## Pennies For Proofs Scholarship

### About Mathematics and Proof

Mathematics has two great traditions: computation and proof. Most of your mathematics studies have been devoted to learning computation skills. Calculus is, for example, one of the finest computation tools we have. If you decide to pursue any of a number of science or engineering degrees at UAF, Calculus II is a required class and you will want to take it early in your studies.

Proofs are different. The goal of a proof is not to arrive at a number, but to show that a mathematical fact is true. For example, the early Egyptians had a rule of thumb that a triangle with side-lengths 3, 4, and 5 has a right angle. By contrast, the early Greeks had a proof that the triangles with right angles are exactly the triangles where the side lengths satisfy a2 + b2 = c2 (Euclid, Book I, Propositions 47 and 48).

Here are some things you can show using proofs:

• At any given time, there are two exactly opposite points on the Earth where it is exactly the same temperature.
• It is impossible to comb the hairs on a sphere so that all the hairs lie flat.
• Fundamental Theorem of Algebra: every polynomial equation has at least one complex solution. Maybe someone mentioned this to you in an algebra class. Is it true? How would you know?

In Math 215, Introduction to Mathematical Proofs, you learn the skills needed to put a proof together and to analyze someone else's proof. These are the skills you need to have before tackling any of results mentioned above. By learning about this side of mathematics, you'll gain a better understanding of all your mathematics courses.