MATH3560/GENS2005 HISTORY OF MATHEMATICS

Transcription

1 THE UNIVERSITY OF NEW SOUTH WALES SCHOOL OF MATHEMATICS OCTOBER/NOVEMBER 2008 MATH3560/GENS2005 HISTORY OF MATHEMATICS (1) TIME ALLOWED 2 Hours (2) TOTAL NUMBER OF QUESTIONS 19 (3) CANDIDATES SHOULD ATTEMPT QUESTIONS WORTH 100 MARKS (4) THE QUESTIONS ARE NOT OF EQUAL VALUE (5) THIS PAPER MAY BE RETAINED BY THE CANDIDATE (6) CALCULATORS WILL BE PROVIDED All answers must be written in ink. Except where they are expressly required pencils may only be used for drawing, sketching or graphical work.

2 OCTOBER/NOVEMBER 2008 MATH3560/GENS2005 Page 2 Please see over...

3 Please see over... OCTOBER/NOVEMBER 2008 MATH3560/GENS2005 Page 3 1. [10 marks] i) Use the Babylonian method of finding square roots to obtain first and second rational approximations to 12. ii) Square your approximation to see how close it is to [10 marks] i) Use the Egyptian method of doubling to multiply 37 by 41. ii) Use the Egyptian method to find [10 marks] Let x = i) Express x in (modern) sexagemisal notation x = 0; s 1, s 2. ii) Which ancient civilization used sexagemisal notation? iii) Express x as a sum of unitary fractions. iv) Which ancient civilization represented fractions in this form. 4. [10 marks] Let p n denote the nth pentagonal number. i) Draw a diagram illustrating the first 3 pentagonal numbers. ii) Write down the first 4 pentagonal numbers. iii) Use your diagram to write down a recurrence for p n iv) Assuming that the formula for p n is given by a quadratic p n = an 2 +bn+c, find a, b, c. 5. [10 marks] Cardano ( ) had a method for solving cubics. In this question we shall employ a variant of Cardano s method to solve the cubic x 3 6x = 9. (1) i) Use the substitution x = u + v to convert the cubic equation (1) to the form (u 3 + v 3 ) + 3uv(u + v) = 9 + 6(u + v). (2) ii) Equate terms on both sides of (2) to produce values for u 3 + v 3 and uv. iii) Let α = u 3 and β = v 3. Use the fact that we know the numerical values of α+β and αβ to write down a quadratic which has α and β as its roots. iv) Solve this quadratic. Hence find u and v, and finally, x.

4 Please see over... OCTOBER/NOVEMBER 2008 MATH3560/GENS2005 Page 4 6. [10 marks] For many centuries people tried to prove Euclid s fifth postulate until it was realised that it truly was independent of Euclid s other postulates. i) State Euclid s fifth postulate, either in the original form, or else in a more recent equivalent form. ii) Write half a page giving some of the history surrounding this postulate. 7. [10 marks] Let ABCDE be a regular pentagon with side 1. i) Calculate the angles EAB and EAD. Let v = 2 cos π 5. ii) Use the cosine rule in DAC to show that and hence calculate cos π 5. iii) Is the angle π 5 constructible? v 3 2v = 0, 8. [10 marks] There is no rational number x for which x 2 = 2. i) Prove this statement. ii) When was this result first proved? Comment on the historical significance of the proof. 9. [10 marks] Pierre de Fermat lived from 1601 to i) Give 3 areas of mathematics to which Fermat made significant contributions. ii) State Fermat s Last Theorem and Fermat s Little Theorem. iii) Did Fermat give proofs of either of these results? 10. [10 marks] Al-Karkhi (ca. 1020) found a family of rational solutions to x 3 + y 3 = z 2. He took x = with n, m natural numbers. n2, y = mx, z = nx, 1 + m3 i) Show that these formulae do indeed give solutions. ii) Find a rational solution (x, y, z) which is not generated by this formula.

5 Please see over... OCTOBER/NOVEMBER 2008 MATH3560/GENS2005 Page [10 marks] Nasr ad-din al-tusi ( ) is said to have been the first to show that the sum of two odd squares cannot be a square. Let x = 2n + 1 and y = 2m + 1 for two natural numbers n and m. i) Calculate x 2 + y 2 mod 4. ii) Hence or otherwise prove Nasr ad-din al-tusi s result. iii) Give an example of two even squares whose sum is a square. 12. [10 marks] We follow Archimedes in approximating π. i) Inscribe a regular hexagon inside a circle of radius r. What is the perimeter of the hexagon? ii) Do the same with a regular dodecagon. What is its perimeter? Thus, give a lower bound for π. 13. [10 marks] i) Explain the golden rectangle and the golden ratio τ. ii) What are the relationships between τ, τ 2 and 1 τ? iii) How is τ related to the Fibonacci numbers? 14. [10 marks] Suppose a 2 + b 2 = c 2 and let x a y = b. z c i) Show that x 2 + y 2 = z 2 and y x = b a. ii) Starting with (a, b, c) = (5, 12, 13), find two more Pythagorean triads with the difference between the shorter sides equal to [10 marks] i) Define carefully what is meant by Platonic solid. ii) List all the Platonic solids, giving a description of each (give the type of face, number of faces, edges and vertices of each). iii) Explain the duality of the Platonic solids. Which of them is self dual?

6 OCTOBER/NOVEMBER 2008 MATH3560/GENS2005 Page [10 marks] i) (Roughly) when did Évariste Galois live? ii) How did Galois die? iii) What famous problems did Galois Theory solve? 17. [20 marks] Write a short essay (around 500 words) about Arabic mathematics and its influence on later European mathematical development. 18. [20 marks] Write a short essay (around 500 words) about the mathematics of the Babylonians, Egyptians and Ancient Greeks. You should explain i) how we know about this mathematics. ii) the major differences between the styles of mathematical documents between these cultures. 19. [20 marks] In about 500 words, write a short summary of the ideas in your essay.