Student 673 Math 331 Fall Project 1

Transcription

1 Student 673 Math 331 Fall 2017 Project 1 Proofs show that a given math equation or theorem is true by breaking down and showing the logical steps it would take to reach the conclusion that is being proven. This can be done by using reasonable assumptions and/or previously proven mathematical rules. The alternative to this is intuition-based math; using what can be observed about mathematical principles and making assumptions based on these observations. The two may seem similar at first, but a proof has a much more rigid structure it must adhere to. Greek mathematics was highly proof based. An example of Grecian proof based mathematics lies in their method of multiplication. Multiplication in the Greek mathematical system was accomplished by using the distributive law, which shows that multiplying a number by a group of numbers added together is the same as doing each multiplication separately. In contrast, the Egyptians method for multiplying two numbers was accomplished by multiplying/dividing by two and adding. They used things such as number tables for multiplication and division to make math easier. The steps to geometrically prove that n 3 = ( n) 2 are detailed below. This proof is relatively simple compared to some of the other problem studies featured in this class; the proof itself could be said to be done in an intuitive manner. This form of proof is likely easier to understand than a more formulaic proof method, such as a proof by induction. Proof: Arrange n squares that incrementally increase in size (1x1, 2x2, 3x3,..., nxn) diagonally on a grid. This grid will be ( n) x ( n). The area of this grid is equal to ( n) 2, which is the right hand side of the equation. The example below is done for n = 5, but the same exercise could be completed for any positive integer n given enough space.

2 Next, fill in the grid with 1 1x1 square, 2 2x2 squares, 3 3x3 squares, etc. up to n nxn squares.* This can be written as 1 2 * * n 2 * n. This can be reduced down to n 3, which is the left hand side of the equation. Again, the example below is done for n = 5, but the same exercise could be completed for any positive integer n given enough space.

3 The areas of the two grid squares are the same, therefore it has been geometrically proven that n 3 = ( n) 2 *The rectangles of the same color can be combined to form the last needed square for that set (as seen in the 2x2s and 4x4s)

4 Sources: Eves, Howard Whitley. An Introduction to the History of Mathematics. Saunders College Publishing, Emerson, N. Greek Math, faculty.csuci.edu/roger.roybal/teaching/fall17/math331/handout3-greeks.pdf. Bellomo, Carryn. Greek Mathematics. UNLV Math, University of Nevada, Las Vegas Department of Mathematical Sciences, faculty.unlv.edu/bellomo/math714/notes/10_greek.pdf. Gilmore, Maurice E. Egyptian Mathematics. NU Department of Mathematics, Northeastern University Department of Mathematics, 2008, mathserver.neu.edu/~gilmore/u201sp08files/egypt.pdf. Pierce, Rod. "Definition of Distributive Law" Math Is Fun. Ed. Rod Pierce. 11 Jan Oct 2017,

5 Explain in clear, ordinary language the concept of proof. Do not copy a definition, but explain the concept. Discuss how proof based mathematics differs from non-proof based mathematics? Give at least one historical example. How was Greek mathematics helped by using proofs? How was it hindered? Give an example of each. Include lots of mathematical detail. Give a detailed example of a proof; it must be the problem from Problem Set #3 that you worked on. You do not have to include a complete proof, but you must include all major steps. What are the advantages of solving this problem by proof, instead of by intuition? What are the disadvantages? Be specific. For a typical student, would it be better to solve your problem intuitively or by proof? Why?