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

UNDEFINED.

That’s a difficult term 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.

Take a nice friendly number like 10.

What happens if you divide 10 by 10 ten? You get 1.

Let’s mark this on our map.

Stick with 10 and divide it by a smaller number, like 5.

How many 5s in 10? Just 2.

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

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 10 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

See that? As the denominator approaches zero from the negative side, the result approaches *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 anything divided by zero. Instead it gives you the disturbingly beautiful term: undefined.