The proof for √2 and √3
One plan proves that √2 is irrational. The same plan proves √3. An irrational number cannot be written as a fraction. The plan uses one fact, the lemma. If a prime divides n², it divides n. Here n is a whole number.
- Suppose p/q, in lowest terms, squares to a prime k.
- Square it: p² = k q².
- The lemma: k divides p² so k divides p.
- Substitute p = km: q² = k m².
- The lemma again: k divides q.
- Contradiction: k divides p and q, but lowest terms said no.
The plan
Let k be a prime. Suppose √k = p/q. Here p and q are whole numbers, and q is not 0. They have no common factor except 1. Then p/q is in lowest terms. Follow the steps. Each step has a reason.
As k divides p, write p = km. Here m is a whole number. Then p² = k²m². Put this into p² = kq². That gives k²m² = kq². Divide both sides by k: q² = km².
A contradiction means two facts cannot both be true. Here k divides p and q, but lowest terms says no. So the guess was wrong.
| Step | For √2 | For √3 |
|---|---|---|
| Suppose and square | p² = 2q² | p² = 3q² |
| The lemma on p | 2 divides p² so 2 divides p | 3 divides p² so 3 divides p |
| Put in p = km | p = 2m gives q² = 2m² | p = 3m gives q² = 3m² |
| The lemma on q | 2 divides q | 3 divides q |
| Contradiction | 2 divides p and q | 3 divides p and q |
The same steps prove √2, √3 and √5 irrational. They work for any prime k.
Now try it. Give the reason for each line of the √3 proof.
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