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

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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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