# 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).

## Author: mathtuition88

https://mathtuition88.com/

## 8 thoughts on “What is 1 divided by 0?”

1. You’re welcome! I like this demonstration of the value of “1 divided by zero.”

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1. Thanks for your comment! Yeah, this is rather puzzling and mind-boggling, which is part of why it is such an intriguing question to debate on!

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1. suevanhattum says:

You could use a similar process to show that the limit as x->0 of 1/x^2 is infinity.

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2. suevanhattum says:

I am a bit concerned about your use of the language of mathematical proof. It doesn’t make sense to “show that … is undefined.” It sounds like you are trying to use your steps to prove that it cannot be well-defined.

But the idea referred to in your Wikipedia quote is so much simpler: We cannot find an answer for x/0 because division and multiplication have a simple relationship (6/2=3 because 2*3=6, similarly top / bottom = answer is describing the same relationship as bottom * answer = top), and there is no number a such that 0*a = 1.

Infinity is not a number, though. So, it remains a valid question: What is 0 * infinity? Yes, there are situations, like the one you describe, where there can be no sensible answer. And there are other situations where, like the one I describe, where there can be a sensible answer.

Division by zero, though not allowed in simple arithmetic, does lead to some deep ideas, and is worth pondering.

(I must also say that it does not do justice to the math to refer to Wikipedia as an authority. Yes, it often has great explanations. But it can be wrong. The authority is in the math itself and in the logic.)

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