Important note: On the problems that require computations, the steps to find the answer must be included in your submission. An answer with little or no explanantion will receive little or no credit.
- Give a (reasonable) lower bound of the number of cuneiform tablets that have been found. List your sources.
- When and where (approximately) did the first mathematical proof we know of appeared?
- Explain the Proposition I of Euclid's Elements in your own words. 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.
- Choose one proposition of the Elements, not discussed in class or in this homework. Read it, understand it and write down the proof in your own words. You can assume everything that has been proved before that proposition.
- Learn and understand definition of commensurability, incommensurability, axiom (recall that Euclid divided axioms into “common notions” and “postulates”.)
- 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 succesufully, you would have produced a quadrature of an isosceles triangle!)
- On the Proposition 14 of Book 2 of Euclid Elements a construction for squaring a “ rectilinear figure A" is given. Perform the construction in this Geogebra activity. starting with a rectangle BCDE (instead of a general rectilinar figure A). If you follow this translation, start in the fourth sentence of teh proof (“Then, if BE equals ED, then that which was proposed is done, for a square BD has been coequalnstructed equal to the rectilinear figure A.”). Construct a square of side length ED, and check in Geogebra using the Area tool that the areas of rectangle BCDE and the square of side ED are equal. Send a screenshot to grade.
Sample Quiz 3
- Explain the Proposition I of Euclid's Elements in your own words. 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.
- 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 succesufully, you would have produced a quadrature of an isosceles triangle!)
- Show that √5 is incommensurable with 1.
- State the Pythagorean theorem (with words, no points or segment labels and no pictures)