: it cannot be done
• 11 red tokens
• 11 green tokens

Take out, then put in the opposite

Take the tokens out. If there are too few, put in zero pairs first. Then start again and get the same result a second way: put in the inverse.

Two ways, one bag

To take 18 greens out of 7 greens, 11 zero pairs went in first. Then 18 greens came out, and 11 reds were left: −11.

Putting in 18 reds instead gave the same value. A red token is the inverse of a green token.

The additive inverse of 18 is −18. The additive inverse of −18 is 18: −(−18) = 18.

Subtracting an integer is the same as adding its additive inverse: a − b = a + (−b). The additive inverse of *a* is −a, so −(−a) = a.

• Taking away | Adding the inverse | Value
• 7 − 18 | 7 + (−18) | −11
• 4 − (−12) | 4 + 12 | 16

Notes

Subtracting an integer is the same as adding its additive inverse: a − b = a + (−b). The additive inverse of *a* is −a, so −(−a) = a.

Check yourself

Find 9 − 15. Think of it as 9 + (−15).

Answer: -6

9 − 15 = 9 + (−15) = −6.

Find 5 − (−8).

Answer: 13

5 − (−8) = 5 + 8 = 13.

Which is the same as 6 − (−2)?

Which pair are additive inverses of each other?

• 6 + 2 — correct. Yes! The additive inverse of −2 is 2.
• 6 + (−2). That is 6 − 2. The inverse of −2 is 2, not −2.
• 6 − 2. Taking away −2 is not taking away 2. The inverse of −2 is 2.
• 18 and 18. These are the same number. Inverses cancel to 0: 18 + 18 is 36.
• −18 and −18. Two reds do not cancel. An inverse has the opposite sign.
• −18 and 18 — correct. Yes! A green and a red token cancel: −18 + 18 = 0.