230hw1 - n to be not divisible by 3(b Use the method of...

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Math 230 E Fall 2011 Homework 1 Drew Armstrong Problem 1. In the Lilavati , the Indian mathematician Bhaskara (1114–1185) gave a one-word proof of the Pythagorean theorem. He said: “Behold!” Add words to the proof. Your goal is to persuade a high school student who claims he/she doesn’t “get it”. Try to avoid algebra. (Sorry, the two pictures are not quite to scale.) Problem 2. Prove that the interior angles of any triangle sum to 180 . You may use the following two facts without justification. Fact 1: Given a line and a point p not on , it is possible to draw a line through p parallel to . Fact 2: If a line falls on two parallel lines, then the corresponding angles are equal, as in the following figure. Problem 3. Prove that 3 is not a fraction, in two steps. (a) First prove a lemma. Given a whole number n , if n 2 is a multiple of 3, then so is n . (Hint: Use the contrapositive, and note that there are two ways for
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Unformatted text preview: n to be not divisible by 3.) (b) Use the method of contradiction to prove that √ 3 is not a fraction. Quote your lemma in the proof. Problem 4. Use the 2D Pythagorean Theorem to prove the 3D Pythagorean Theorem. That is, prove that the distance between points (0 , , 0) and ( x,y,z ) equals p x 2 + y 2 + z 2 . (Hint: There are two triangles involved.) Problem 5. The dot product of vectors u = ( u 1 ,u 2 ,...,u n ) and v = ( v 1 ,v 2 ,...,v n ) is defined by u · v := u 1 v 1 + u 2 v 2 + ··· + u n v n . The length k u k of a vector u is defined by k u k 2 := u · u . (a) Prove the formula k u-v k 2 = k u k 2 + k v k 2-2 ( u · v ). (b) Use this formula together with the 2D Pythagorean Theorem and its converse to prove the following statement: “the vectors u and v are perpendicular if and only if u · v = 0.” (Hint: Where is the triangle?)...
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