Recall that we denoted the "larger part" by a and the "smaller part" by b, and concluded that This gives a startlingly quick proof that the golden ratio is an irrational number. An irrational number is one that cannot be written as a/b where a and b are integers. If a/b is such a fraction, in lowest terms, then b/(a − b) is in even lower terms — a contradict ...

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The formula can be expanded recursively to obtain a continued fraction for the golden ratio: and its reciprocal: Note that the successive convergents of these continued fractions are ratios of Fibonacci numbers. The equation likewise produces the continued square root form: Also: These correspond to the fact that the length of the diagonal of a regular pentagon is φ times the length of ...

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It has been claimed that the ancient Egyptians knew the golden ratio because ratios close to the golden ratio may be found in the positions or proportions of the Pyramids of Giza. The ancient Greeks already knew the golden ratio from their investigations into geometry, but there is no evidence they thought the number warranted special attention above that for numbers like π (Pi), for example. Studies by psychologists have been devised to test the idea that the golden ratio plays a role in human perception ...

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The number φ turns up frequently in geometry, in particular in figures involving pentagonal symmetry. For instance the ratio of a regular pentagon's side and diagonal is equal to φ, and the vertices of a regular icosahedron are located on three orthogonal golden rectangles. The explicit expression for the Fibonacci sequence involves the golden ratio: The limit of ratios of successive terms of the Fibonacci sequence (or any Fibonacci-like sequence) equals the golden ratio; therefore, when a number in the F ...

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Two quantities are said to be in the golden ratio, if "the whole (i.e., the sum of the two parts) is to the larger part as the larger part is to the smaller part", i.e. if where a is the larger part and b is the smaller part. Equivalently, they are in the golden ratio if the ratio of the larger one to the smaller one equals the ratio of the smaller one to their difference, i.e. if After multiplying the first equation with a/b or the second equation with (a − b)/b, both of these equations are se ...

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(sequence A001622 in OEIS) 1.6180339887 4989484820 4586834365 6381177203 0917980576 2862135448 6227052604 6281890244 9707207204 1893911374 8475408807 5386891752 1266338622 2353693179 3180060766 7263544333 8908659593 9582905638 3226613199 2829026788 0675208766 8925017116 9620703222 1043216269 5486262963 1361443814 9758701220 3408058879 5445474924 6185695364 8644492410 4432077134 4947049565 8467885098 7433944221 2544877066 4780915884 6074998871 2400765217 0575179788 3416625624 9407589069 7040002812 1042762177 11177780 ...

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