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This is one of the cool things about math, there are a lot of aspects of math which stem from thinking, "well what if I took a square root of negative one?" or

This is one of the cool things about math, there are a lot of aspects of math which stem from thinking, "well what if I took a square root of negative one?" or "what if geometry worked this way?" and then someone does the thought exercise and fleshes it out into something interesting. And then there just happen to be aspects of reality that end up being very nicely described by these what if questions

thaumasiotes 5y ago
> there are a lot of aspects of math which stem from thinking, "well what if I took a square root of negative one?" I don't think that's what imaginary numbers stem from. If I recall, they came from the much more specific question of "what if I apply the formula for roots of a cubic polynomial, even when that formula produces some nonsensical intermediate values that end up canceling each other out?". It's less about "what if we do this?" and more about "we have this method, and it seems to work even in cases where it's not obvious that it should work".
- emj 5y ago
  That is the reverse of this subject in botany, we have this method that we know works and is logical lets make our language more logical. The problem with that is the same reason we don't use imaginary numbers everywhere we can use them. Veritasium has a video description [1] with links to references about the history of imaginary numbers and polynomials. There are some great writing on this and understanding not the history but the problems that gave rise to imaginary numbers, this connection to taxonomy makes me want to read them again. There must be a study of this; failed attempts of taxonomy. [1] https://www.youtube.com/watch?v=cUzklzVXJwo https://www.youtube.com/watch?v=cUzklzVXJwo
betterunix2 5y ago
Historically the square root of negative one arose more naturally than you might think; it is basically impossible to avoid it when you derive a cubic formula. Historically the work on complex numbers started as soon as cubic and quartic formulas were derived, because complex numbers emerge naturally and unavoidably from those formulas (even for polynomials that have only real roots). At first mathematicians were generally dismissive of the imaginary units and viewed them as "nonsense" terms that sometimes appear in some formulas and there was a lot of doubt about dealing with such terms (the same doubt you might have if someone divided by zero while solving some formula, even if they wind up finding the correct answer). Those doubts were completely eliminated by the proof of the fundamental theorem of algebra. For what it's worth, you can avoid "imaginary" numbers entirely if you want -- you can do arithmetic on real polynomials modulo x^2+1, or equivalently on 2x2 real matrices of a special form.

thaumasiotes 5y ago

> there are a lot of aspects of math which stem from thinking, "well what if I took a square root of negative one?" I don't think that's what imaginary numbers stem from. If I recall, they came from the much more specific question of "what if I apply the formula for roots of a cubic polynomial, even when that formula produces some nonsensical intermediate values that end up canceling each other out?". It's less about "what if we do this?" and more about "we have this method, and it seems to work even in cases where it's not obvious that it should work".

emj 5y ago
That is the reverse of this subject in botany, we have this method that we know works and is logical lets make our language more logical. The problem with that is the same reason we don't use imaginary numbers everywhere we can use them. Veritasium has a video description [1] with links to references about the history of imaginary numbers and polynomials. There are some great writing on this and understanding not the history but the problems that gave rise to imaginary numbers, this connection to taxonomy makes me want to read them again. There must be a study of this; failed attempts of taxonomy. [1] https://www.youtube.com/watch?v=cUzklzVXJwo https://www.youtube.com/watch?v=cUzklzVXJwo

betterunix2 5y ago

Historically the square root of negative one arose more naturally than you might think; it is basically impossible to avoid it when you derive a cubic formula. Historically the work on complex numbers started as soon as cubic and quartic formulas were derived, because complex numbers emerge naturally and unavoidably from those formulas (even for polynomials that have only real roots). At first mathematicians were generally dismissive of the imaginary units and viewed them as "nonsense" terms that sometimes appear in some formulas and there was a lot of doubt about dealing with such terms (the same doubt you might have if someone divided by zero while solving some formula, even if they wind up finding the correct answer). Those doubts were completely eliminated by the proof of the fundamental theorem of algebra. For what it's worth, you can avoid "imaginary" numbers entirely if you want -- you can do arithmetic on real polynomials modulo x^2+1, or equivalently on 2x2 real matrices of a special form.