Comment by seanhunter

> I have a real philosophical problem with complex numbers

> I believe real numbers to be completely natural

I have to say I find this perspective interesting but completely alien.

We need to have a way to find x such that x^2-2 = 0, and Q won’t cut it so we have R. (Or if you want, we need a complete ordered field so we have R)

We need to have a way to find x such that x^2+2 = 0, and R won’t cut it so we have C. (Or if you want, we need algebraic closure of R by the fundamental theorem of algebra so we need C)

I don’t really think any numbers (even “natural” numbers) are any more natural than any other kind of numbers. If you start to distinguish, where do you stop. Negative numbers are ok or not? What about zero? Is that “natural”? Mathematicians disagree about whether 0 is in N at least.

It reminds me of the famous quote from Gauss:

   That this subject [imaginary numbers] has hitherto been surrounded by mysterious obscurity, is to be attributed largely to an ill adapted notation. If, for example, +1, -1, and the square root of -1 had been called direct, inverse and lateral units, instead of positive, negative and imaginary (or even impossible), such an obscurity would have been out of the question.