Concept 1.3
Proving √2 Is Irrational
Source: NCERT Mathematics, Class 1012 minute lesson
Formal statement
A number is irrational if it cannot be written as p/q with p and q integers and q ≠ 0. √2, √3 and √5 are irrational; NCERT proves this by contradiction.
The intuition
A proof by contradiction is a trap you set for yourself. Assume √2 = p/q in lowest terms, follow the algebra honestly, and watch the assumption fall apart.
Squaring gives p² = 2q², so p is even; write p = 2c and you get q² = 2c², so q is even too. Both even means they share the factor 2, but we said lowest terms. Contradiction.
The attention
Look at the assumption the proof is built to break.
√2 = p/q with p, q coprime ⇒ p² = 2q² ⇒ 2 divides p and q
Look hereNot here
- Look here
- The assumption: p and q share no factor. The whole proof aims at one sentence: p and q have no common factor. Everything else is a road to the place where that sentence fails.
- Not here
- The algebra in the middle. Squaring and substituting are mechanical. Students copy them perfectly and still can't say why the proof works.
- What it teaches
- In a proof by contradiction, find the sentence that gets contradicted first. Then every step has a direction.