Comment by chongli
12 days ago
The simple fact you wanted to write this:
i=0+1*i
Makes i a number. Since * is a binary operator in your space, i needs to be a number for 1*i to make any sense.
Similarly, if = is to be a binary relation in your space, i needs to be a number for i={anything} to make sense.
Comparing i with a unary operator like - shows the difference:
i*i=-1 makes perfect sense
-*-=???? does not make sense
i is a complex number, complex numbers are of the form real + i*real... Don't you see the recursive definition ? Same with 0 and 1 they are not numbers until you can actually define numbers, using 0 and 1
This is one definition of i. Or you could geometrically say i is the orthogonal unit vector in the (real,real) plane where you define multiplication as multiplying length and adding angles
There's no issue with recursive definitions. That's how arithmetic was original formalized by Peano's axioms [1].
[1] https://en.wikipedia.org/wiki/Peano_axioms
i is also a quaternion. So by this logic we could say complex numbers are made up of quaternions. But we don’t say such things because they wouldn’t be a good mental model of what we want to talk about.
4 replies →