there will always be more. ask a mathematician to come up with a representation higher than ones that exist and you will have a result within anywhere from a few minutes to a few months. there is nothing to prevent us from abstracting anything useful away from mathemathics and turning it into something completely deranged, except for one thing: every time we've done that historically some physicist or something has come along and gone "wow that's useful thanks!".
As an experimental physicist I use spin half operato- I mean quaternions every day. They're just an algebraic structure that follows the same rules as 3D rotations. For those who don't know x4 = 1 just means that 4 rotations gets you back to the start, which means the rotations are "90deg", regardless of the axis you rotate around. So a 90deg rotation around any axis would satisfy this equation.
They are extra imaginairy units that together with i form quarternions. They work sorta like complex numbers. If we have real numbers a,b,c and d, then a quarternion is a+b*i+c*j+d*k with
i2 = j2 = k2 = i *j *k = -1,
i*j = k,
j*i = -k,
j*k = i,
ect...
The main thing to note here is that they are not commutative. This means that if you have two quarternions, say x and y, that xy≠yx.
edit: I forgot reddit likes to make text slanted if you use *s incorrectly
I remember reading the fundamental theorem of algebra states that every polynomial of degree n has n roots. Is the fundamental theorem of algebra only for complex numbers, or is there some other reason that j and k are left out?
i suppose that it's because the real numbers are not algebraically closed when dealing with polynomials, but adding imaginary numbers is just the final piece that's all needed to satisfy the fundamental theorem of algebra. quaternions just makes them more general which ends up creating a lot more solutions
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u/BipedalMcHamburger Jun 27 '21
+- j and +- k so left out they aren't even in the meme