Hiw to prove that additive inverse is not equal to multiplicative inverse?

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Consider the rational number field. Then prove that additive inverse can not be equal multiplicative inverse.

It is easy to prove that the identity elements are different by the following reasoning.

Say if $0=1$

$a=a*1=a*0=0$ hence $a=0=1$ So the set contains exactly one element.

But how can I prove the same for inverse elements ?

In case of duplicate,please give the link of similar posts.

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1 Answer

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For $a\ne 0$ the multiplicative inverse is $1/a$

$$a+1/a=0$$ does not have a real solution.

Thus the multiplicative inverse of a real number is different from its additive inverse.

For complex numbers $i$ and $-i$ two inverses are the same.

$$i+(-i)=0$$ and $$i(-i)=1$$

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