Mathematical Proof Research Paper

Mathematical proof is strict and rigorous, however by using diagrams and graphs, it can also be beautiful. Some mathematical identities and relationships can be proven without words, which are simple and logical. Like Martin Gardner wrote, "There is no more effective aid in understanding certain algebraic identities than a good diagram. One should, of course, know how to manipulate algebraic symbols to obtain proofs, but in many cases a dull proof can be supplemented by a geometric analogue so simple and beautiful that the truth of a theorem is almost seen at a glance."

One example is about Pythagorean Theorem. The Pythagorean Theorem can be proved simply by the following diagram. From the diagram (pic.1) below, we can easily find out that

I chose this topic because it helps answer several concerns that arise at attempts to teach and to learn about proofs. Through a diagram that made the statement obvious, the result may be sensed or discovered intuitively. Hence it helps the viewer internalize the idea by gaining an insight into why the idea was correct, and it makes more discernable relationships between parts or parameters of a mathematical statement. Proof without words can be one step proof, or even a proof that does not start with fundamental axioms. It is very effective not only for learning mathematical statements, but also for developing a feeling for mathematics as a discipline. In the project, I came up with my own proofs with the aid of elegant diagrams. Proof by simple diagrams is not only beautiful and effective, it also grabs the interest of the beginners in

There is lower chances to make mistakes with the aid of diagrams, while careless mistakes may happen quite frequently in tedious intermediate steps of an algebraic proof. However, transforming formulas into graphs challenges us during our research.

Eventually, I have overcome these problems along the way. By reading some books, and doing some research. I successfully proved some mathematical identities by constructing effective diagrams. Through the study, I also developed the interests and senses for mathematics.

In conclusion, I have learnt how to illustrate the statements resorting to diagrams, and am able to create simple diagrams to seek for further proofs. Furthermore, I use different diagrams to prove one statement, and some general strategies and results are also concluded in the discussion of the proof. In the end, I have achieved the aim of the project, which is learning