Real Numbers

The real numbers split into two families. A rational number can be written as p/q (q ≠ 0) — its decimal either terminates or repeats. An irrational number cannot — its decimal goes on forever without repeating (like √2 or π). Explore both:

The Fundamental Theorem of Arithmetic says every composite number can be written as a product of primes in exactly one way (apart from the order). Primes are the "atoms" of the number system.

From prime factorisations: the HCF takes the smallest power of each common prime; the LCM takes the greatest power of every prime. A handy check: HCF × LCM = a × b — try it on the calculator at the top of the page.

To prove √2 is irrational, we use proof by contradiction:

• Assume √2 is rational, so √2 = p/q in lowest terms (p, q have no common factor).
• Then 2q² = p², so p² is even → p is even → write p = 2m.
• Then 2q² = 4m², so q² = 2m² → q is also even.
• But then p and q share the factor 2 — contradicting "lowest terms".
• So our assumption was wrong: √2 is irrational.

The same method proves √3, √5, and numbers like 3 + 2√5 are irrational.