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How much math have you learned? When I first saw Euler's Identity I kind of laughed and said WTF? We "discovered" Euler's number while calculating compound in

How much math have you learned? When I first saw Euler's Identity I kind of laughed and said WTF? We "discovered" Euler's number while calculating compound interest continously. We knew pi from grade school. Learned about square root of -1 around 9th grade. But when I learned Euler's Identity I didn't even understand what it was trying to say. Then someone took an hour to derive it through a Taylor series and my mind was blown. I still find it amazing. Then in quantum physics class we learned about the wave function and Schrödinger equation. Those "imaginary" numbers influence the real world. I'm not religious but when someone sees some kind of divine beauty in math I kind of feel it too.

yw3410 4y ago
If you haven't done it, then the statistical proofs for the gas equations and the proofs for Maxwell's equations also rank up there for me as well.
- ivansavz 4y ago
  Thanks for the recommendations, here are the three first links that come up for the "statistical proof" of the gas equations: https://www.one-tab.com/page/UhuMmBwCTma12IwYdMb54Q https://www.one-tab.com/page/UhuMmBwCTma12IwYdMb54Q Bookmarked to use as my just-in-time curriculum for the afternoon coffee break...
ianai 4y ago
Partially, those imaginary numbers offer ways to embed more geometric/topological reasoning into the number system than being imaginary. Like how adding the point at infinity makes the 2d plane more like a sphere.
- lizknope 4y ago
  I'm an electrical engineer so we did a lot of math in circuits class where we used Euler's identity. We did stuff like e^(i * theta) = cos(theta) + i * sin(theta) Then our teacher said "that part is imaginary" and just crossed out the isin(theta) part so e^(itheta) = cos(theta) and use that to solve circuit problems. I kind of laughed but some other people have told me that it is more of a math shortcut rather than "i" having a real world impact. As I said I have taken a quantum physics class but I'm not a physicist. But it seems like "i" really is necessary for quantum mechanics and manifests itself in the real world. https://www.quantamagazine.org/imaginary-numbers-may-be-essential-for-describing-reality-20210303/ https://www.quantamagazine.org/imaginary-numbers-may-be-esse...
  - whatshisface 4y ago
    The imaginary unit shows up in both quantum mechanics and EE as a way to embed i*i = -1 in the equations. Any differential equation that involves it can be written without it by separating real and imaginary parts into different variables. Furthermore, if you let "one" be represented by the 2x2 identity matrix, the matrix [[0,1],[-1,0]] can act like i in every respect, because that matrix squares to -1 times the identity. In fact, even -i can stand in for i. The "nouns" of math might be real, but whether they are is a metaphysical question; all we observe in science are what you might call the verbs of math. FWIW, the "crossing out the imaginary part" works like this: if the system of equations is linear and doesn't have complex coefficients, every solution has a counterpart solution that is its complex conjugate, and you can add the conjugate (remember, it's linear) to get another, real-valued, solution. You can do that with time-invariant (definite-energy) solutions to the Schrodinger equation as well, because the energy eigenvalue equation doesn't have complex coefficients.
bigbillheck 4y ago
I have a phd in mathematics and to me, ultimately, it's just a tool. There's plenty of things that make me say 'oh, that's neat' but it's all mundane, never sacred.
radomir_cernoch 4y ago
For me, it's the halting problem (and related proofs).
StrictDabbler 4y ago
Descartes called them "imaginary" numbers because he didn't understand them well and didn't like them. He thought they were ugly and therefore false. If sine, cosine, and tangent had instead been named divine, codivine and tantric we'd be having ridiculous conversations about how magical they are too. What do you think the universe would look like if things didn't fit together? Sit with that thought for a little while, really let it sink in. Oh, hey, it wouldn't be a universe. “This is rather as if you imagine a puddle waking up one morning and thinking, 'This is an interesting world I find myself in — an interesting hole I find myself in — fits me rather neatly, doesn't it? In fact it fits me staggeringly well, must have been made to have me in it!' This is such a powerful idea that as the sun rises in the sky and the air heats up and as, gradually, the puddle gets smaller and smaller, frantically hanging on to the notion that everything's going to be alright, because this world was meant to have him in it, was built to have him in it; so the moment he disappears catches him rather by surprise. I think this may be something we need to be on the watch out for.” -Douglas Adams Our sense of beauty is formed by the universe. It is not remarkable that the universe is made of systems that seem elegant to the beings that live in it.

