The mirror of a fraction

The idea

Every positive fraction has an additive inverse called a negative fraction; its minus sign may be placed before the fraction, with the numerator, or with the denominator.

Inverses of fractions

Every natural number has an additive inverse: 3 has −3 and 19 has −19. Adding the two gives 0. Do fractions have one too?

What do you think the additive inverse of 3/4 is?

• −3/4
• 4/3
• 3/4 again

Mirror at zero

Mirror the fraction across 0. Then write the minus sign in the three places it can go.

One point, three ways

Every positive fraction has an additive inverse, called a negative fraction: −3/4 for 3/4, −19/7 for 19/7. The minus sign can go before the fraction, with the numerator, or with the denominator.

All three ways name the same point, the same distance from 0 on the other side.

Notes

Every positive fraction has an additive inverse, a negative fraction. Its minus sign can go before the fraction, with the numerator, or with the denominator.

Check yourself

What is the additive inverse of 5/7?

Where can the minus sign of a negative fraction go?

What is 3/4 + (−3/4)?

Answer: 0

0. That is what additive inverse means: the two numbers cancel.

Compared with 3/4, where is −3/4 on the number line?

• −5/7 — correct. Yes. 5/7 + (−5/7) = 0.
• 5/7. Adding 5/7 to 5/7 doubles it. It does not give 0.
• 7/5. 7/5 is the fraction turned upside down. Adding it to 5/7 does not give 0.
• Only before the fraction. The numerator and the denominator can carry it too.
• Only with the numerator. It can go before the fraction or with the denominator too.
• Before the fraction, with the numerator, or with the denominator — correct. Yes. All three place the number at the same point.
• The same distance from 0, on the other side — correct. Yes. It is the mirror image of 3/4 across 0.
• On the same side, further from 0. A negative number lies on the left of 0, on the other side.
• At 0. Only 0 is at 0. −3/4 is three quarters of a step left of it.