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Unformatted text preview: Lecture 2: Is an algorithm sufficient to solve a problem? Daniel Frances c 2012 Contents 1 Introduction 2 2 Fibonacci numbers and the Golden Ratio 2 3 Computing Fibonacci Numbers 3 4 Exponential Algorithms 4 5 Conclusion 5 1 1 Introduction Professor Scott Rogers, now retired, asked me once about the introduction of the decimal numerical system into Europe. At first I did not understand, didn’t we always know that 2x2=4 and that 972x345=335,340? Apparently not. Have you tried to multiply with Roman Numerals CMLXXII times CCCXLV? It’s not easy. Hopw did they calculate interest on loans? Maybe that’s why usury was forbidden? Apparently only a few ”spets” could do such advanced math. It turns out that our numerical system was invented in India, transferred to the Middle East, unto North Africa, and introduced into Europe through the conquest of the Iberian peninsula (Spain and Portugal) by the Moors. I happened to have read a book published in Spain around the year 1000, praising the powers of this new ”decimal system” with which anyone could perform the most advanced arithmetic. What was so new about the system was a new symbol “0” (zero), and the significance of the location of each digit. The book had six chapters: Addition, subtraction, multiplication, division, powers and square roots. What was provided was a new algorithm , a new method, for getting something done more efficiently than before. You may want to explore how you would compute CMLXXII times CCCXLV, without using the decimal system. The question here is: Does having an algo rithm for solving a problem mean the problem is solved? To investigate this question we will need an example, which requires a brief introduction. 2 Fibonacci numbers and the Golden Ratio Consider a sequence of numbers 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, ... By now you should see the pattern F n = F n 1 + F n 2 . This sequence of numbers is called the Fibonacci sequence, and F n is the n th Fibonacci number. It turns out these numbers appear in many different contexts: in the shapes of flowers, in optimization search methods, and search “Fibonacci And The Golden Ratio” in Investopedia for their use by investment traders.“For reasons that are unclear, these ratios seem to play an important role in the stock market, just as they do in nature, and can be used to determine critical points that cause an asset’s price to reverse.”. Figure 1: Golden Ratio For interest, these numbers are also closely related to the Golden Ratio, which arises often 2 in relation to Fibonacci numbers, although the relation is not immediately apparent. If you take any length and split it according to the figure, then the ratio a/b is an irrational number often denoted as a b = φ = 1 . 61803 ... . This is not a mystery....
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This note was uploaded on 03/31/2012 for the course MIE 335 taught by Professor Frances during the Spring '12 term at University of Toronto.
 Spring '12
 Frances

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