A Little History of Some Familiar Mathematics

Transcription

1 A Little History of Some Familiar Mathematics Gary Talsma Mathematics and Statistics Department Calvin College Grand Rapids, MI First-Year Seminar November 30,

2 I. The Quadratic Formula(s) A. Muhammad ibn Musa al-khwarizmi (ca ) 1. Persian mathematician 2. Worked at the Bayt al-hikma (House of Wisdom) in Baghdad 3. Books a) Algoritmi de numero Indorum (Al-Khwarizmi on the Hindu Art of Reckoning) Explained the basics of the decimal number system from India This Latin translation was later very influential in popularizing Hindu- Arabic numeration in Europe Our word algorithm originated with al-khwarizmi s name in the title b) al-kitab al-mukhtasar fi hisab al-jabr w'al-muqabala (The Compendious Book on Calculation by Restoring and Comparing) Al-jabr: moving a subtracted quantity to the other side of an equation Al-muqabala: applying operation to both sides of an equation Translated into Latin in 1140 as Liber algebrae et almucabala, from which our word algebra comes B. al-khwarizmi s classification of quadratic equations 1. Involved quantities he called squares, roots, and numbers 2. Negative quantities were not considered; all coefficients and roots had to be positive (that s where al-jabr was useful) 3. Approach was purely rhetorical no algebraic symbolism, everything (even the numbers!) written out in words 4. Six cases Case 1: squares equal to roots (ax 2 = bx) Case 2: squares equal to number (ax 2 = c) Case 3: roots equal to number (bx = c) Case 4: squares and roots equal to number (ax 2 + bx = c) Case 5: squares and number equal to roots (ax 2 + c = bx) Case 6: squares equal to roots and number (ax 2 = bx + c) al-khwarizmi presented a general method for each case, and a specific example; he also included a geometric proof of each method. al-khwarizmi s claim to fame was his set of quadratic formulas to solve the last 3 cases. 2

3 Al-Khwarizmi s Quadratic Formulas In modern symbolic notation, al-khwarizmi s 3 cases that include all three relevant quantities (squares, roots, and numbers) are: Case 4: square and roots equal number: x 2 + bx = c; Case 5: square and number equal roots: x 2 + c = bx; and Case 6: square equals roots and number: x 2 = bx + c Note that al-khwarizmi prescribes that we always start by making the coefficient of the squared term 1: Whether there are many or few squares, they will have to be reduced in the same manner to the form of one square However many squares are proposed all are to be reduced to one square. Similarly also you may reduce whatever numbers or roots accompany them in the same way in which you have reduced the squares. That step has already been taken for the equations shown at the top of this page, compared to the corresponding cases on the previous page. Below are 3 sets of instructions al-khwarizmi gives for solving these cases (translated into English for those of you who, like me, don t read Arabic). Translate each rule into an algebraic formula involving the coefficients b and c from the equations above, and match the formula to the appropriate case. a) Halve the number of roots; multiply this by itself; subtract from this the number, and take the square root of the difference; subtract this from, or add it to, half the number of roots. Formula: x = Equation: Case b) Halve the number of roots; multiply this by itself; add this to the number, and take the square root of the sum; subtract from this half the number of roots. Formula: x = Equation: Case c) Halve the number of roots; multiply this by itself; add this to the number, and take the square root of the sum; add this to half the number of roots. Formula: x = Equation: Case 3

4 Al-Khwarizmi s Geometric Proofs of His Quadratic Formulas 4

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6 II. The Pythagorean Theorem The Babylonian Box We ve just seen how this was used to justify one of al-khwarizmi s quadratic formulas. We ve probably all used it as a visual representation of (a + b) 2 = a 2 + 2ab + b 2. How can it be used to prove the Pythagorean Theorem? There s not a right triangle in sight! Does this seem more promising? The square has the same dimensions as the one above. Let s say each side has length a + b. This diagram, along with the algebra above, is often used to prove the Pythagorean Theorem. Can you see how to use the two diagrams together to produce an algebra-free proof? 6

7 If we take the previous diagram and fold in the corner triangles, we get this diagram. The 12 th century Indian mathematician Bhaskara used this diagram to prove the Pythagorean Theorem. One can use an algebraic analysis of the diagram: c 2 = 4(ab/2) + (a b) 2 = 2ab + a 2 2ab + b 2 = a 2 + b 2 But I like the geometric, or dissection, approach: slide the upper right triangle to the lower left, and slide the upper left triangle to the lower right to produce this diagram. Can you see a square of side a and a square of side b sitting there? 7

