Dear Uncle Colin,

What is $\frac{1}{\infty}$?

- Calculating A Number, Though Outside Reals

Hi, CANTOR, and thanks for your message!

The short answer: it’s undefined.

The longer answer: Infinity is not a number. It’s not something you’re allowed to divide by. The calculation doesn’t make sense, and writing it down is a bad thing to do.

Some wrong answers

(This question came up on facebook recently, and there were some wrong answers in the thread I wanted to discuss, too.)

The answer is not 0.

If the answer was 0, then $0 \times \infty$ would equal 1. It most definitely doesn’t. It’s undefined.

The answer is not “A zero, a decimal point, an infinite number of zeros and then a 1.”

“Infinite”. You keep using that word. I do not think it means what you think it means. Also, TIL, infinity is a power of 10.

Less sardonically, you can’t have “then” after an infinite number of anything. The whole point of an infinite number of anything is that it goes on forever without anything afterwards.

The answer is not “something approaching zero”.

There is no limit being taken here, nothing is changing, so there’s nothing to approach. Had the question been “What is the limit of $\frac{1}{x}$ as $x \to \infty$?”, this would be correct - however, that’s not the question.

The answer is not “an asymptotic decay to zero”

There is no function being graphed here, there is no curve, therefore there is no asymptote. Nothing is decaying.

The moral of the story

If you’re doing maths, it’s a good general rule to sound an alarm any time you see an $\infty$ symbol. The exceptions I can think of:

  • In or as a limiting value (such as $\lim_{x \to +\infty} e^x = +\infty$)
  • As a limit for integration or summation (e.g. $\int_0^\infty e^{-x}\dx = 1$)
  • Physics. They put up with any old nonsense and pretend it’s maths.
  • Set theory - and even then, I’d expect to see symbols other than $\infty$ when talking about cardinals.
  • The Riemann Sphere, or extended complex plane, which uses a point at infinity, $\tilde{\infty}$; points at infinity in general ((Thanks to Adam Atkinson.))
  • Combinatorial game theory and surreal numbers *do* allow sums involving infinity ((usually denoted $\omega$ rather than $\infty$)) - but only when treated very carefully ((Thanks again to Adam)).

In short, CANTOR, one needs to be very careful when working with infinity. As well you know.

- Uncle Colin

* Edited 2017-09-07. Thanks to Adam Atkinson for suggesting some extra uses for infinity. * Edited 2017-09-09 after further discussion about points at infinity with Adam.