Perhaps the numbers most easily proved to be irrational are certain logarithms. Here is a proof by reductio ad absurdum that log23 is irrational: Assume log23 is rational. For some positive integers m and n, we have log23 = m/n. It follows that 2m/n = 3. Raise each side to the n power, find 2m = 3n. But 2 to any power greater than 0 is even (because at least one of its prime factors is 2) and 3 to any power greater than 0 is odd (because none of its prime factors is ...

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It is often erroneously assumed that mathematicians define "irrational number" in terms of decimal expansions, calling a number irrational if its decimal expansion neither repeats nor terminates. No mathematician takes that to be the definition, since the choice of base 10 would be arbitrary and since the standard definition is simpler and more well-motivated. Nonetheless it is true that a number is of the form n/m where n and m are integers, if and only if its decimal expansion repeats or terminates. When the lon ...

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A 2-dimensional system of linear difference equations that describes the Fibonacci sequence is or The eigenvalues of the matrix A are and , and the elements of the eigenvectors of A, and , are in the ratios and . Note that this matrix has a determinant of −1, and thus it is a 2×2 unimodular matrix. This property can be understood in terms of the continued fraction representation for the golden mean: φ = [1; 1, 1, 1, 1, …]. The Fibonacci numbers occur as the rat ...

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The Parthenon was elaborately decorated with marble sculptures both internally and externally. They survive only in part, but there are good descriptions of most of those parts that have been lost. On the eastern pediment (the triangular area above the columns on the "front" and "back" of the temple) was a depiction of the birth of Athena. The western pediment showed Athena's battle with Poseidon for possession of the land of Attica. Metopes ran along the outer frieze of all four sides of the temple, above the lines of columns and below the ...

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By the late eighteenth century, many more Europeans were visiting Athens, and the picturesque ruins of the Parthenon were much drawn and painted, helping to arouse sympathy in Britain and France for Greek independence. In 1801, the British ambassador at Constantinople, the Earl of Elgin, obtained a firman (permit) from the Sultan to make casts and drawings of the antiquities on the Acropolis, to demolish recent buildings if this was necessary to view the antiquities, and to remove sculptures from them. He took this as permission to co ...

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The earliest known use of irrational numbers was in the Indian Sulba Sutras composed between 800-500 BC. The first proof of irrational numbers is usually attributed to Pythagoras, more specifically to the Pythagorean Hippasus of Metapontum, who produced a (most likely geometrical) proof of the irrationality of the square root of 2. The story goes that Hippasus discovered irrational numbers when trying to represent the square root of 2 as a fraction (proof below). However Pythagoras believed in the absoluteness of numbers, and could not accep ...

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In 1975, the Greek government began a concerted effort to restore the Parthenon and other Acropolis structures. The project later attracted funding and technical assistance from the European Union. An archaeological committee thoroughly documented every artifact remaining on the site, and architects assisted with computer models to determine their original locations. In some cases, prior re-construction was found to be incorrect. Particularly important and fragile sculptures were transferred to the Acropolis Museum. A crane was installed for ...

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The sum of the reciprocals of the positive Fibonacci numbers converges: A079586 Convergence is easy to show with the ratio test. This value has been proven irrational by André-Jeannin, R. No closed form is currently known. See also the Mathworld article on the subject. ...

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Computing Fibonacci numbers by computing powers of the golden mean is not very practical except for small values of n, since rounding errors will accrue and floating point numbers usually do not have enough precision. The straightforward recursive implementation of the Fibonacci sequence definition is also not advisable, since it computes many values repeatedly (unless the programming language has a feature which allows the storing of previously computed function values, such as memoization). Therefore, one usually computes the ...

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The Parthenon survived as the most important temple of the Ancient Greek religion for close to a thousand years. It was certainly still intact in the 4th century AD, by which time it was already as old as Notre Dame Cathedral in Paris is now, and far older than St. Peter's Basilica in Rome. But by that time Athens had been reduced to a provincial city of the Roman Empire, albeit one with a glorious past. Sometime in the 5th century, the great statue of Athena was looted by one of the Emperors, and taken to Constantinople, where it was later destroyed, possibly during the sack of ...

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The power series has a simple and interesting closed-form solution for x < 1/φ: This function is therefore the generating function of the Fibonacci sequence. In particular, math puzzle-books note the curious value s(1/10)/10=1/89. The sum is easily proved by noting that and then explictly evaluating the sum. ...

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In analogy to its numerical counterpart, a Fibonacci string is defined by: , where + denotes the concatenation of two strings. The sequence of Fibonacci strings starts: b, a, ab, aba, abaab, abaababa, abaababaabaab, … The length of each Fibonacci string is a Fibonacci number, and similarly there exists a corresponding Fibonacci string for each Fibonacci number. Fibonacci strings appear as ...

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Fibonacci number - Vector space. The term Fibonacci sequence is also applied more generally to any function g where g(n + 2) = g(n) + g(n + 1). These functions are precisely those of the form g(n) = aF(n) + bF(n + 1) for some numbers a and b, so the Fibonacci sequences form a vector space with the functions F(n) and F(n + 1) as a basis. ...

