What is the strangest math that turned out to be useful?
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You can, they map, but complex numbers are much much easier to deal with
wrote on last edited by [email protected]In quantum mechanics, there are times you divide two different complex numbers, and complex multiplication/division is the thing two real numbers can't really replicate. That's how the Bloch 2-sphere in 3D space is constructed from two complex dimensions (which maps to 4 real ones).
It's peripheral, though. Nothing in the guts of the theory needs it AFAIK - the Bloch sphere doesn't generalise much and is more of a visualisation. So, jury's still out on if it's us or if it's nature that likes seeing it that way.
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I mean, it's endorsed by the same people. He has a page on their website and everything.
The same site says things like:
Between 1901 and 2024, the Nobel Prizes and the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel were awarded 627 times to 1,012 people and organisations.
which pretty clearly makes a distinction between the Nobel Prizes and the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel.
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A complex number is just two real numbers stitched together. It's used in many areas, such as the Fourier transform which is common in computer science is often represented with complex numbers because it deals with waves and waves are two-dimensional, and so rather than needing two different equations you can represent it with a single equation where the two-dimensional behavior occurs on the complex-plane.
In principle you can always just split a complex number into two real numbers and carry on the calculation that way. In fact, if we couldn't, then no one would use complex numbers, because computers can't process imaginary numbers directly. Every computer program that deals with complex numbers, behind the scenes, is decomposing it into two real-valued floating point numbers.
That's like saying negative numbers or fractional numbers is just two while numbers stitched together because that's how computers deal with it
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Integration.
Integration was literally developed to be useful
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The same site says things like:
Between 1901 and 2024, the Nobel Prizes and the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel were awarded 627 times to 1,012 people and organisations.
which pretty clearly makes a distinction between the Nobel Prizes and the Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel.
wrote on last edited by [email protected]Yeah, you're not wrong about the history. It just still seems like it counts.
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Would that make it less true? Complex numbers can be seen as a weird subgroup of the 2x2 real matrices. (And you can "stack" the two representations to get 4x4 real quaternions)
Furthermore, octonions are non-associative, and so can't be a subgroup of anything (although you can do a similar thing using an alternate matrix multiplication rule). They still show up in a lot of the same pure math contexts, though.
I just think complex vector spaces are a great place to stop your abstraction
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I just think complex vector spaces are a great place to stop your abstraction
Stopping while we're ahead? Never!
/s, but also I'm sort of in this picture.
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Stopping while we're ahead? Never!
/s, but also I'm sort of in this picture.
Well who wants constraints anyway? The most inconvenient constraints in the wrong place can make certain things much more complicated to deal with... Now a nice, sensible normal Hilbert space, isn't that lovely?
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There have been a number of Scientific discoveries that seemed to be purely scientific curiosities that later turned out to be incredibly useful. Hertz famously commented about the discovery of radio waves: “I do not think that the wireless waves I have discovered will have any practical application.”
Are there examples like this in math as well? What is the most interesting "pure math" discovery that proved to be useful in solving a real-world problem?
Riemann went nuts working on higher dimensional mathematics and linear algebra. At the time there was not a clear use case for math higher than like 3 or 4 dimensions, but he drove himself crazy discovering it anyways. Today, this kind of math underlies all of artificial intelligence
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George Boole introduced Boolean algebra, not Charles. Charles, according to this site on the Boole family, he had a career in management of a mining company.
Man, I've had the wrong Boole in my head this whole time? Guess I have a 0 somewhere that I should have a 1
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Man, I've had the wrong Boole in my head this whole time? Guess I have a 0 somewhere that I should have a 1
This is such a fucking cheeky pun I cannot even be mad at it.
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That's not quite accurate because the two numbers have a relationship with each other. i^2 = - 1, so any time you square a complex number or multiply two complex numbers, some of the value jumps from one dimension to the other.
