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If you take this tack, then 0 and 1 are not numbers either.
by chongli 8mo ago
If you take this tack, then 0 and 1 are not numbers either.
- ttoinou 8mo agoi 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
- chongli 8mo agoThe 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
- ttoinou 8mo agoi 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 i*i=-1 makes perfect sense 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
- chongli 8mo agoThere'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 https://en.wikipedia.org/wiki/Peano_axioms
- ttoinou 8mo agoi 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.
- srean 8mo ago> i is also a quaternion Yeah, so ? The fraction 1/2 is a rational and also a real and also a complex numbers and also a quaternion also an octonion .... We use a number along with the minimal abstraction that is sufficient for our purpose.