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What is mathematical proof?

A proof is a mathematical argument that shows that something is absolutely true, without a doubt. Mathematicians begin their process of mathematical inquiry by suggesting that something might be true. This is called a conjecture. Then they must prove that it is true all the time.

Why is proof important?

Certainty in mathematics is vital and it's one of the aspects that students most love about the subject. Either it's right or it's wrong. Mathematical proof is the way mathematicians prove unequivically that what they have proposed is correct.

What types of proof are there?

Basically there are two types of proof:

Direct Proof

Direct Proof

A direct proof uses a sequence of logical steps to arrive at its conclusion. All steps must be based on proven facts not assumptions. Additionally we cannot use what we are trying to prove as part of the proof.

There are several methods:

  • Proof by deduction
  • Proof by exhaustion (cases)
  • Proof by induction
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Indirect Proof

Indirect Proof

Indirect proof arrives at the conclusion by an alternate path, such as:

  • Proof by contradicton
  • Proof by contrapositive
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Hilbert's Problems

David Hilbert published twenty-three mathematical problems in 1902, many of which still remain unproven today. These formed the basis for the Millenium problems, proposed at the end of the 20th century. The proofs are each worth $1 000 000 USD to the solver.