4 ms·

The analogy to complex numbers is really useful. Of course sqrt(-1) doesn't exist. But if we just pretend that is does exist, we can build a rigorous theory out

The analogy to complex numbers is really useful. Of course sqrt(-1) doesn't exist. But if we just pretend that is does exist, we can build a rigorous theory out of these imaginary numbers. Once we do that we notice that these numbers are really useful for calculating real physical things. So maybe imaginary numbers aren't so imaginary. Same with these infinite sums. By the math you learn in middle school, you can't have infinite sums. But break the rules for just a second and again we have something that is helpful with real physics.

chowells 10y ago
I don't know what you mean by sqrt(-1) not existing. What does it even mean to exist? Far better to talk about whether a number is defined in a particular numerical system. In the real numbers, sqrt(-1) isn't defined. But why privilege the real numbers as "existing"? Despite an official-sounding designation, they're very deeply weird. The real numbers are famously uncountable. But any subset of them that can be enumerated is by definition countable. Think about the consequences of that for a moment. No matter what you do, the subset of the reals you can enumerate is countable, meaning the subset you can't enumerate is uncountable. In a rather flippant way, you could describe the real numbers as "mostly useless." Most of them exist to make some theorems work, rather than being a number that you could ever use to describe anything - solely because describing the number would require an infinite amount of information. In a pretty significant sense, it's valid to say that the real numbers are mostly figments of analysts' imagination. If they "exist", might as well say complex numbers exist too. They're actually more useful in physics than real numbers are.
- quantumhobbit 10y ago
  I'll admit that the concept of numbers "existing" here is quite poorly defined. I was trying to capture the effect of people upon encountering complex numbers for the first time to just sort of shut down and refuse accept that they are "real" ( pun intended). It is hard to remember how weird complex numbers feel after years of middle school math teachers saying that you can't take the square root of a negative. Similarly weird feeling is encountering zeta(-1) = -1/12 after years of calculus teachers telling you to ignore divergent sums because they are infinite.
  - sebastos 10y ago
    I think the point is that if that's true, the concept of complex numbers is being taught incorrectly. If you tell a student that this number is fake but useful, what are they to make of that? It just starts to make mathematics seem spooky and unpredictable. When you're first learning about imaginary numbers in 8th or 9th grade, the answer to "what is sqrt(-1)?" _should_ be undefined. If you claim otherwise, you're pulling the rug out from under their feet, because the number system that they are familiar with indeed has sqrt(-1) undefined. Instead, the teacher should go on to introduce a new system of mathematical objects that have certain rules, and the students could play around with them and see how they have two components, how you can plot those two components in 2 dimensions, how you can think of them as arrows sticking out of the origin, how you can combine their components to rotate each other, etc. Then work backwards into showing that we can call these objects complex numbers for short, because those operations are similar to addition, multiplication, etc. And finally, just as a curiosity, you can see that sqrt(z) = i for z = -1 + 0i. There's no need to introduce this whole concept of an imaginary number line that points off in a direction nobody can see or measure. The whole takeaway should be that you can't just square real numbers get negatives. If you have something that can "multiply" by itself to get its own inverse, then you have either overloaded the multiplication operator with something very very different, or you're dealing with an object that can "rotate" through another dimension. It's an ordinary two dimensional space, and the only difference between the two axes is their name, just like "x" and "y". In my opinion, this lesson should actually be reassuring to a young mathematical intuition: there's only so many ways to skin this cat.
    - nkurz 10y ago
      You're likely aware of it, but in case others aren't, there's a classic online presentation that makes the same case: https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ https://betterexplained.com/articles/a-visual-intuitive-guid... And here's some "ancient" HN commentary on that article: https://news.ycombinator.com/item?id=2712575 https://news.ycombinator.com/item?id=2712575
witty_username 10y ago
Infinite sums do not require any breaking of rules. Infinite sums are formally defined as the limit of partial sums.

chowells 10y ago

I don't know what you mean by sqrt(-1) not existing. What does it even mean to exist? Far better to talk about whether a number is defined in a particular numerical system. In the real numbers, sqrt(-1) isn't defined. But why privilege the real numbers as "existing"? Despite an official-sounding designation, they're very deeply weird. The real numbers are famously uncountable. But any subset of them that can be enumerated is by definition countable. Think about the consequences of that for a moment. No matter what you do, the subset of the reals you can enumerate is countable, meaning the subset you can't enumerate is uncountable. In a rather flippant way, you could describe the real numbers as "mostly useless." Most of them exist to make some theorems work, rather than being a number that you could ever use to describe anything - solely because describing the number would require an infinite amount of information. In a pretty significant sense, it's valid to say that the real numbers are mostly figments of analysts' imagination. If they "exist", might as well say complex numbers exist too. They're actually more useful in physics than real numbers are.

quantumhobbit 10y ago
I'll admit that the concept of numbers "existing" here is quite poorly defined. I was trying to capture the effect of people upon encountering complex numbers for the first time to just sort of shut down and refuse accept that they are "real" ( pun intended). It is hard to remember how weird complex numbers feel after years of middle school math teachers saying that you can't take the square root of a negative. Similarly weird feeling is encountering zeta(-1) = -1/12 after years of calculus teachers telling you to ignore divergent sums because they are infinite.
- sebastos 10y ago
  I think the point is that if that's true, the concept of complex numbers is being taught incorrectly. If you tell a student that this number is fake but useful, what are they to make of that? It just starts to make mathematics seem spooky and unpredictable. When you're first learning about imaginary numbers in 8th or 9th grade, the answer to "what is sqrt(-1)?" _should_ be undefined. If you claim otherwise, you're pulling the rug out from under their feet, because the number system that they are familiar with indeed has sqrt(-1) undefined. Instead, the teacher should go on to introduce a new system of mathematical objects that have certain rules, and the students could play around with them and see how they have two components, how you can plot those two components in 2 dimensions, how you can think of them as arrows sticking out of the origin, how you can combine their components to rotate each other, etc. Then work backwards into showing that we can call these objects complex numbers for short, because those operations are similar to addition, multiplication, etc. And finally, just as a curiosity, you can see that sqrt(z) = i for z = -1 + 0i. There's no need to introduce this whole concept of an imaginary number line that points off in a direction nobody can see or measure. The whole takeaway should be that you can't just square real numbers get negatives. If you have something that can "multiply" by itself to get its own inverse, then you have either overloaded the multiplication operator with something very very different, or you're dealing with an object that can "rotate" through another dimension. It's an ordinary two dimensional space, and the only difference between the two axes is their name, just like "x" and "y". In my opinion, this lesson should actually be reassuring to a young mathematical intuition: there's only so many ways to skin this cat.
  - nkurz 10y ago
    You're likely aware of it, but in case others aren't, there's a classic online presentation that makes the same case: https://betterexplained.com/articles/a-visual-intuitive-guide-to-imaginary-numbers/ https://betterexplained.com/articles/a-visual-intuitive-guid... And here's some "ancient" HN commentary on that article: https://news.ycombinator.com/item?id=2712575 https://news.ycombinator.com/item?id=2712575

witty_username 10y ago

Infinite sums do not require any breaking of rules. Infinite sums are formally defined as the limit of partial sums.