A mathematical proof is a logical argument that demonstrates the truth of a mathematical statement. It uses definitions, axioms, and previously established theorems to show that a conclusion follows necessarily from the premises. Proofs are essential in mathematics because they provide certainty and validation for mathematical concepts and results. There are various types of proofs, including direct proofs, indirect proofs, and proof by contradiction. Each method has its own approach to establishing the validity of a statement. Mathematicians rely on proofs to ensure that their findings are accurate and universally accepted within the mathematical community.