Why can’t you invent an imaginary number for division by zero like you can for a square root of a negative?
In: Mathematics
You can, and people have.
Creating an imaginary number that breaks the rules of squaring a negative produces a useful concept that can be used to explore things like imaginary roots of polynomials, which helps with things like typing solutions to differential equations. So far, there has been absolutely no useful application of a concept that allows you to divide by zero, and it also comes with the added complication of creating many more problematic hiccups in your math, such as allowing 1=2.
I challenge you to define a new symbol, &, where &/0=1, and produce a useful result with it.
As a math major, the closest I’ve come to dealing with division by zero is using limits. This means you divide by numbers approaching zero and see how the result responds. You must do this both from the negative (very small negative numbers) and positive very small positive numbers. There are basically 4 cases:
1. Both sides approach positive infinity. Therefore the limit of x/0 = infinity
2. Both sides approach negative infinity => negative infinity
3. Both sides approach 1 (since it’s essentially 0/0) => 1
4. The sides approach different values (infinity on one side, negative infinity on the other side, etc). In this case, it’s undefined
To put it simply, if we do, all starts of funky things can start happening, like proving that 1 = 2.
Since 1 isn’t equal to 2 we can’t have anything for division by zero
My favorite explanation for this is the t shirt dilemma. Say you have $100 and t shirts cost $5. Then you can buy 100/5=20 t shirts. But what if t shirts are free? You could buy 1, or 10, or 50, or literally any number of shirts. So we say that 100/0 is “undefined” because there are multiple possible correct answers. It’s not that we can’t invent something to represent the value of 100/0, it’s that there is no single correct such value.
As others have said, you can also see this algebraically using an equation. But that’s not appropriate for most five year olds.
To put it in non-math terms. Take a pizza and slice it up into 8 pieces. You divided it by 8 and every piece is 1/8th of the whole. But if you divided it into zero pieces, there are no pieces at all, the pizza can’t exist if there are zero pieces. The result can’t be represented by anything because it no longer exists.
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.
You can. In fact there are multiple, conflicting ways to do it. It is sometimes convenient to extend the real line by compactification in such way that infinity = -infinity, like an infinite ‘circle’. Sometimes it is convenient to keep these separate. So we don’t take any of them as *default*, and in the reals themselves we leave 1/0 as undefined.
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