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.