of History Math Number: 369
Math 331
Project 1- Proofs Galore
A proof is a combination of statements and ideas that can be put together to prove a mathematical idea. Proofs can be long and complicated, but sometimes they are not nearly so long or complicated. In my experience, the length and difficulty of the proof is tied to the difficulty of the mathematical idea. To put together a proper proof, one must find and use other statements and ideas that can explain how to get from "point a" to "point b" and in the end you should prove whatever it is you are trying to prove so that someone who has never seen or heard anything about that mathematical idea could follow your steps and understand how you reached your conclusion. For most people I think non-proof based mathematics is a simpler way to solve ideas. Non-proof based mathematics is more a way of using intuition to solve something where proof based mathematics uses detailed step but step ideas to get to the end goal. I think the best example of this is the Greeks and the Babylonians and their ways of dealing with the Pythagorean Theorem. The Babylonians, as far as we know, don't have the exact proof of the Pythagorean
Theorem. They seemed to know that it worked and they had lists of the Pythagorean triples, but it doesn't seem that they had the detailed proof that the
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Greeks had. [E. Chapter 3] The Greeks seemed to use proof based mathematics
In almost every aspect they studied, but they did extensive work on the
Pythagorean Theorem from Pythagoras all the way to Euclid. They had a long and detailed proof for the Pythagorean Theorem that not only showed their mastery of the proof, but that they also understood exactly why it worked. [E. Chapter 3 and Chapter 5] I think the Greeks were able to be as advanced as they were because they were so diligent about using proofs. The Greeks were able to prove so much about algebraic identities, geometry, conies, the calculation of pi, volume, area, etc. [E. Chapter 4 and Chapter 5] I think that proof based mathematics can sometimes become more complicated because it can be so easy to miss one or two steps. You know something works, but you can't quite tie all of the pieces together, so you are stuck until you can do that. I'm sure that created problems for the Greeks at some point, especially since they didn't have all of the technological conveniences we are able to enjoy today to help them. The proof I chose to work on was from section 5.8. [E Chapter 5] It stated: the "Assuming equality of alternate interior angles formed by a transversal cutting a pair of parallel lines, prove that the sum of the angles of a triangle is equal to a straight angle." I will start by explaining the assumption.
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The assumption states that in the above drawing angles 1 and 3 are equal to each
other and angles 2 and 4 are equal to each other also based on the transversal that was drawn through the two parallel lines. For the next step I just want to remind everyone what a "straight angle" is. It is an angle that equals 180 degrees as diagrammed below:
For the next few steps, please refer to the drawing below:
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Since we can see that two of the legs of the triangle (lines AC and AB) represent transversals of two parallel lines, we can then conclude that angles 1 and 4 are equal to each other and angles 2 and 5 are equal to each other. We can also now see that angles 1, 2, and 3 make a straight angle when they are put together. Therefore, angles 1 + 2 + 3 = 180 degrees. We already showed that: Angle 1 = Angle 4 and Angle 2 = Angle 5.
Now we can substitute 4 and 5 in where 1 and 2 were.
Hence, angles 3 4 - 4 + 5 = 180. Since those are the three angles inside triangle ABC, we have proven that the three angles inside the triangle equal 180 degrees
I don't think I would've been able to solve this problem without proving it.
I have known for many years that the interior angles of a triangle equaled 180 degrees but u...