The proof for √5
A square of area 5 has side √5. No fraction can be that side.
Step 1 of 6
- Suppose p/q, in lowest terms, squares to 5.
- Square it: p² = 5q².
- The lemma: 5 divides p² so 5 divides p.
- Substitute p = 5m: q² = 5m².
- The lemma again: 5 divides q.
- Contradiction: 5 divides p and q, but lowest terms said no.
Why each line holds
Here p and q are whole numbers, and q is not 0. Lowest terms means p and q share no factor except 1.
- Line 2 squares both sides of line 1.
- Line 3: 5 is prime and divides p². By the lemma, 5 divides p.
- Line 4: 5 divides p, so p = 5m. Here m is a whole number. Then p² = 25m². Put this into p² = 5q². That gives 25m² = 5q², so q² = 5m².
- Line 5: 5 divides q². By the lemma, 5 divides q.
- Line 6: 5 divides p and q. Lowest terms says they share no factor but 1.
So the guess fails. No fraction squares to 5.
Each line needs the line before it.
Now try it. Put the six lines in order.
Loading lab…