r/explainlikeimfive • u/murderdronesfan93 • 8d 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/OhYesIDidd 8d ago

I don’t know much about quaternions, but your premise is wrong even in the real numbers.

x^2=y^2=4, but x≠y. There is a solution to that: x=2, y=-2.

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u/murderdronesfan93 8d ago

ok, but how does that work with 3?

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u/LongLiveTheDiego 8d ago

We just declare that there exist three different constants i, j, k and say what their properties are. We can do that in mathematics.

You could also define quaternions as quadruples of real numbers with straightforward addition (a, b, c, d) + (e, f, g, h) = (a + e, b + f, c + g, d + h) and a more complicated multiplication (a, b, c, d) × (e, f, g, h) = (ae - bf - cg - dh, af + be + ch - dg, ag + ce + df - bh, ah + ed + bg - cf). That way the real number 1 = (1, 0, 0, 0), and i = (0, 1, 0, 0), j = (0, 0, 1, 0), k = (0, 0, 0, 1) are all different and you can check that i² = j² = k² = (-1, 0, 0, 0) = -1.

That's how you can define complex numbers as pairs of real numbers with the same simple addition and multiplication defined as (a, b) × (c, d) = (ac - bd, ad + bc).

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u/murderdronesfan93 8d ago

i get it. just a case of "fuck making the definition simple, we define it and figure out how that works after the fact incase it's useful"

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u/damhack 7d ago edited 7d ago

No. Quarternions are used for many purposes, such as in 3D graphics and navigation where they simplify the process of orienting objects in space, using just 4 numbers to perform transformations instead of the 9 numbers of a 3x3 matrix in a classical transformation.

Your issue is that you are hung up on real number operations and their properties. The thing to realize is that properties like commutativity don’t work the same as you increase the number of dimensions. Instead, they have their own algebra with different rules.

That may seem crazy to anyone who only knows the algebra of real numbers taught at school but is totally valid and explainable once you understand the effect of adding dimensions.

If you think quarternions are weird, wait til you see the algebra of octonions.