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Enter any number into a calculator and divide it by zero. What do you get?


That’s a difficult word to understand. After all, how do you define undefined? What in the name of Turing’s testicles does your calculator mean by undefined?

Undefined is not a real number. It is not even an irrational number. It is a mind-blowing abstraction that’s not infinity, negative infinity, nor anywhere in between.

And yet, in the spirit of scientific enquiry, I’m going to try to illustrate it for you.


Calculator: Undefined


Take a nice friendly number like ten.

What happens if you divide ten by ten? You get one.

Let’s mark this on our map.


10 Divided by 10


Stick with ten and divide it by a smaller number, like five.

How many fives in ten? Two.

Map it, Delilah.


10 Divided by 5


Now rinse and repeat with smaller and smaller numbers:

10 / 5 = 2
10 / 2 = 5
10 / 1 = 10
10 / 0.5 = 20
10 / 0.1 = 100
10 / 0.001 = 10,000
10 / 0.000000001 = 10,000,000,000


Divide by Zero


As you can see: the smaller the denominator, the bigger the result. In other words, as the denominator approaches zero, the result approaches infinity. So if we were to actually divide ten by zero, the result should be infinity, right?

Not so fast! Look at what mind-fizzing symmetry occurs if we approach zero from the other direction:

10 / -10 = -1
10 / -5 = -2
10 / -2 = -5
10 / -1 = -10
10 / -0.5 = -20
10 / -0.1 = -100
10 / -0.001 = -10,000
10 / -0.000000001 = -10,000,000,000


Divide by Negative Numbers


See that? As the denominator approaches zero from the negative side, the result approaches negative infinity.


Infinity or Negative Infinity?


This puts us in quite a pickle. The diminishing positive denominators suggest dividing by zero would give a result of infinity – and yet the diminishing negative denominators suggest the exact opposite: a result of negative infinity.

And that’s why your calculator – a veritable pocket deity in all things mathematical – struggles to answer the seemingly simply sum of ten divided by zero. Instead it gives you the disturbingly beautiful term: undefined.

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