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Question Number 93044 by fath035990 last updated on 10/May/20

is (8/9) is the multiplicative inerse of −1 (1/8)? why or why not?

$$\mathrm{is}\:\frac{\mathrm{8}}{\mathrm{9}}\:\mathrm{is}\:\mathrm{the}\:\mathrm{multiplicative}\:\mathrm{inerse}\:\mathrm{of}\:−\mathrm{1}\:\frac{\mathrm{1}}{\mathrm{8}}? \\ $$$$\mathrm{why}\:\mathrm{or}\:\mathrm{why}\:\mathrm{not}? \\ $$

Commented by prakash jain last updated on 10/May/20

−1(1/8)=−(9/8) (8/9)×(−(9/8))=−1 So (8/9) is not a multiplicative inverse of −(9/8)

$$−\mathrm{1}\frac{\mathrm{1}}{\mathrm{8}}=−\frac{\mathrm{9}}{\mathrm{8}} \\ $$$$\frac{\mathrm{8}}{\mathrm{9}}×\left(−\frac{\mathrm{9}}{\mathrm{8}}\right)=−\mathrm{1} \\ $$$$\mathrm{So}\:\frac{\mathrm{8}}{\mathrm{9}}\:\mathrm{is}\:\mathrm{not}\:\mathrm{a}\:\mathrm{multiplicative} \\ $$$$\mathrm{inverse}\:\mathrm{of}\:−\frac{\mathrm{9}}{\mathrm{8}} \\ $$

Commented by fath035990 last updated on 10/May/20

thanks

$$\mathrm{thanks} \\ $$$$ \\ $$

Answered by Rio Michael last updated on 10/May/20

a multiplicative inverse of q is that number p such that pq = 1. here q = (8/9) and p = −1(1/8) = −(9/8) ⇒ pq = ((8/9))×(−(9/8)) = −1 hence −1(1/8) cannot be a multiplicative inverse of (8/9).

$$\mathrm{a}\:\mathrm{multiplicative}\:\mathrm{inverse}\:\mathrm{of}\:{q}\:\:\mathrm{is}\:\mathrm{that}\:\mathrm{number}\:{p}\: \\ $$$$\mathrm{such}\:\mathrm{that}\:{pq}\:=\:\mathrm{1}. \\ $$$$\mathrm{here}\:{q}\:=\:\frac{\mathrm{8}}{\mathrm{9}}\:\mathrm{and}\:{p}\:=\:−\mathrm{1}\frac{\mathrm{1}}{\mathrm{8}}\:=\:−\frac{\mathrm{9}}{\mathrm{8}} \\ $$$$\Rightarrow\:{pq}\:=\:\left(\frac{\mathrm{8}}{\mathrm{9}}\right)×\left(−\frac{\mathrm{9}}{\mathrm{8}}\right)\:=\:−\mathrm{1} \\ $$$$\mathrm{hence}\:−\mathrm{1}\frac{\mathrm{1}}{\mathrm{8}}\:\mathrm{cannot}\:\mathrm{be}\:\mathrm{a}\:\mathrm{multiplicative}\:\mathrm{inverse}\:\mathrm{of}\:\frac{\mathrm{8}}{\mathrm{9}}. \\ $$

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