Times 1 and times −1

The idea

For every integer a, 1 × a = a, and – 1 × a = – a, that is, multiplying by – 1 gives the additive inverse.

Once, and once taken out

Put the tokens of a into an empty bag once: the bag holds a. That is 1 × a.

Now take the tokens of a out of the bag once (zero pairs first). That is (−1) × a.

If a = 5, what does (−1) × 5 leave in the bag?

• 5 green tokens
• 5 red tokens
• an empty bag

Press × 1 and × (−1)

The coin sits at a number. Press × 1, then press × (−1) twice. Where does the coin go?

× 1 stays, × (−1) flips

Multiplying by 1 puts the tokens in once, so 1 × a = a. Multiplying by −1 takes them out once. A coin at 6 goes to −6; a coin at −4 goes to 4. It jumps over 0 to its additive inverse.

Pressing × (−1) twice brings the coin home: −(−a) = a.

For every integer a, 1 × a = a, and (−1) × a = −a. Multiplying by −1 gives the additive inverse.

• a | 1 × a | (−1) × a
• 6 | 6 | −6
• −4 | −4 | 4
• 9 | 9 | −9

Notes

For every integer a, 1 × a = a and (−1) × a = −a: multiplying by −1 gives the additive inverse.

Check yourself

Find (−1) × (−17).

Answer: 17

(−1) × (−17) = −(−17) = 17.

Find 1 × (−46).

Answer: -46

1 × (−46) = −46.

a is an integer. Is −a always a negative number?

(−1) × a = 12. What is a?

Answer: -12

(−1) × a is the inverse of a. The inverse of −12 is 12, so a = −12.

• Yes, because it has a minus sign. The minus sign means “the additive inverse of a”. If a is negative, −a is positive.
• No: if a = −3, then −a = 3 — correct. Yes! −a is the additive inverse of a. Its sign is the opposite of a’s sign.
• Only when a is 0. When a = 0, −a = 0, which is not negative. For a positive a, −a is negative.