8 How did the Pythagoreans prove the theorem? (Please label the intersection of AB and CJ as K) Find 3 similar triangles. Use them to set up 2 proportions that can be used to express a 2 and b 2 in terms of c and x. Explain how this shows that a 2 + b 2 = c 2. How did Euclid prove the Pythagorean Theorem (Elements I.47)? Show that the area of triangle AGB is half the area of the square on side b. Show that the area of triangle AHC is half the area of AHJK. Show that the shaded triangles AGB and AHC are congruent. What can we conclude about the areas of the square on side b and AHJK? Use a similar approach to show that the area of BIJK is a 2. Explain how this all shows that a 2 + b 2 = c 2. 8

9 Liu Hui, a Chinese mathematician of the 3 rd century AD, wrote a commentary on the Jiuzhang Suanshu (Nine Chapters on the Mathematical Arts, originally published in the first century BC) that came to be considered part of that book. He provided explanations and brief justifications for the rules in the Nine Chapters, taking it beyond a mere book of recipes. This diagram resembles the one Liu Hui used to justify the Gougu rule (what the Chinese called the theorem). a) Show that all the triangles in the diagram are similar to one another. b) Find 3 triangles that are inside the squares on the legs of triangle ABC but outside the square on the hypotenuse of ABC. c) Find 3 triangles that are outside the squares on the legs of triangle ABC but inside the square on the hypotenuse of ABC. d) Show that each triangle from part b) has a congruent partner from part c). e) Explain how this shows that a 2 + b 2 = c 2. 9

10 III. Archimedes Estimate of π Archimedes used the method of exhaustion to find his approximation for π: he trapped the circumference of a circle of radius r between the perimeters of inscribed and circumscribed regular polygons with increasingly larger numbers of sides: P inscribed < 2"r < P circumscribed (Archimedes is often credited with foreshadowing integral calculus, since the method he used is very similar to the approach used in developing the Riemann integral, where the area under the graph of a function is! trapped between sums of rectangles under the graph and rectangles over the graph with increasingly more (and skinnier) rectangles.) a) Archimedes started by using regular hexagons inscribed and circumscribed on a circle of radius r. Without loss of generality, and to simplify our calculations, let s use a unit circle with r = 1 in what follows. Show that for this unit circle the perimeter of the inscribed hexagon is 6 and the perimeter of the circumscribed hexagon is 4 3, and that this leads to an interval estimate of 3 < " < 2 3 # (See Figure a) on next page.)! b) We have an advantage over Archimedes, because we know trigonometry and he didn t. Use trigonometric methods to show that the length of one side of a regular n-gon inscribed in a unit circle is 2sin(180/n), and thus " n sin 180 % $ ' < (. (See Figure b) on next page.) # n & c) Similarly, use trigonometric methods to show that the length of one side of a regular n-gon # circumscribed! around a unit circle is 2tan(180/n), and thus " < ntan 180 & % (. (See Figure c) on next $ n ' page.) d) Using amazing ingenuity and effort, given the number system and computational methods! available to him, Archimedes eventually found the perimeters of inscribed and circumscribed regular 96-gons, which resulted in his famous approximation for π: < " < 3 1. Use your results from 7 parts b) and c) to get an interval estimate for π based on n = 96. How does it compare with Archimedes approximation?! e) Use your results from parts b) and c) to generate interval estimates for π, taking n to be multiples of 100. How large must n be to get π accurate to 4 decimal places (3.1416)? 10

11 Figure a) Regular hexagons inscribing and circumscribing unit circle. How large are the central angles for each hexagon? If GC = 1, how long is DC, and what s the perimeter of the inscribed hexagon? What are the measures of the angles of triangle GDJ? If GD = 1, how long is DJ? How long is KJ? What s the perimeter of the circumscribed hexagon? If GC = 1, what s the circumference of the circle? Use the three perimeters you found to estimate π. Figure b) BC is one side of a regular n-gon inscribed in a unit circle with center A; D is midpoint of BC. What s the measure of angle BAD? If AB = 1, how long is BD? How long is BC? What is the perimeter of the regular n-gon? Use this perimeter to get an underestimate for π. Figure c) EF is one side of a regular n-gon circumscribing a unit circle with center A; D is midpoint of EF. What s the measure of angle EAD? If AD = 1, how long is ED? How long is EF? What is the perimeter of the regular n-gon? Use this to get an overestimate for π. 11