Homework 1 from Lecture 1 to Lecture 10

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1 Homework from Lecture to Lecture 0 June, 207 Lecture. Ancient Egyptians calculated product essentially by using additive. For example, to find 9 7, they considered multiple doublings of 7: Since 9 = , we can calculate Here 349 = = = ( )7 = 9 7. Calculate 8 25 by using the same method as above.

2 2 2. Ancient Egyptians calculated division essentially by using the reverse of additive. For example, to find 9 7, they considered multiple doublings of 7: Here ( )7 = = 9 so that 9 7 = 3. The above example may be too simple, and division can be more complicated. For example, to find 35 8, they not only considered multiple doublings of 8, but also considered a number of cutting by half : Hence 35 8 = Calculate 7 8 by using the above method. 3. Make a table for the values of the expression n 3 + n 2 where n =, 2, 3,..., 0. Use this table to solve the cubic equation 44x 3 + 2x 2 = 48. Lecture 2. Ancient Greeks classified numbers as triangular, square, pentagonal, and so on, according to the shapes made by the arrangement of the dots. This idea goes back at least to the golden age of Greek mathematics ( B.C.). A triangular number counts the objects that can form an equilateral triangle. For example, T =, T 2 = 3, T 3 = 6,...,. In general, n n(n + ) T n = j = 2 Hint: Multiply the given equation by 2. j=

3 3 Prove that if T n is a triangular number, so is 9T n Verify that 225 and 4, 66 are simultaneously square and triangular numbers. 2 Lecture 3. From Figure 3.3 of the lecture notes, suppose Thales found that at the time a stick of length 6 feet cast a shadow of 9 feet, there was a length of 342 feet from the edge of the pyramid s side to the tip of its shadow. Suppose further that the length of a side of the pyramid was 756 feet. Find the height of the pyramid. 2. Thales is said to have invented a method of finding distance of ships from shore by the use of the angle-side-angle theorem. Here is a possible method: Suppose A is a point on shore and S is a ship. Consider a line segement AC which is perpendicular to ASwith the midpoint B. Draw CE at right angles to AC and pick point E on it in a straight line with B and S. Show that EBC = SBA and therefore that SA = EC. A E C B S Lecture 4. James Garfield was the 20th president of the United States. Before becoming president, he found a proof of the Pythagorean theorem, published in the New England Journal of Education in 876. Search James Garfield and Pythagorean theorem in Google to read about his proof. Understand and write down his proof. 2. Prove that 3 is an irrational number. 3. By Pythagorean triple (a, b, c), we mean that it satisfies a b c with a 2 + b 2 = c 2. Pythagoras and the Babylonians gave a formula for generating Pythagorean triples as (2m, m 2, m 2 + ). Prove this formula. 2 Hint: Finding an integer n such that T n = n(n+) 2 = 225 is equivalent to solving a quadratic equation.

4 4 4. The following formula was discovered by Archimedes ( B.C.): n 2 = Prove it (by mathematical induction). n(n + )(2n + ). 6 Lecture 5. Suppose that Achilles runs 0 times as fast as the tortoise and that Achilles runs 0 yards a second. Suppose that the initial distance between Achilles and the tortoise is 00 yards. Then Achilles runs successively 00 yards, 0 yards, yard, yard, 0 and so on. Find the total number of yards Achilles must travel in order to catch his competitor and find the total number of yards the tortoise must travel, by using the geometric series that you learned in Calculus. Lecture 6. The Greek mathematician Menaechnus (350 B.C.) obtained a purely theoretical solution to the duplication problem based on finding the point of intersection of certain conic sections. In modern notation, he drew two curves x 2 = ay and y 2 = 2ax. Then he found two intersection points, (0, 0) and (x, y). Show this x gives an answer to the doubling the cube problem. Lecture 7. Read Socrates in the Appendices. 2. Read the entire passage from Plato s Meno. Here is some information about Plato s Meno: Meno is a Socratic dialogue written by Plato, which is one of Plato s earliest dialogues. See for the whole Meno. For some analysis, summary and information on Meno, see 3. Answer the following question: What is Socrate s method of convincing the slave boy that he knows how to construct a square double that of a given square?

5 5 Lecture 8. We review from Eudoxus definition: Magnitudes are said to be of the same ratio, the first to the second and the third to the fourth, when, if any equimultiples whatever be taken of the first and the third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or are alike less than the latter equimultiples taken in corresponding order. Equivalently, Eudoxus definition is that a : b = c : d means the following: if na > mb, then nc > md; if na = mb, then nc = md; if na < mb, then nc < md, for any positive integers m and n. In today s notation, a : b = c : d means the following: If a > m, then c > m; if a = m, b n d n b n then c = m; if a < m, then c < m for any positive integers m and n. d n b n d n Let us recall some Propositions in Book V, Eucilid s Elements: Proposition 5: a : b = ma : mb; Proposition : If a : b = c : d and c : d = e : f, then a : b = e : f; and Proposition 4: If a : b = c : d and a >=< c, then b >=< d. Use the above Propositions and Eudoxus s definition to prove the following statement, which was in Proposition V-6 in his work: If a : b = c : d, then a : c = b : d. Lecture 9. Read the internet article: 2. The argument if all monkeys are primates, and all primates are mammals, then it follows that all monkeys are mammals, exemplifies one type of syllogism, whereas the argument if all Catholic are Christians and no Christians are Muslim, then it follows that no Catholic is Muslim, exemplifies a second type. Give two more examples of each of the two types of syllogisms mentioned above. 3. The negation of statement p is not p. For example, for the statement The number 9 is odd, its negation is the number 9 is not odd. Another example, for the statement A can buy anything, its negation is there is at least one thing such that A cannot buy it.

6 6 Consider the following statement: given a sequence {a n }, there exists a positive number M, such that a n M for any n. What is the negation of the above statement? Lecture 0. Take a look at Euclid s Elements in the appendices. 2. The following is Proposition 35 from Book I of Euclid s Elements: Two parallelograms that have the same base and lie between the same parallel lines are equal in area to one another. Prove this statement. 3. Is there a prime number that is the largest? The answer is no. The proof was given by Euclid (Proposition 20, Book IX) in his Elements. By Euclid s argument with the modern notation, we write the primes 2, 3, 5, 7,... in ascending order. For any prime p, it is sufficient to prove that there is a new prime q such that q > p. To prove this, write N = ( p) +. If N is prime, we are done. Otherwise, N is divisible by some prime number q. Prove that q > p.

Greece In 700 BC, Greece consisted of a collection of independent city-states covering a large area including modern day Greece, Turkey, and a multitu

Chapter 3 Greeks Greece In 700 BC, Greece consisted of a collection of independent city-states covering a large area including modern day Greece, Turkey, and a multitude of Mediterranean islands. The Greeks