It's like a vector, where sure, certain operations can be treated as if the dimensions of the vector are distinct, like a translation or scale. But other operations can have one dimension affecting the other, like rotation.
wrote on last edited by [email protected]uh... broski... you do realize a vector of two real numbers can be rotated... right? Please give me a single example for a supposed impossible operation to do on a vector of two real numbers that you can do on complex numbers. I can just define v² where v is a vector (a,b) as (a,b)²=(a²-b²,2ab). Okay, now I've succeeded in reproducing your supposedly mathematically impossible operation. Give me another one.
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That's like saying negative numbers or fractional numbers is just two while numbers stitched together because that's how computers deal with it
wrote on last edited by [email protected]Negative numbers are just real numbers with a symbol attached. Yes, that's literally true. In computer code we only ever deal with 0s and 1s. We come up with a convention to represent negative numbers, they are still ultimately zeros and ones but we just say "zeros and ones in this form represent a negative number," usually just by having the most significant bit 1. They are no physical negative numbers floating out there in the world like in a Platonic sense. What we call "negative" is contextual. It depends upon how we frame a problem and how we interpret a situation. You can lose money at a casino and say your earnings are now negative, or you can say your losses are now positive. Zeus isn't going to strike you down for saying one over the other. There is nothing physically dictating what convention you use. You just use which convention you find most intuitive and mathematically convenient given the problem you're trying to describe.
Yes, when we are talking about how computers work, we are talking about how numbers actually manifest in objective, physical reality. They are not some magical substance floating out there in the Platonic realm. Whenever we actually go to implement complex numbers or even negative in the real world, whenever we try to construct a physical system that replicates their behavior and can perform calculations on a physical level, we always just use unsigned real numbers (or natural numbers), and then later establish signage and complexity as conventions combined with a set of operations on how they should behave.
I'm not sure your point about fractional numbers. If you mean literally a/b, yes, there is software that treats a/b as just two natural numbers stitched together, but it's actually a bit mathematically complicated to always keep things in fractional form, so that's incredibly rare and you'd only see it in very specialized math software. Usually it's represented with a floating point number. In a digital computer that number is an approximation as it's ultimately digital, but I wouldn't say that means only digital numbers are physical, because we can also construct analogue computers that can do useful computations and are not digital. Unless we discover that space is quantized and thus they were digital all along, then I do think it is meaningful to treat real numbers as, well, physically real, because we can physically implement them.
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I don't think this is really an accurate way of thinking about them. Yes, they can be mapped to a 2d plane, so you can represent them with their two real-numbered coordinates along the real and imaginary axes, but certain operations with them (eg. multiplication) can be done easily with complex numbers but are not obvious how to carry out with just grid points. (3,4) * (5,6) isn't well-defined, but (3+4i) * (5+6i) is.
wrote on last edited by [email protected]I am having genuine difficulty imagining in your head how you think you made a point here. It seems you're claiming that given if two vectors have the same symbols between them, they should have identical output, such as (a,b) * (c,d) should have the same mathematical definition as (a+bi) * (c+di), or complex numbers are not reducible to real numbers.
You realize mathematical symbols are just conventions, right? They were not handed down to us from Zeus almighty. They are entirely human creations. I can happily define the meaning of (a,b) * (c,d) to be (ac-bd,ad+bc) and now it is mathematically well-defined and gives identical results.
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Negative numbers are just real numbers with a symbol attached. Yes, that's literally true. In computer code we only ever deal with 0s and 1s. We come up with a convention to represent negative numbers, they are still ultimately zeros and ones but we just say "zeros and ones in this form represent a negative number," usually just by having the most significant bit 1. They are no physical negative numbers floating out there in the world like in a Platonic sense. What we call "negative" is contextual. It depends upon how we frame a problem and how we interpret a situation. You can lose money at a casino and say your earnings are now negative, or you can say your losses are now positive. Zeus isn't going to strike you down for saying one over the other. There is nothing physically dictating what convention you use. You just use which convention you find most intuitive and mathematically convenient given the problem you're trying to describe.
