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An Idea
A mathematical proof starts with an idea or theory. Similar to the scientific method used by scientists to prove the validity of a hypothesis, a mathematical proof is used to verify a mathematical notion. Without proof to back it up, an algebraic statement does not have any weight behind it. The proof is what gives mathematicians the evidence they need to accept or deny a given principle.
Foundation of Math
The proof is an important foundation for the entire science of mathematics. According to professor Steven G. Krantz of Washington University in St. Louis, no other science relies on proof as much as math does. Other sciences have theories and then try to validate them, but theories that become scientific laws are very difficult to establish. In math, however, proof is the backbone on which the entire discipline is built. Concrete, irrefutable proof is a regular occurrence in the work.
How Proofs Are Written
Proofs are written using mathematical statements. Each adjustment to the equation through simple mathematics or algebra is completed and expressed as a new sentence in the proof. Reading a proof should be logical and make sense to anyone familiar with basic algebraic and mathematical principles, even if they are unfamiliar with the theory being tested. Details not critical to the outcome of the proof should be omitted, as a good proof should not be cluttered.
Uses
An algebraic proof has more uses than simply proving a theory to be true. They are also valuable teaching tools. Telling a student of mathematics that a given principle is true does not explain to him why it is true. The proof demonstrating the validity of the theory will explain why people believe that mathematical idea to be true. Proofs written by different mathematicians are a way for the science to grow, be shared and be reviewed, the way journal articles are in different disciplines. Proofs also are useful in disproving ideas by showing a statement is false.