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50 51 Theorem 2 The nth fraction is f n1fn where f n is the nth Fibonacci number. Proof The proof is by induction, as described in Mathematical Dominos. Let s denote the nth fraction g n. We first note that g 1, the first term of our sequence, is indeed f 2f1 11 1. Next, we assume that the nth term g n f n1fn. Then gn1 1 1gn 1 fn f n1 fn1 fn fn1. With the defining equation of the Fibonacci numbers, f n1 f n fn2, this implies that gn1 fn2fn1, as desired. Q.E.D. Theorem 3 The Fibonacci numbers f n, fn1, f n2, fn3 form the Pythagorean triple Proof If a f n, b fn1, then fn2 ab, fn3 a 2b and 2bab2 aa2b2 equals b2 ab2 2. Q.E.D. For example, choosing n 1, 2, and 3 gives the triples 3 4 5, 5 12 13, and 16 30 34. From the proof it is clear that we can replace the Fibonacci numbers by any a, b, a b, a 2b. Delving deeply into the secrets of mathematics, one gets the feeling that everything is somehow connected to everything else by a network of beautiful relationships. For example, the Fibonacci numbers 1, 1, 2, 3, 5, and the golden ratio f 5 12 appear alongside many of the other topics covered in this little book. Connections with regular figures and Pascal s triangle have already been touched upon earlier. Here are some more in the TheoremProof style favoured by mathematicians. Theorem 1 Proof Call the infinite fraction x. Clearly, x 1 1x, or x2 x 1 0. This equation has the solutions f and 1 f. Since 1 f is negative and both x and f are not, x equals f. Q.E.D. Hence, just as 0.99... gets approximated by the numbers 0, 0.9, 0.99, , f gets approximated by the fractions aPPendIX II all For one and one For all Theorem 5 Proof Call the infinite expression y. Then, because of its selfsimilarity y 1 y, or y2 y 1 0. Thus y f. Q.E.D. We finish with another beautiful connection between the Fibonacci numbers and f. Theorem 6 The nth Fibonacci number is Proof In the proof of Theorem 2, we saw that both f and 1 f satisfy the equation x2 x 1. We conclude that f2 1f 1, f3 f 2 f 2f 1, f 4 f3 f2 3f 2. It follows by induction that f n f n f fn 1 and, similarly, n f n fn 1. Subtracting the second equation from the first gives f n n f nf f n2f 1. Finally, dividing both sides of this equation by 2f 1, gives the desired formula. Q.E.D. Since 1 fn 0.6180...n is tiny, the nth Fibonacci number is the closest natural number to f n 2f 1. For example, f 10 2f 1 55.0036 and the 10th Fibonacci number is 55. Theorem 4 f is an irrational number. Proof As in Nature of Numbers, we assume that f is a fraction a b of two natural numbers. Thus we have a golden rectangle with side lengths a and b. Using the self similarity property of golden rectangles see page 40, we cut off a square to produce another golden rectangle. Repeating this for the smaller golden rectangle produces a third, and so on. We notice that the side lengths of these golden rectangles form the infinitely decreasing sequence of natural numbers a b a b 2b a . Since any decreasing sequence of natural numbers must end 1 being the lower limit, this is a contradiction to our assumption. Hence f is irrational. Q.E.D. The infinite expression is the main part of our formula for the value of p see page 9, bottom.
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