Why is it ok to divide by 0.0?

It is a relatively small number. It has nine digits and one can easily write it down on a small sheet of paper.

If we're ok with some small error, let's say 0.1%, then we can say that the speed of light is approximately this:

It's the same number, the same power of ten, it just takes fewer symbols to write it down.

But 3e8 with the 0.1% error also represents a lot of other numbers. In fact, it represents every number in the range of [3e8 - 3e5, 3e8 + 3e5]. That's 600 001 integer numbers. And an infinite number of real ones.

Digital computers mostly keep numbers in a similar way. They store a small amount of binary digits for the number and even a smaller amount of binary digits to indicate the e-something analog which is the multiplying power of two. In this way, they can operate both very large and very small numbers. But of course, the number of digits is finite so there are numbers so large they can not be written at all. We just don't have enough digits to write them down.

And the same goes for the small numbers. There are numbers so small that we can't write them down in engineering notation.

Not the integer zero but the floating-point zero. That's two different animals. Actually, these are three different animals since there is also -0.0 that represents all the tiny negative numbers.

Anyway, just like 3e8 represents a range of numbers, a 0 represents not a single number but every number that is so small you don't have enough digits to write down how small it is.

So the number representing zero might be an actual zero but it might very well not be one. So when we ask a computer to divide a number by zero we're actually telling it to divide a number by some arbitrary real small number. And it does. And the result is the arbitrary very large number.

In floating-point numbers, there is a special number to represent all the arbitrary large numbers. It's called “infinity” but just as the 0.0 doesn't only represent zero but all the ultra-small numbers, floating-point infinity doesn't represent infinity but rather all the numbers that are so large you don't have enough digits to write down how large they are.

But why do we have to keep our numbers small? Why is it important to fit things into some relatively small amount of digits? Cant, we just have all the digits we want?

In computing, especially in high-performant computing when you measure latencies in nanoseconds, the light suddenly doesn't seem to go travel all that fast.

How far do you think light travels in one nanosecond given the optimal conditions?

In one nanosecond, the light travels only about 30 centimeters. This means that if you want something computed in under 1 nanosecond, the signal starting your computation has to go through all the transistors that do the computation by the path that is only 30 centimeters in length.

And that's it. That's the essential limitation of electronic computers. Of course, for most of the computing history, we've been successfully making our processors faster by making the transistors smaller but this also has its limit. We can't make subatomic transistors. At least not in this universe.

So our computational capabilities are essentially limited by the speed of light.

And that's why we have to be resourceful with our numbers. Not with the real numbers but with the digital representations we crunch with computers. And having division by zero allowed, as a subsequence of that resourcefulness, is a small price to pay for all the fast computations.