That's not true, and that is the point. To prove the main theorems of calculus, you don't need complex numbers, but you do need irrationals.

When you first learn calculus, you do one or two epsilon-delta proofs, and then your teacher gets a little hand wavy about limits and you move on to the real work of derivatives and integrals, cause the limits stuff intuitively makes sense. When you continue on in Real Analysis or Topology of the Real Line, you discover that your intuition lied to you, and concepts like open and closed sets and intersections and accumulation points are important and are in general non-obvious.

That's not true, and that is the point. To prove the main theorems of calculus, you don't need complex numbers, but you do need irrationals.

When you first learn calculus, you do one or two epsilon-delta proofs, and then your teacher gets a little hand wavy about limits and you move on to the real work of derivatives and integrals, cause the limits stuff intuitively makes sense. When you continue on in Real Analysis or Topology of the Real Line, you discover that your intuition lied to you, and concepts like open and closed sets and intersections and accumulation points are important and are in general non-obvious.

Counterexamples In Analysis

https://www.amazon.com/Counterexamples-Analysis-Dover-Books-...

(Pdf version) https://pdfs.semanticscholar.org/a4e7/eb352e4c44bf75d8fabaf7...

https://www.amazon.com/Counterexamples-Analysis-Dover-Books-...

These counterexamples are sometimes a bit involved, but I find they are often useful for understanding the purpose of the technical assumptions that accompany many theorems.