Mathematical proof

An argument that shows why a statement is true in every case, not just the ones checked. Checking examples, however many, is never enough, and one counterexample is enough to break a claim.

Depth 1 · Foundation

You know that checking examples can't show something is true for every number, that one counterexample breaks an "every" claim, and you can follow a proof with dots that covers every number at once.

Depths of understanding

1
Foundation
You know that checking examples can't show something is true for every number, that one counterexample breaks an "every" claim, and you can follow a proof with dots that covers every number at once.
Opens after
Even and odd numbers, depth 2Showing with pairs of dots why odd + odd is always even is the first proof in Number Theory Level 1.CounterexampleOne counterexample is enough to break a claim; a proof shows that none can ever turn up.

Your path here

What's still between you and Mathematical proof, depth 1, in the order you can learn it.

Builds on

Opens up

ConjecturesA conjecture is a statement that has been checked but not proved, so you need to know what a proof is.

Makes these deeper

Once you know Mathematical proof, come back to these — they'll mean more.

Prime numbersDepth 2That the primes never run out is not something you can check; it has to be proved.