284 CHAPTER 5 Number Theory and the Real Number System Fibonacci Numbers and the Golden Ratio In 1753, while studying the Fibonacci sequence, Robert Simson, a mathematician at the University of Glasgow, noticed that when he took the ratio of any term to the term that immediately preceded it, the value he obtained remained in the vicinity of one specific number. To illustrate this, we indicate in Table 5.2 the ratio of various pairs of sequential Fibonacci numbers. The ratio of the 50th term to the 49th term is 1.6180. Simson proved that the ratio of the n( 1) + term to the nth term as n gets larger and larger is the irrational number + ( 5 1)/2, which begins 1.61803.… This number was already well known to mathematicians at that time as the golden number. Many years earlier, Bavarian astronomer and mathematician Johannes Kepler wrote that for him the golden number symbolized the Creator’s intention “to create like from like.” The golden number + ( 5 1)/2 is frequently referred to as “phi,” symbolized by the Greek letter .Φ The ancient Greeks, in about the sixth century b.c., sought unifying principles of beauty and perfection, which they believed could be described by using mathematics. In their study of beauty, the Greeks used the term golden ratio. To understand the golden ratio, let’s consider the line segment AB in Fig. 5.11. Example 1 Determining a Fibonacci-Type Sequence Determine if the sequence is a Fibonacci-type sequence. If it is, determine the next two terms of the sequence. a) 2, 3, 6, 18, 108, 1944,… b) 1, 3, 4, 7, 11, 18,… Solution a) A Fibonacci-type sequence is a sequence in which each term, after the first two terms, is the sum of the two preceding terms. The third term, 6, is not the sum of the two preceding terms, 2 and 3. Therefore, this sequence is not a Fibonaccitype sequence. b) Each term is the sum of the two preceding terms. Thus, this sequence is a Fibonacci-type sequence. The next term is + 11 18, or29. The following term is + 18 29, or 47. 7 Now try Exercise 7 Table 5.2 Numbers Ratio 1, 1 = 1 1 1 1, 2 = 2 1 2 2, 3 = 3 2 1.5 3, 5 = … 5 3 1.666 5, 8 = 8 5 1.6 8, 13 = 13 8 1.625 13, 21 ≈ 21 13 1.615 21, 34 ≈ 34 21 1.619 34, 55 ≈ 55 34 1.618 55, 89 ≈ 89 55 1.618 A B C Figure 5.11 When this line segment is divided at a point C such that the ratio of the whole, AB, to the larger part, AC, is equal to the ratio of the larger part, AC, to the smaller part, CB, each ratio AB AC / and AC CB / is referred to as a golden ratio. The proportion these ratios form, = AB AC AC CB / / , is called the golden proportion. Furthermore, each ratio in the proportion will have a value equal to the golden number, + ( 5 1)/2. AB AC AC CB 5 1 2 1.618 = = + ≈ The Great Pyramid of Giza in Egypt, built about 2600 b.c., is the earliest known example of use of the golden ratio in architecture. The ratio of any side of the square base (775.75 ft) to its altitude (481.4 ft) is about 1.611. In medieval times, people referred to the golden proportion as the divine proportion, reflecting their belief in its relationship to the will of God. Twentieth-century architect Le Corbusier developed a scale of proportions for the human body that he called the Modulor (Fig. 5.12). Note that the navel separates the entire body into golden proportions, as do the neck and knee. m The Great Pyramid of Giza 1 1 1 1.618 1.618 1.618 Figure 5.12 Dan Breckwoldt/Shutterstock
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