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Is there an intuitive explanation or a mathematical principle that explains the following equality:

1-(1/x)=(x-1)/x

Basically, how come the difference between 1 and the inverse of x is equal to the difference between x and 1 multiplied by the inverse of x?

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  • $\begingroup$ It's the same as $\,\dfrac{1}{x}+\dfrac{x-1}{x}=1\,$, or $\,(x-1) + 1 = x\,$, if you find those more intuitive. $\endgroup$
    – dxiv
    Sep 10 '18 at 22:15
  • $\begingroup$ $1$ is equal to $\frac{x}{x}$ $\endgroup$
    – edcharlie
    Sep 10 '18 at 22:16
  • $\begingroup$ Thank you for this $\endgroup$
    – od320
    Sep 11 '18 at 5:41
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I'll give you two explanations:

  1. Math

$$1-\frac1x = \frac xx-\frac1x=\frac{x-1}x$$

  1. Intuitive

When $x$ gets larger, $\frac1x$ gets smaller, so $1-\frac1x$ gets closer to $1$. On the other hand, when $x$ gets larger, $$\frac{x-1}x\sim\frac xx = 1$$

Another way to see it is to multiply both sides by $x$.

$$x-1 = x-1$$

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  • $\begingroup$ Thanks for the very detailed answer. I get it now. $\endgroup$
    – od320
    Sep 11 '18 at 5:42
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1 = $\frac{x}{x}$, and then you add the fractions on the left hand side.

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  • $\begingroup$ Thanks very much. I get it now. $\endgroup$
    – od320
    Sep 11 '18 at 5:42

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