As a layperson who is interested in math, imaginary numbers always fascinated me. Like in the real world you taking the square of a negative makes no sense whatso ever, but in theoretical math you can just invent new imaginary numbers, make it so that *i*^2 = -1 and suddenly you have just revolutionized math. If this is useful, why can’t you break other rules and account for them with new imaginary symbols?

So let’s pretend that we call them made up numbers and use *m* to represent them. Why is *m*=1/0 impossible when something like *i*^2 = -1 is not?

In: 27

As a computer engineer that became an educator, one of the things you need to realize is that math is just a language, but it’s the most precise way of saying things. It’s a way of saying most precisely exactly one thing. No matter how many words you use, it doesn’t make sense to take nothing, and divide it into smaller pieces of nothing and no amount of nothing becomes something. That’s what nothing is.

The reason is that the basic operations that we define on real numbers and complex numbers (and fields in general – there are more than just these two) can’t fulfill the basic properties we want them to have if m=1/0 was defined.

It’s not too hard to see this:

What’s 0? 0 is the element that you can add to any number without changing it (the “neutral element”): For all numbers a, a+0=a

From this and our idea of how addition and multiplication should be compatible, it follows that any number multiplied by 0 is 0 again: For all a, a*0 = 0 (you can see this by multiplying the above equation with any number b: ab + a*0 = a*b, so a*0=0).

But now, m=1/0 can’t be defined, because m*0 should be 0, but if m=1/0, m*0= 1/0*0, it should also be 1, and that’s not possible at the same time.

The one easy way to illustrate why dividing by zero is problematic is to literally see what happens to 1/x as it approaches 0.

If you approch from the right (positive numbers), it races off to positive infinity. If approached from the left (negatives) it races off to negative infinity. In other words, it’s two literally opposite things at once. A contradiction. It just doesn’t behave nicely.

If I recall properly, there are attempts to define 1/0 as a new number similarly to i, but it keeps causing problems. I honestly don’t know the details, though.

In comparison, i has well defined properties and complex numbers not only don’t introduce contradictions, but solve a few other problems along the way.

We could define m to equal 1 / 0, but the problem is, this new number forces us to add weird exceptions to some of the rules of arithmetic.

For example, if m = 1 / 0, then presumably, 0 x m = 1. But this breaks the rule that 0 x [anything] = 0. We also have to sacrifice one of the rules used in the following argument: 2 = 1 + 1 = 0 x m + 0 x m = (0 + 0) x m = 0 x m = 1.

Either 0+0 isn’t 0 any more, or a x c + b x c isn’t always (a + b) x c, or 2 isn’t 1 + 1…

It would be fine to sacrifice those rules if the benefits of having this new number were worthwhile, but the ability to divide by 0 doesn’t actually gain us all that much.

When Complex numbers were introduced, they faced a lot of skepticism – hence their derogatory names: “complex numbers” and “imaginary numbers” as opposed to “real numbers”

Gradually, though, people found more and more uses for complex numbers, until they’re an indispensible tool for mathematics. We do lose some of the normal rules of arithmetic, for example, we no longer have sqrt(a x b) = sqrt(a) x sqrt(b) always: otherwise 1 = sqrt(1) = sqrt(-1 x -1) = sqrt(-1) x sqrt(-1) = i x i = -1. But we gain others: for example, an n-degree polynomial always has n roots (some maybe repeated), instead of some number between 0 and n. Or this: if a function is differentiable once, it’s differentiable as many times as we like.

Complex numbers are useful and interesting enough that it’s worth getting used to a new set of rules. Not so much with allowing division by 0.

Here’s the thing – you have to make enough math to make it make sense.

It turns out that inventing the idea *i* so that *i*^(2)=-1 wasn’t obvious, and was the result of a lot of people doing things trying to avoid making up new math before finally saying “We can keep working around it – or we can just work out the rules of this thing”; and someone did it.

Dividing by zero, on the other hand, doesn’t work that way. Instead of getting consistent answers, you get answers that get weird in different ways depending on how you divide by zero. For example, take the following three equations:

– y=1/(x^(2)). As you get close to x=0; y approaches infinity.

– y=1/(-x^(2)). As you get close to x=0; y approaches negative infinity.

– y=1/x. In this case, y approaches either infinity or negative infinity, depending on whether you get close to x from the positive side or the negative side.

Which means that if I try to say 1/0; there are reasons the answer could be “infinity” or “negative infinity”.

It’s worse when you try to divide zero by zero.

…

To explain why this is the case, let’s go back to what division is: division is just a multiplication problem with the product and one factor given: saying “what is 1 divided by 0” is really asking “what number, times 0, equals 1”: the two following equations are mathematically equivalent:

– 1 / 0 = ___

– 0 * ___ = 1

And the problem there is that there is no answer we can put in that works, and generates consistent answers for 2/0, 3/0, etc. The problem is reversed with 0/0:

– 0 / 0 = ___

– 0 * ___ = 0

Any number I put there works. 0/0 equals any number. There’s no way to know what the one answer is because any answer works.

This lack of consistency is why nobody has made a system.

…

And to come full circle, *i* works because it IS consistent. When we start from *i*^(2)=-1; everything else we understand about *i* can be worked out, mathematically, from that starting point. Anything you know about *i*, I can show you how to get starting from that starting point.

There’s a lot of helpful stuff here about why 1/0 causes problems in “regular” math — but you’re exactly right that we can define it anyway! That’s what happened with negative numbers (from positives), rational numbers (from integers), irrational numbers (from rationals), and complex numbers (from reals) — none of them fit with all the axioms of previous, simpler systems of numbers, but mathematicians found them all useful enough that they (eventually) figured out new axioms that could correctly describe the behavior of these new numbers. Take a look here to see how adding 1/0 to the club changes things: [https://en.wikipedia.org/wiki/Projectively_extended_real_line#Dividing_by_zero](https://en.wikipedia.org/wiki/Projectively_extended_real_line#Dividing_by_zero)

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