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I think I might be overthinking this question and I need some help

  1. Assume only the knowledge of addition and multiplication of real numbers. What do we mean by $u^{-1}=v$ ? Can this equality be verified?
  2. Assume only the knowledge of addition and multiplication of real numbers. What do we mean by $u/v=w$ ? Can this equality be verified?
Vincenzo Tibullo
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user245858
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  • Maybe you are looking for something like: $u^{-1}=v$ means that $uv=1$ and $u/v=w$ means $vw=u$. I am not sure if that is what you are asking. –  Jun 11 '15 at 17:02

2 Answers2

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In the context of real numbers, "$u^{-1}$" denotes the multiplicative inverse of $u$. That is, it is the number such that, when multiplied to $u$, you obtain $1$. For example, $5^{-1}$ is the number $\frac{1}{5}$ (or $0.2$ if you like).

Hence, if it is known that $u^{-1}=v$, then $v \cdot u = 1$.

Now, "$\frac{u}{v}$" can be written as "$u \cdot \frac{1}{v}$" which in turn can be written as "$u \cdot v^{-1}$". If you have that $\frac{u}{v} = w$, then you know that $u \cdot v^{-1} = w$. Now, if you multiply both sides by $v$, you can obtain $u \cdot v^{-1} \cdot v = w \cdot v$, and so $u \cdot 1 = w \cdot v$, and hence $u = w \cdot v$.

Ken
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$u^{-1}=v \implies uv=1$

$u/v=w \implies u=wv$

RowanS
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