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The Fibonacci numbers are important in the run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers. Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his original solution of Hilbert's tenth problem. The Fibonacci numbers occur in a formula about the diagonals of Pascal's triangle (see binomial coefficient). Every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci numbers ...

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Fibonacci number, Fibonacci number - Origins, Fibonacci number - The bee ancestry code, Fibonacci number - Relation to the golden ratio, Fibonacci number - Matrix form, Fibonacci number - Computation, Fibonacci number - Applications, Fibonacci number - Fibonacci numbers in nature, Fibonacci number - Identities, Fibonacci number - Common factors, Fibonacci number - Power series, Fibonacci number - Reciprocal sum constant, Fibonacci number - Generalizations, Fibonacci number - Vector space, Fibonacci number - Similar integer sequences, Fibonacci number - Other generalizations, Fibonacci number - Fibonacci primes, Fibonacci number - Fibonacci strings, Fibonacci number - Fiction, Fibonacci number - Journals

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The Fibonacci numbers are important in the run-time analysis of Euclid's algorithm to determine the greatest common divisor of two integers. Yuri Matiyasevich was able to show that the Fibonacci numbers can be defined by a Diophantine equation, which led to his original solution of Hilbert's tenth problem. The Fibonacci numbers occur in a formula about the diagonals of Pascal's triangle (see binomial coefficient). Every positive integer can be written in a unique way as the sum of one or more distinct Fibonacci nu ...

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Fibonacci number, Fibonacci number - Origins, Fibonacci number - The bee ancestry code, Fibonacci number - Relation to the golden ratio, Fibonacci number - Matrix form, Fibonacci number - Computation, Fibonacci number - Applications, Fibonacci number - Fibonacci numbers in nature, Fibonacci number - Identities, Fibonacci number - Common factors, Fibonacci number - Power series, Fibonacci number - Reciprocal sum constant, Fibonacci number - Generalizations, Fibonacci number - Vector space, Fibonacci number - Similar integer sequences, Fibonacci number - Other generalizations, Fibonacci number - Fibonacci primes, Fibonacci number - Fibonacci strings, Fibonacci number - Fiction, Fibonacci number - Journals

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The earliest known reference to Fibonacci numbers is contained in a book on meters by an Indian mathematician named Pingala called Chhandah-shastra (500 BC). As documented by Donald Knuth in The Art of Computer Programming, this sequence was described by the Indian mathematicians Gopala and Hemachandra in 1150, who were investigating the possible ways of exactly bin packing items of length 1 and 2. In the West, it was first studied by Leonardo of Pisa, who was also known as Fibonacci (c. 1200), to describe the growth of an idea ...

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Fibonacci is also stated as having described the sequence "encoded in the ancestry of a male bee." This turns out to be the Fibonacci sequence. One can derive this truth by taking the following facts: If an egg is laid by a single female, it hatches a male. If, however, the egg is fertilized by a male, it hatches a female. Thus, a male bee will always have one parent, and a female bee will have two. If one traces the ancestry of this male bee (1 bee), he has 1 female parent (1 bee). This female ha ...

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These identities can be proven using many different methods. But, among all, we wish to present an elegant proof for each of them using combinatorial arguments here. In particular, F(n) can be interpreted as the number of ways summing 1's and 2's to n − 1, with the convention that F(0) = 0, meaning no sum will add up to −1, and that F(1) = 1, meaning the empty sum will "add up" to 0. Here the order of the summands matters. For example, 1 + 2 and 2 + 1 are considered two different sums and are counted tw ...

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The golden ratio is defined as given the conditions above. This provides an interesting relationship. Let f be the fraction of the circumference subtended by the golden angle, or equivalently, the golden angle divided by the angular measurement of the circle. Hence, we see that This is equivalent to saying that φ2 golden angles can fit in a circle. It can also be shown that See also:

Golden angle, Golden angle - Derivation, Golden angle - Golden angle in nature

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If the other parameters of the Solow model are taken as 'given' (beyond the policy-maker's control), but the savings rate can be set exogenously to maximize steady state consumption the solution can be derived from the steady state equations. Per capita consumption (c) is the difference between output per capita (y) and savings per capita: c(k) = y(k)[1 − s] where s is the propensity to save, c = per-capita consumption and k = the capital/labour ratio (i.e. c ...

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Golden Rule savings rate, Golden Rule savings rate - Derivation of the Golden Rule savings rate, Golden Rule savings rate - Measuring the Golden Rule savings rate, Golden Rule savings rate - Policy that can change the savings rate, Golden Rule savings rate - Private and public saving, Golden Rule savings rate - Golden rule taxes within economic models, Golden Rule savings rate - Notes

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All of the following J examples generate . Fibonacci number program - Double recursion. f0a and f0b use the basic identity . f0c uses a cache of previously computed values. f0d depends on the identity , whence and obtain by substituting n and n + 1 for k. f0a=: 3 : 'if. 1<y. do. (y.-2) +&f0a (y.-1) else. y. end.' f0b=: (-&2 +&$: -&am ...

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