I just saw Eric Weinstein talk about them on Joe Rogans podcast, but I couldn’t quite keep up. What exactly are they? Why are they significant?
In: Mathematics
The complex numbers are a sort of two-dimensional extension of the real number line, with an imaginary axis (i, 2i, 3i, …) along with the ordinary real axis (1, 2, 3, …). It turns out that you can extend this idea even farther to have four distinct axes (“quaternions”), or eight (“octonions”), or sixteen (“sedenions”), and so on for any power of 2. While a real number might just look like 3, and a complex number might look like 3+2i, a quaternion could look like 3+2i+4j+6k, where i, j, and k are distinct values with the property that i^2 = j^2 = k^2 = -1. Octonions would then have eight distinct components, one “real” component and seven distinct “imaginary” components.
The farther you go with this, the more nice properties about numbers you lose. When you go from the real numbers to the complex numbers, you lose the fact that numbers have a neat linear ordering. When you move from the complex numbers to the quaternions, you lose the fact that multiplication is commutative: for quaternions, it need not be the case that x*y = y*x. When you move from the quaternions to the octonions, you lose the fact that multiplication is associative: for octonions you aren’t even guaranteed to have x*(y*z) = (x*y)*z.
Despite the fact that the octonions are both rather abstract and lack many of the properties we expect number systems to have, they show up occasionally in theoretical physics.
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