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:
I’m curious, couldn’t we define z as only 1/0? Then 2/0 would have to be factored to 2*(1/0) first and it would solve this specific example of things breaking. I haven’t done advanced math in a while but your comment picked my curiosity haha
I remember 1/0 is pretty important in limits and stuff, it just seemed to me that this specific example seems not too hard to resolve
I’m fuzzy on the deeper details. I think you can do something like this, but you have to be very careful, in ways where you don’t have to be so careful with ✓-1.
One of the more obvious ways to consider: plot a graph of y = 1 / x. Note how as x approaches zero from the right, the graph shoots up, asymptotically approaching the y-axis and shooting up to infinity. It’s very tempting to say that 1 / 0 is “infinity”. “Infinity” is not a real number, but nothing is stopping you from defining a new kind of number to represent this singularity if you want to. But at that point you have left the real numbers. Which is fine, right? Complex numbers aren’t real numbers either, after all…
But look at the left side of the graph. You have the same behavior, but the graph shoots down, not up. It suggests that the limit of approaching from the left is “negative infinity”. Quite literally the furthest possible imaginable thing from the “infinity” we had to define for the right side. But this is supposed to be the same value, at x = 0. Just by approaching it from different directions, we don’t just get two different answers, we get perhaps the most different answers possible.
I think it’s not hard to intuit a handwavey answer that this simply represents the curve of y = 1 / x “wrapping around through infinity” or some notion like that. Sure, perhaps that is what’s going on. But dancing around a singularity like that mathematically isn’t simple. The very nature of mathematical singularities is to give you nonsensical results. Generally, having them at all tends to be a sign that you have the wrong model for something.
You can mostly avoid this problem by snipping off the entire left half of the x-axis. Shrink your input domain to only non-negative numbers. Then, I believe, you can just slap “infinity” on it and run with it and be mostly fine. But that’s a condition you have to be upfront about. This becomes a special case solution, not a generalized one.
I haven’t looked into it, but I believe this singularity gets even more unweildy if you try to extend it to complex numbers. All the while, complex numbers “just work”. You don’t need doctor’s gloves to handle them. √-1 isn’t a mathematical singularity, it’s a thing with an answer, the answer just isn’t a real number.
Seriously u/pixelscript that’s a really good explanation and I hope it’s right. My understanding of infinities is pretty limited but I’d like to subscribe to your newsletter.
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.
I’m curious, couldn’t we define z as only 1/0? Then 2/0 would have to be factored to 2*(1/0) first and it would solve this specific example of things breaking. I haven’t done advanced math in a while but your comment picked my curiosity haha
I remember 1/0 is pretty important in limits and stuff, it just seemed to me that this specific example seems not too hard to resolve
I’m fuzzy on the deeper details. I think you can do something like this, but you have to be very careful, in ways where you don’t have to be so careful with ✓-1.
One of the more obvious ways to consider: plot a graph of y = 1 / x. Note how as x approaches zero from the right, the graph shoots up, asymptotically approaching the y-axis and shooting up to infinity. It’s very tempting to say that 1 / 0 is “infinity”. “Infinity” is not a real number, but nothing is stopping you from defining a new kind of number to represent this singularity if you want to. But at that point you have left the real numbers. Which is fine, right? Complex numbers aren’t real numbers either, after all…
But look at the left side of the graph. You have the same behavior, but the graph shoots down, not up. It suggests that the limit of approaching from the left is “negative infinity”. Quite literally the furthest possible imaginable thing from the “infinity” we had to define for the right side. But this is supposed to be the same value, at x = 0. Just by approaching it from different directions, we don’t just get two different answers, we get perhaps the most different answers possible.
I think it’s not hard to intuit a handwavey answer that this simply represents the curve of y = 1 / x “wrapping around through infinity” or some notion like that. Sure, perhaps that is what’s going on. But dancing around a singularity like that mathematically isn’t simple. The very nature of mathematical singularities is to give you nonsensical results. Generally, having them at all tends to be a sign that you have the wrong model for something.
You can mostly avoid this problem by snipping off the entire left half of the x-axis. Shrink your input domain to only non-negative numbers. Then, I believe, you can just slap “infinity” on it and run with it and be mostly fine. But that’s a condition you have to be upfront about. This becomes a special case solution, not a generalized one.
I haven’t looked into it, but I believe this singularity gets even more unweildy if you try to extend it to complex numbers. All the while, complex numbers “just work”. You don’t need doctor’s gloves to handle them. √-1 isn’t a mathematical singularity, it’s a thing with an answer, the answer just isn’t a real number.
Awesome, thank you for taking the time to explore it more, that makes sense to me
Seriously u/pixelscript that’s a really good explanation and I hope it’s right. My understanding of infinities is pretty limited but I’d like to subscribe to your newsletter.
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 the first time I’ve seen this spelled out and it makes perfect sense.