Yes, when we are talking about how computers work, we are talking about how numbers actually manifest in objective, physical reality. They are not some magical substance floating out there in the Platonic realm. Whenever we actually go to implement complex numbers or even negative in the real world, whenever we try to construct a physical system that replicates their behavior and can perform calculations on a physical level, we always just use unsigned real numbers (or natural numbers), and then later establish signage and complexity as conventions combined with a set of operations on how they should behave.
I'm not sure your point about fractional numbers. If you mean literally a/b, yes, there is software that treats a/b as just two natural numbers stitched together, but it's actually a bit mathematically complicated to always keep things in fractional form, so that's incredibly rare and you'd only see it in very specialized math software. Usually it's represented with a floating point number. In a digital computer that number is an approximation as it's ultimately digital, but I wouldn't say that means only digital numbers are physical, because we can also construct analogue computers that can do useful computations and are not digital. Unless we discover that space is quantized and thus they were digital all along, then I do think it is meaningful to treat real numbers as, well, physically real, because we can physically implement them.
Talking about how numbers actually manifest in objective, physical reality - imaginary number appears in quantum hamiltonian, so maybe that's more real?
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I am having genuine difficulty imagining in your head how you think you made a point here. It seems you're claiming that given if two vectors have the same symbols between them, they should have identical output, such as (a,b) * (c,d) should have the same mathematical definition as (a+bi) * (c+di), or complex numbers are not reducible to real numbers.
You realize mathematical symbols are just conventions, right? They were not handed down to us from Zeus almighty. They are entirely human creations. I can happily define the meaning of (a,b) * (c,d) to be (ac-bd,ad+bc) and now it is mathematically well-defined and gives identical results.
wrote on last edited by [email protected]Right, but you need to specify that additional definition. Imaginary numbers are useful because they come pre-loaded with all those additional definitions about how to handle operations that use them.
I also find your hostile confusion unwarranted, given two other commenters have pointed out the same flaw in your argument that I have.
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Talking about how numbers actually manifest in objective, physical reality - imaginary number appears in quantum hamiltonian, so maybe that's more real?
wrote on last edited by [email protected]And you can also just write it out using real numbers if you wish, it's just more mathematically concise to use complex numbers. It's a purely subjective, personal choice to choose to use complex-valued notation. You are trying to argue that making a personal, subjective, arbitrary choice somehow imposes something upon physical reality. It doesn't. There isn't anything wrong with the standard formulation, but it is a choice of convention, and conventions aren't physical. If I describe my losses in a positive number, and then later change convention and describe my winnings with a negative number, the underlying physical reality has not changed, it's not going to suddenly transmute into something else because of a change in convention in how I describe it.
The complex numbers in quantum theory are not magic. They are also popular in classical mechanics as well, and are just quite common in wave mechanics in general (classical or quantum). In classical wave mechanics, in classical computer science, we use the Fourier transform a lot which is typically expressed as a complex number. It's because waves have two degrees of freedom, and so you could describe them using a vector of two real numbers, or you could describe them using complex numbers. People like the complex-valued notation because it's more concise to write down and express formulas in, but at the end of the day it's just a convention, a notation created by human beings which many other mathematically equivalent notations can describe the same exact thing.
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Right, but you need to specify that additional definition. Imaginary numbers are useful because they come pre-loaded with all those additional definitions about how to handle operations that use them.
