Real Numbers for Math Olympiad: Complete Preparation Guide

Real numbers form the deepest number theory topic in the school-to-Olympiad bridge. Euclid's division algorithm, the fundamental theorem of arithmetic, and irrationality proofs are all fair game.

For Class 10 students, Olympiad problems test your understanding of divisibility at a theoretical level. Can you use Euclid's algorithm creatively? Can you prove that a number is irrational?

Mistakes:

* Euclid's algorithm direction — Divide the larger by the smaller. Continue until remainder is 0.
* FTA uniqueness — The factorization is unique only up to the order of factors.
* Irrationality proof structure — Assume $\sqrt{p}$ is rational, derive that $p$ divides both numerator and denominator, contradicting coprimality.
* Decimal representations — Terminating decimals have denominators with only 2 and 5 as prime factors.

Try these!