eli5: Why haven’t mathematicians invented a symbol for x/0 like they have pi and imaginary numbers?
In: 277
Because it serves no useful purpose.
Please don’t think, like everyone does, in “imaginary” numbers. They are *complex* numbers which contain a component in another dimension, we call it imaginary because you can’t really see it, but it’s DEFINITELY there and the maths works all the way through it and back to the “real” axis all the time for almost everything – AC electrics, astrophysics, and even your MP3, JPG and MPG files (using Fourier transforms). There’s nothing imaginary about them. They are useful, they exist in nature, they crop up all the time, and they are rigorous and predictable.
Same as pi – it exists in nature, crops up all the time, all over the place (not just circles), and its application is rigorous and predictable.
However, division by zero is meaningless.
Think of it like this:
If you were dividing 30 by 5, say, then that’s the same as asking “how many 5’s would I need to add together to get 30?”. 5. 10. 15. 20. 25. 30.
If you were dividing 30 by 3, it’s “how many 3’s would I need to add together to get 30?”. 3, 6, 9, etc.
If you are dividing by 0, it’s “how many 0’s would I need to add together to get 30?”
0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. 0. …..
There is no answer. There is NO number of zeroes that you can add together to get 30. Ever. No matter how long you go for or how hard you try. The question literally has no answer.
Alternatively, if you choose, say, 0 divided by 0:
“How many 0’s would I need to add together to get 0?”
0. One zero suffices. But hold on.
0 + 0. So does that. Two zeroes also answers the question.
0 + 0 + 0. So does that.
In fact, every possible number of zeroes will add up to 0.
So dividing by zero only ever gives you “no answer whatsoever” or, in a single special case, “every possible number in existence”.
As such it’s useless, and technically it’s undefined – there is no universally correct answer, way of determining it, etc. Having a symbol, constant or formula for it wouldn’t help at all. And it has almost no real world relevance because – in precisely the same way that there is no possible answer – it doesn’t crop up in nature. Nature avoids ever doing it too. No useful use of division by zero has been found.
Which is totally different to complex numbers (formed by two dimensional components, called “real” and “imaginary” but both equally valid), and pi.
It’s like asking why there isn’t a symbol for “and then the purple crocodile eats this number”. What areas of crocodilatry have you invented that requires such a symbol? Is it a shorthand for a longer Alligator formula that gets tedious to write out all the time? Does it allow you to analogise the crocodile maths to areas of spotted-chicken maths utilising the “and the hen sits on this number” operator, and thus make a breakthrough connection between them? Does it open up new areas of Crocodithmetic which we need to start teaching kids?
For all intents and purposes, division by zero is undefined and the answer doesn’t actually exist.
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