## What is 1 divided by 0?

What is 1 divided by 0? Is it infinity?

Contrary to popular opinion, 1 divided by 0 is not infinity! Wikipedia states that “the expression has no meaning, as there is no number which, multiplied by 0, gives a (assuming a≠0), and so division by zero is undefined“.

## How to show that division by zero is undefined

$\displaystyle \lim_{x\to 0^+} \frac{1}{x}=+\infty$

The limit of 1/x as x approaches zero from the right is positive infinity.

However, $\displaystyle \lim_{x\to 0^-} \frac{1}{x}=-\infty$

The limit of 1/x as x approaches zero from the left is negative infinity.

Since the left limit and right limit are different, the limit of 1/x as x approaches infinity does not exist!

Note: There are mathematical structures in which a/0 is defined for some a (see Riemann sphere, real projective line, and section 4 for examples); however, such structures cannot satisfy every ordinary rule of arithmetic (the field axioms).