Asking as a layman, I've always puzzled over imaginary numbers and how they can be used to solve problems involving real numbers or quantities only (e.g. contour integration methods or Fourier ...
The question is the title of a recent piece in the Notices of the American Mathematical Society, by twelve authors (of which I am one). The contention is that traditional history of mathematics is ...
Let me take up some details in the answer of another question. Submitted by user hyg17: Heading: All real numbers can be expressed as a limit of rational numbers? The question was: Let $C$ be a set ...
Recently I stumbled upon the website of Prof. N. J. Wildberger, who has an written a thought-provoking article about set theory and the real numbers. In his opinion, real numbers are actually "a ...
I am currently reading Jeremy Gray's "Plato's Ghost", and I run into the following passage (Chapter 5, page 332). The point is, it seems to me that it contains two very elementary mistakes that feel ...
Michael Spivak in “Calculus” asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by “exist”?
Michael Spivak in "Calculus" asserts that $\sqrt2$ cannot be proven to exist, and that such a proof is impossible. What does he mean by "exist"? How are you to prove that any number "exists"? Why ...
Continuity is an intuitive concept. I will not dwell on the precise definitions of continuity and the rest here. Note that differentiability is a more restrictive condition than continuity, while ...