yw3410 4y ago

If you haven't done it, then the statistical proofs for the gas equations and the proofs for Maxwell's equations also rank up there for me as well.

ivansavz 4y ago
Thanks for the recommendations, here are the three first links that come up for the "statistical proof" of the gas equations: https://www.one-tab.com/page/UhuMmBwCTma12IwYdMb54Q https://www.one-tab.com/page/UhuMmBwCTma12IwYdMb54Q Bookmarked to use as my just-in-time curriculum for the afternoon coffee break...

ianai 4y ago

Partially, those imaginary numbers offer ways to embed more geometric/topological reasoning into the number system than being imaginary. Like how adding the point at infinity makes the 2d plane more like a sphere.

lizknope 4y ago
I'm an electrical engineer so we did a lot of math in circuits class where we used Euler's identity. We did stuff like e^(i * theta) = cos(theta) + i * sin(theta) Then our teacher said "that part is imaginary" and just crossed out the isin(theta) part so e^(itheta) = cos(theta) and use that to solve circuit problems. I kind of laughed but some other people have told me that it is more of a math shortcut rather than "i" having a real world impact. As I said I have taken a quantum physics class but I'm not a physicist. But it seems like "i" really is necessary for quantum mechanics and manifests itself in the real world. https://www.quantamagazine.org/imaginary-numbers-may-be-essential-for-describing-reality-20210303/ https://www.quantamagazine.org/imaginary-numbers-may-be-esse...
- whatshisface 4y ago
  The imaginary unit shows up in both quantum mechanics and EE as a way to embed i*i = -1 in the equations. Any differential equation that involves it can be written without it by separating real and imaginary parts into different variables. Furthermore, if you let "one" be represented by the 2x2 identity matrix, the matrix [[0,1],[-1,0]] can act like i in every respect, because that matrix squares to -1 times the identity. In fact, even -i can stand in for i. The "nouns" of math might be real, but whether they are is a metaphysical question; all we observe in science are what you might call the verbs of math. FWIW, the "crossing out the imaginary part" works like this: if the system of equations is linear and doesn't have complex coefficients, every solution has a counterpart solution that is its complex conjugate, and you can add the conjugate (remember, it's linear) to get another, real-valued, solution. You can do that with time-invariant (definite-energy) solutions to the Schrodinger equation as well, because the energy eigenvalue equation doesn't have complex coefficients.

bigbillheck 4y ago

I have a phd in mathematics and to me, ultimately, it's just a tool. There's plenty of things that make me say 'oh, that's neat' but it's all mundane, never sacred.

radomir_cernoch 4y ago

For me, it's the halting problem (and related proofs).

StrictDabbler 4y ago

Descartes called them "imaginary" numbers because he didn't understand them well and didn't like them. He thought they were ugly and therefore false. If sine, cosine, and tangent had instead been named divine, codivine and tantric we'd be having ridiculous conversations about how magical they are too. What do you think the universe would look like if things didn't fit together? Sit with that thought for a little while, really let it sink in. Oh, hey, it wouldn't be a universe. “This is rather as if you imagine a puddle waking up one morning and thinking, 'This is an interesting world I find myself in — an interesting hole I find myself in — fits me rather neatly, doesn't it? In fact it fits me staggeringly well, must have been made to have me in it!' This is such a powerful idea that as the sun rises in the sky and the air heats up and as, gradually, the puddle gets smaller and smaller, frantically hanging on to the notion that everything's going to be alright, because this world was meant to have him in it, was built to have him in it; so the moment he disappears catches him rather by surprise. I think this may be something we need to be on the watch out for.” -Douglas Adams Our sense of beauty is formed by the universe. It is not remarkable that the universe is made of systems that seem elegant to the beings that live in it.