1
Make the assumption: Assume that √2 is rational. Then we can write √2 = p/q, where p and q are relatively prime natural numbers (in other words, the fraction cannot be reduced any further).
2
Square both sides: 2 = p²/q², so p² = 2q². Since the right-hand side is a multiple of 2, p² is also a multiple of 2 — that is, it is even.
3
Show that p is even: If p² is even, then p itself must be even (the square of an odd number is always odd). Therefore, we can write p = 2k.
4
Substitute and simplify: Substituting p = 2k into p² = 2q² gives 4k² = 2q², or q² = 2k². By the same reasoning, q² is even, so q is also even.
✗
Contradiction! If both p and q are even, they are both divisible by 2. But we initially assumed that p and q were "relatively prime (cannot be reduced any further)" — a contradiction! This means our original assumption (that "√2 can be written as p/q") must be false, so √2 is a number that cannot be expressed as a fraction (an irrational number).