quarternions vs octonions

octet of 8 real numbers (e0,e1,…e8)
x = sigma(xiei) where i = 0 to 8

multiplication is distributive over addition.

Octonions == pairs of Octonions
Quarternions == pairs of complex numbers

non-commutative division algebra
non-commutative multiplication algebra
i^2 = j^2 = k^2 = i*j*k = -1
H = a*1 + b*i + c*j + d*k

So moral of the story? If you think complex numbers literally mean numbers with real and imaginary parts try naming these two group of numbers. The point really being the imaginary part of Complex numbers are not really considered imaginary in the sense of “not existent in the world” anymore. They both are considered a inevitable part of the world as science understands it. See Scott Aaronson’s quantum theory lecture here.


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