I also find your hostile confusion unwarranted, given two other commenters have pointed out the same flaw in your argument that I have.
wrote on last edited by [email protected]Because your arguments are just bizarre. Imaginary numbers do not have a priori definitions. Humans have to define imaginary number and define the mathematical operations on them. There is no "hostile confusion" or "flaw," there is you making the equivalent of flat-earth arguments but for mathematics. You keep claiming things that are objectively false and so obviously false it is bizarre how anyone could even make such a claim. I do not even know how to approach it, how on earth do you come to believe that complex numbers have a priori definitions and they aren't just humans defining them like any other mathematical operation? There are no pre-given definitions for complex numbers, their properties are all explicitly defined by human beings, and you can also define the properties on vectors. You at first claim that supposedly you can only do certain operations on complex numbers that you cannot on vectors, I point out this is obviously false and you can't give a single counter-example, so now you switch to claiming somehow the operations on complex numbers are all "pre-given." Makes zero sense. You have not pointed out a "flaw," you just ramble and declare victory, throwing personal attacks calling me "confused" like this is some sort of competition or something when you have not even made a single coherent point. Attacking me and downvoting all my posts isn't going to somehow going to prove that you cannot decompose any complex-valued operations into real numbers, nor is it going to prove that complex numbers somehow don't have to have their properties and operations on them postulated just like real numbers.
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Because your arguments are just bizarre. Imaginary numbers do not have a priori definitions. Humans have to define imaginary number and define the mathematical operations on them. There is no "hostile confusion" or "flaw," there is you making the equivalent of flat-earth arguments but for mathematics. You keep claiming things that are objectively false and so obviously false it is bizarre how anyone could even make such a claim. I do not even know how to approach it, how on earth do you come to believe that complex numbers have a priori definitions and they aren't just humans defining them like any other mathematical operation? There are no pre-given definitions for complex numbers, their properties are all explicitly defined by human beings, and you can also define the properties on vectors. You at first claim that supposedly you can only do certain operations on complex numbers that you cannot on vectors, I point out this is obviously false and you can't give a single counter-example, so now you switch to claiming somehow the operations on complex numbers are all "pre-given." Makes zero sense. You have not pointed out a "flaw," you just ramble and declare victory, throwing personal attacks calling me "confused" like this is some sort of competition or something when you have not even made a single coherent point. Attacking me and downvoting all my posts isn't going to somehow going to prove that you cannot decompose any complex-valued operations into real numbers, nor is it going to prove that complex numbers somehow don't have to have their properties and operations on them postulated just like real numbers.
wrote on last edited by [email protected]I'm being combative because I don't get how you don't understand our argument, and because I view claims like "You keep claiming things that are objectively false" to be hostile when they stem from a misunderstanding rather than a fault on my part.
Let me restate my main point: complex numbers can be defined as vectors with the necessary rules to define various operations, such as multiplication over them and how they relate to sqrt(-1). Those additional rules are just as important to their definition as their appearance as two real-numbered values is. Both vectors and complex numbers are defined by humans, but we have chosen to give them separate definitions, because each definition includes the rules defining these operations and relationships, and they are different between the two types of mathematical object.
And, for the record, I downvoted your posts that were hostile (not all of them) and responded in kind. It's a separate effort than trying to prove my point here.
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And you can also just write it out using real numbers if you wish, it's just more mathematically concise to use complex numbers. It's a purely subjective, personal choice to choose to use complex-valued notation. You are trying to argue that making a personal, subjective, arbitrary choice somehow imposes something upon physical reality. It doesn't. There isn't anything wrong with the standard formulation, but it is a choice of convention, and conventions aren't physical. If I describe my losses in a positive number, and then later change convention and describe my winnings with a negative number, the underlying physical reality has not changed, it's not going to suddenly transmute into something else because of a change in convention in how I describe it.
The complex numbers in quantum theory are not magic. They are also popular in classical mechanics as well, and are just quite common in wave mechanics in general (classical or quantum). In classical wave mechanics, in classical computer science, we use the Fourier transform a lot which is typically expressed as a complex number. It's because waves have two degrees of freedom, and so you could describe them using a vector of two real numbers, or you could describe them using complex numbers. People like the complex-valued notation because it's more concise to write down and express formulas in, but at the end of the day it's just a convention, a notation created by human beings which many other mathematically equivalent notations can describe the same exact thing.
Yeah, I agree it can all be mapped onto positive reals. How about zero? Would that be needed to be having a physical reality ?
