"Proof surfaced in 1898 that the reals, complex numbers, quaternions and octonions are the only kinds of numbers that can be added, subtracted, multiplied and divided." Quanta's math coverage is usually very good but this is ouch wrong
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Replying to @St_Rev @AnnKSterzinger
I think the key word is “number”. What’s a number? Is the ring Z_p numbers (where p is prime)?
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Which I recognize is pretty redundant with your subsequent tweets that went into the true statement that Quanta mangled, but eh
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By the way, no one has ever explained "i" to me in a way that makes sense. I think my calculus teacher was just like "'i' stands for imaginary, OK?" and then grunted and went back to staring out the window.
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Replying to @AnnKSterzinger @St_Rev
i shows up in enough contexts that there are multiple ways to go about explaining it. The standard, I think, is i = sqrt(-1), but there’s also a geometrical explanation (complex numbers as isomorphic to vectors) and a quaternion-style explanation (i^2 = -1, 1 * i = i).
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First way: no real number squared is ever negative. This means that the square root of -1 doesn’t exist. But if we pretend it does, it has to be a new number, not an existing one. Following the standard rules of numbers, we can add and multiply between normal numbers and i.
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It's a bit bizarre to me that you can invent a number and it still works with the standard rules of numbers.
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Replying to @AnnKSterzinger @TWakalix
Sometimes you can, sometimes you can't! Like, I could say "Let z be a real number such that z = 1 and z = 0" and this would be Extremely Bad because the universe would collapse.
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This was actually a plot point on Rick and Morty, come to think of it. Anyway, inventing a number and throwing it in the bag is what mathematicians call 'adjoining an element'. You just have to check that everything still works and figure out the consequences.
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So with 'i' the process goes, informally, something like... ok we still have all the real numbers like 3 and pi. 'let a be a real number' Multiplication means we have stuff like 12i and -i/4. 'bi, where b is a real number' Addition means we have pairs like 3 + 12i and pi - i/4.
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So we at least have to handle everything that looks like 'a + bi, where a and b are real numbers' It turns out that this is all you need -- you can add, subtract, multiply and divide these guys and you don't get anything hairier than 'a + bi'.
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It takes some checking -- proving division works is a bit of a slog. But this is the kind of thing we algebraists do for a living.
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. Banned in Sweden. SubGenius, Zhuangist, white-hat troll. Defrocked mathematician. Brain problems.