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?

Question

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?

Answer 1

I'll give you two explanations:

1. Math

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

2. 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 $x$.

$$x-1 = x-1$$

Answer 2

1 = $\frac{x}{x}$, and then you add the fractions on the left hand side.