Homework 4

Reminder: You can (and are encouraged to) discuss problems with your classmates. Then write down the answers by yourself.

In problems 1, 2 and 3, you need to explain the proof of the propositions from Euclid’s in your own words. Try to minimize the number of symbols. (Versions of Euclid’s elements can be found in the Useful Links)

  1. Proposition I.1 of Euclid's Elements. For this proposition, do not use labels of points of segments (For instance, you can write, give a segment, draw a circle with center one of its endpoints and…") Explain whether there is any gap in the proof, from a modern point of view.
  2. Proposition I.47, Pythagorean Theorem (In lecture 1, we will discuss it on Tuesday Oct 26)
  3. One one proposition of Euclid’s Elements of your choice.
  4. Divide isosceles triangle with base of length 8 and height of length 9 into polygonal pieces from which a square can be assembled. What is the side length of this square? (You can use cm or any other unit of our choice) “Make” your solution in paper or Geogebra. Explain how you obtained solution and provide photos of the pieces you made or in Geogebra. (If you complete this successfully, you would have produced a quadrature of an isosceles triangle!)
  5. Check that the Euler characteristic of a pyramid with pentagonal base is 2.
  6. Euclid’s Element influenced at least two US presidents. One is mentioned here. Find both of these presidents and write a short paragraph describing how they were influenced by the Elements.
  7. In the proof of "Proposition IX.20: Prime numbers are more than any assigned multitude of prime numbers.", given three primes a new one is defined. Find four concrete examples of this procedure, assuming that one of the given primes is 41 and the other one is 2. (You need to choose the third prime).
  8. Explain when and where (approximately) did the first mathematical proof we know of appeared.

    9. Sample quiz 4

    1. Proposition I.1 of Euclid's Elements. For this proposition, do not use labels of points of segments (For instance, you can write, give a segment, draw a circle with center one of its endpoints and…") Explain whether there is any gap in the proof, from a modern point of view.
    2. Divide isosceles triangle with base of length 8 and height of length 9 into polygonal pieces from which a square can be assembled. What is the side length of this square? (You can use cm or any other unit of our choice) . Make a drawing of your solution.
    3. Check that the Euler characteristic of a pyramid with pentagonal base is 2.
    4. In the proof of "Proposition IX.20: Prime numbers are more than any assigned multitude of prime numbers.", given three primes a new one is defined. Find four concrete examples of this procedure, assuming that one of the given primes is 41 and the other one is 2. (You need to choose the third prime).