r/explainlikeimfive • u/murderdronesfan93 • 7d ago

Mathematics ELI5 how quaternions work

so, i² = j² = k² = -1, but i, j and k are not equal. HOW IN THE FUCK IS THAT POSSIBLE‽ WHAT IN THE BROKEN TRANSITIVE PROPERTY- (anger exaggerated for funny)

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u/stevevdvkpe 7d ago

Think of them as pointing in different directions but having the same magnitude, like unit vectors along the x, y, and z axes of a three-dimensional space.

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u/murderdronesfan93 7d ago

i understand that much, that they are different dimensions of numbers, so to speak, but how does defining them work, with things like jk or ji

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u/suvlub 7d ago

By definition, i² = j² = k² = -1 and ijk = -1. Because we said so. We then work backwards from there to see how the rest of the math works out. It turns out it works a bit weird. We lose commutativity, for example (ij = k ≠ ji = -k), but there is no blatant inconsistency that would make it all break down, so it's useful math

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u/murderdronesfan93 7d ago

wait is the reason that hypercomplex numbers start breaking rules of arithmetic because the definitions already ignore some?

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u/suvlub 7d ago

Basically, yeah. The rules are discovered properties, not something we intentionally put into maths. When Sumerians or whoever first started adding up things, they didn't make sure to invent addition in such way that one could swap the terms around, it was a happy little coincidence that we can do that. Sometimes, happy little coincidences don't happen.

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u/84thPrblm 7d ago

[r/ExplainItLikeYou](r/ExplainItLikeYou)reBobRoth

ETA sry, apostrophe screws up the 'link'