Comment by ttoinou
10 days ago
Yeah i is not a number. Once you define complex numbers from reals and i, i becomes a complex numbers but that's a trick
10 days ago
Yeah i is not a number. Once you define complex numbers from reals and i, i becomes a complex numbers but that's a trick
i is not a "trick" or a conceit to shortcut certain calculations like, say, the small angle approximation. i is a number and this must be true because of the fundamental theorem of algebra. Disbelieving in the imaginary numbers is no different from disbelieving in negative numbers.
"Imaginary" is an unfortunate name which gives makes this misunderstanding intuitive.
https://youtube.com/playlist?list=PLiaHhY2iBX9g6KIvZ_703G3KJ...
However, what's true about what you and GP have suggested is that both i and -1 are used as units. Writing -10 or 10i is similar to writing 10kg (more clearly, 10 × i, 10 × -1, 10 × 1kg). Units are not normally numbers, but they are for certain dimensionless quantities like % (1/100) or moles (6.02214076 × 10^23) and i and -1. That is another wrinkle which is genuinely confusing.
https://en.wikipedia.org/wiki/Dimensionless_quantity
Yes I agree, I was just arguing against "i=0+1*i so it's by definition a complex number" which is a self referential definition
If you take this tack, then 0 and 1 are not numbers either.
i is not a real number, is not an integer, is not a rational etc.
You need a base to define complex numbers, in that new space i=0+1*i and you could call that a complex number
0 and 1 help define integers, without {Empty, Something} (or empty, set of the empty, or whatever else base axioms you are using) there is no integers
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
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