# Properties of Rational Numbers - Negative Or Additive Inverse of Rational Numbers

## Notes

### Negative Or Additive Inverse of Rational Numbers:

2 + (– 2) = (– 2) + 2 = 0.

So, we say 2 is the negative or additive inverse of – 2 and vice-versa.

In general, for an integer a, we have, a + (– a) = (– a) + a = 0; so, a is the negative of – a and – a is the negative of a.

11/7 + (-11/7) = (-11/7) + 11/7 = 0

So we say 11/7 is the negative or additive inverse of -11/7 and vice-versa.

In general, for a rational number a/b, "we have", a/b + (-a/b) = (-a/b) + a/b = 0

We say that (- a)/b "is the additive inverse of" a/b and a/b "is the additive inverse of" (- a)/b.

## Example

Write the additive inverse of the following: (-7)/19

7/19 "is the additive inverse of" (- 7)/19 "because" (- 7)/19 + 7/(19) = (- 7 + 7)/19 = 0/19 = 0.

## Example

Write the additive inverse of the following:  21/112.

The additive inverse of (21)/(112) "is" (-21)/112.

## Example

Verify that – (-x) is the same as x for "x" = 13/17.

x = 13/17

The additive inverse of x = 13/17  "is" - x = (-13)/17 "since" 13/17 + ((-13)/17) = 0.
The same equality 13/17 + ((-13)/17) = 0,
shows that the additive inverse of (-13)/17  "is"  13/17 "or" -((-13)/17) = 13/17 i.e., -(- x) = x.

## Example

Verify that – (-x) is the same as x for x = (-21)/31.

Additive inverse of "x" = (-21)/31 "is -x" = 21/31 "since" (-21)/31 + 21/31 = 0.

The same equality (-21)/31 + 21/31 = 0,

shows that the addiditive inverse of 21/31 "is" (-21)/31, i.e., -(-x) = x.

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