This is a question I see from time to time, and it’s a good question to ask.
Your question as I understand it can be phrased another way as:
The square root of -1 has no defined answer. So we put a mask on it and pretend that’s the answer. We do math with the masked number and suddenly everything is fine now. Why can’t we do the same thing to division by zero?
The difference is that, if you try to put a funny mask on the square root of -1 and treat it like a number, nothing breaks, but if you try the same thing with a division by zero, all sorts of things break.
If you define i = √-1, that is the only thing i can ever be. That specific quantity. You can factor it out of stuff, raise it to that exponent, whatever. And if it is ever convenient to do so, you can always unmask it back into that thing, e.g. i^2 = (√-1)^2 = -1. All the while, all the already existing rules of math stay true.
A division by zero isn’t like this, because if you tried it, every number divided by zero would equal to the same thing. If we give it a name, say, 1 / 0 = z, then it would also be true that 2 / 0 = z. We could then solve both sides for zero:
Oh man, I knew I had asked this question in the right place.
Thank you!
With imaginary numbers, I visualize something like a needle popping up and moving through cartesian space in new directions or dimensions in all sorts of unexpected and intricate ways, and eventually they find utterly extraordinary and elegant things like the Mandelbrot set. So I wondered if there are other “hacks” or “cheats” that open up new types of progressions and behaviors for study.
Someone else in the thread also mentioned Dirac doing something along the lines of (a)(0) ≠ 0 to handle some of the infinities that pop up in physics.
This is a question I see from time to time, and it’s a good question to ask.
Your question as I understand it can be phrased another way as:
The difference is that, if you try to put a funny mask on the square root of -1 and treat it like a number, nothing breaks, but if you try the same thing with a division by zero, all sorts of things break.
If you define i = √-1, that is the only thing i can ever be. That specific quantity. You can factor it out of stuff, raise it to that exponent, whatever. And if it is ever convenient to do so, you can always unmask it back into that thing, e.g. i^2 = (√-1)^2 = -1. All the while, all the already existing rules of math stay true.
A division by zero isn’t like this, because if you tried it, every number divided by zero would equal to the same thing. If we give it a name, say, 1 / 0 = z, then it would also be true that 2 / 0 = z. We could then solve both sides for zero:
1 / z = 0
2 / z = 0
then set them equal:
1 / z = 2 / z
then multiply both sides by z:
1 = 2
which is a contradiction.
i doesn’t have this problem.
Oh man, I knew I had asked this question in the right place.
Thank you!
With imaginary numbers, I visualize something like a needle popping up and moving through cartesian space in new directions or dimensions in all sorts of unexpected and intricate ways, and eventually they find utterly extraordinary and elegant things like the Mandelbrot set. So I wondered if there are other “hacks” or “cheats” that open up new types of progressions and behaviors for study.
Someone else in the thread also mentioned Dirac doing something along the lines of (a)(0) ≠ 0 to handle some of the infinities that pop up in physics.