From conjecture to justification — Mathematics, 11–13
Classify, spot a regularity, state a conjecture and tell testing from validating — with counterexamples, exhaustion and generic examples.
A hunch you can test
Classify 2/4, 3/6, 4/8, 5/10 and 3/5. Four are worth 1/2 and 3/5 is the odd one out: classifying is grouping by a characteristic. In the four you see a regularity: the numerator is always half the denominator. From it comes a conjecture: “if the numerator is half the denominator, the fraction is worth 1/2”. Testing is trying cases (6/12, 7/14…). Validating is showing it is true every time. They are different: cases can support it, but only a justification settles it.
Four ways to justify
There are different ways to justify. Counterexample: “all prime numbers are odd” is false, because 2 is prime and even. Exhaustion: with few cases you check them all. Up to 20 the primes are 2, 3, 5, 7, 11, 13, 17 and 19, eight in all, and only 2 is even. Generic example: a case chosen so that the reasoning works for any other. Logical coherence: you chain facts you already know. Each fits its own situation: a counterexample brings a statement down, but to prove “always” you need more than examples.
A case: the sum of two consecutive odd numbers
Conjecture: the sum of two consecutive odd numbers is a multiple of 4. Tests: 3 + 5 = 8, 7 + 9 = 16, 11 + 13 = 24, 99 + 101 = 200, all multiples of 4, but that only tests. To validate, use a generic example: 19 + 21 = 40. The number between them is 20, and 19 + 21 = (20 − 1) + (20 + 1) = 2 × 20. Since 20 is even, 20 = 2 × 10 and the sum is 2 × 2 × 10 = 4 × 10. Nothing depends on 19: with a letter, (n − 1) + (n + 1) = 2 × n, and n is even. So it is always a multiple of 4.
Testing only cases of the same kind
Mariana tried 6 × 3 = 18, 5 × 4 = 20, 8 × 2 = 16 and 10 × 7 = 70 and concluded: “multiplying never gives a result smaller than the starting number”. Four correct tests, and the conjecture is false: she only tried natural numbers bigger than 1. One counterexample is enough: 12 × 1/3 = 4, and 4 is smaller than 12. To avoid the trap, go looking for the awkward case: fractions, 0, 1, very big numbers. A counterexample brings a conjecture down, however many tests it has passed.
Where you see it: what nobody has proved yet
Some conjectures are easy to test and nobody has managed to validate them. Goldbach's says: every even number greater than 2 is the sum of two prime numbers. 4 = 2 + 2, 6 = 3 + 3, 8 = 3 + 5, 10 = 5 + 5, 12 = 5 + 7, 14 = 7 + 7. Computers have tested it for enormous numbers, and it is still unproved: testing a great many cases is not justifying. Whenever someone says “always” or “never”, ask: is this a test or a justification?
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