>>12763080I looked at your link.
>When complex numbers were first introduced, it was understood that real numbers were, in particular, complex numbers; The people who first introduced complex numbers are the ones who know what they are. They are the owners of their own invention forever. Just like you people say to me, "That's great if you want to invent your own magic number system but don't call it R," that goes for Spivak's magic pair system as well.
>if our definition is to be taken seriously then this is not true—a real number is not a pair of real numbers, after all.You reveal your foolishness when you read Spivak's words and decide that Spivak's definition is to be taken more seriously than the hundred or more years of definitions which preceded him.
> complex number as an ordered pair of real numbers: z=(a,b) with a,b?R.If you choose this definition, then what Spivak says about real not being pairs is true. However, since reals were already defined as a subset for C before he wrote that, he should have written "z=(a,b) with a,b?{R union EMPTY} but for simplicity we will not consider the case of (x, NULL) != (x,0) in this undergraduate textbook." In this way, he would not have contradicted the preexisting definition of C as a superset of R. Likely he chose the definition he chose because he didn't want to have to add caveats about the difference between (x,0) and (x, NULL) in his undergraduate textbook. This person:
>>12762984>Even if it looks like, at first blush, the reals are a simple punctured great circle in the Riemann sphere, this is wrong.is a complete idiot to say, "Spivak was right and all the people who had already been doing complex analysis for a hundreds of years were all wrong. Out of all the people I could have picked to say, 'This is the real one and all the other ones are fake,' I picked Spivak. Specifically between Spivak and Euler, I picked Spivak."
