Why can’t you invent an imaginary number for division by zero like you can for a square root of a negative?
In: Mathematics
First off, I’d like to say that we didn’t invent imaginary numbers, they were with us the whole time. “i” was just called imaginary by an extremely prominent mathematician of the time named Rene Descartes as a derogatory way of telling another person that they were asking stupid questions and inventing stupid answers and the term stuck. (Do not judge Descartes to harshly here, he made a mistake on this, but has done a lot of good like cartesian coordinates which are named for him.)
Now to get into things a bit deeper on what is actually going on with division by zero.
There are three types of numbers, prime composite and identities. 0 is the additive identity for the real numbers. This is why it behaves quite oddly in the first place. It isn’t truly meant to be part of multiplication, and as such does not have a multiplicative inverse to reach 1, the multiplicative identity. Basically
0 × r =/= 1 for any real number r.
This also has the effect of any number times 0 equalling 0. And since division is multiplication by the inverse we get
0 × r = 0 for any real number r
=> 0 ÷ (1/r) = 0/r = 0 for any real number r.
So if I divide some number by 0 and then run it through the basic process to turn that into a multiplication problem rather than a straight division problem I get
1 ÷ 0
= 1 ÷(0/r)
= 1× (r/0)
= r/0
But r can be literally any number at any time. And the problem also just recycled back on itself by dividing r by zero, which will just happen over and over again because zero does not have a multiplicative inverse to stop the process.
If you’re still interested in learning more about what’s going on here, this is an idea in group theory. Work at it and keep asking questions like you are, and you too many end up in a theoretical math program asking some really neat questions and getting some truly fascinating answers.
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