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
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.
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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.
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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.
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