What are some basic math proofs

“What do we need evidence for?
“How does a proof work?

What do we need evidence for?

A lot of math is done in school at school. The science of mathematics as taught at university is much more than that. This picture tries to explain that:


Image sketch for the mathematical building.svg from Wikimedia Commons, which was created by Golmman and is licensed under CC BY 4.0.

The foundation for mathematics are the axioms. These are basic mathematical statements that are taken to be true. An example of this is "Every natural number \ (n \) has exactly one successor \ (n + 1 \)." Based on this, mathematical propositions (also called theorems) are formulated. These have to be proven in order to be accepted as true by mathematicians. So the proofs are the mortar on the house of mathematics; they hold everything together.

How does a proof work?

Image Structure of a proof.svg from Wikimedia Commons, which was created by Stephan Kulla and is licensed under CC BY 3.0.

The boxes represent statements, the arrows represent mathematical conclusions. Everything starts with the premises, the statements from which we start. These can be axioms or other mathematical theorems that have already been proven. Starting from these, only logical conclusions are allowed in order to arrive at the statement of the sentence with a few intermediate steps. For example, a corollary might look like this: We know that the implication "If \ (A \) then \ (B \)" is true. The statement \ (A \) is one of our premises. So \ (B \) is also true, and we can use \ (B \) to prove further statements.