#SabineHossenfelder - Do #ComplexNumbers Exist?

"Do complex #Numbers exist or are they just a convenient, #MathematicalTool that we use in science? With the exception of #QuantumMechanics, it is easy to get rid of complex numbers. But can you do quantum mechanics without complex numbers? A recent paper says no, you can't."

https://www.youtube.com/watch?v=ALc8CBYOfkw&ab_channel=SabineHossenfelder

#Math #Maths #Mathematics #Science #Physics #Equations #ComplexPlane #TheComplexPlane #ComplexNumber #e #i #Pi #QM #QuantumPhysics

Is the order of the #imaginaryNumber in a #complexNumber just a matter of preference? #math #maths #mathematics

(w+ix)+j(y+iz) = w+ix+jy−kz
(w+xi)+(y+zi)j = w+xi+yj+zk

Either it matters or −kz = zk. Wouldn't the same be true for i & j? So those #quaternions still aren't equal?
That would mean (a+ib)* = (a+bi). And so |a+ib|² = (a+ib)(a+bi)= a²+ibbi+abi+iba = a²+b²-aib+iba, given -ib=bi.
Consistency would prevent you noticing i doesn't commute with reals, either.

That's... disturbing.

Is the order of the #imaginaryNumber in a #complexNumber just a matter of preference? #math #maths #mathematics

(w+ix)+j(y+iz) = w+ix+jy−kz
(w+xi)+(y+zi)j = w+xi+yj+zk

Either it matters or −kz = zk. Wouldn't the same be true for i & j? So those #quaternions still aren't equal?
That would mean (a+ib)* = (a+bi). And so |a+ib|² = (a+ib)(a+bi)= a²+ibbi+abi+iba = a²+b²-aib+iba, given -ib=bi.
Consistency would prevent you noticing i doesn't commute with reals, either.

That's... disturbing.