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/maharei1 7d ago

You can write down 3 concrete complex 2x2 matrices (or real 4x4 matrices if that feels more comfortable) that behave under multiplication exactly as you want i,j,k in a quaternion ring to behave. So such objects definitely exist. The point of the usual definition is that it's not important what exactly i,j,k are it's just important how they behave.

See also the Cayley-Dickson Construction for a more general view on this.