What a proof settles
Testing fractions one by one cannot rule them all out.
Step 1 of 4
- A line of fractions between 0 and 3.
- A search checks three. Each one misses 5.
- Three misses say nothing about a fourth fraction.
- A proof covers the whole line. No fraction squares to 5.
Search or proof
A search tries some fractions. It reports only on those it tried. A proof covers every fraction at once.
A decimal is not a proof. 2.236 × 2.236 = 4.999696. It misses 5. A decimal that stops is a fraction. The proof says no fraction squares to 5.
The proof for √2 does not cover √5 yet. Put 5 in place of 2. The same steps then work for any prime.
One example refuses a claim
Claim: the product of two irrational numbers is always irrational. Try √2 × √2. It is 2, a whole number.
One example is enough to refuse a claim.
Now try it. Read each claim. Tap True or False.
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