how mathematicians “prove” that mathematical proofs, such as the Pythagorean Theorem, are true for any input?
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The idea is to show that even if you don’t know the inputs, you can still know a certain thing for certain. Usually we do this by assigning the unknown inputs variable names, like A B and C for the side lengths of the triangle in the case of the Pythagorean theorem. Then we write down some initial observations about how those variables are related, and do some math to show that the result we wanted to prove must follow from those initial observations. The Pythagorean Theorem is a geometry theorem so it’s much easier to prove using visuals. If you want a good example I suggest just looking it up, there are lots of great accessible proofs online.
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