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Golden ratio - A startlingly quick proof of irrationality | | Golden ratio - A startlingly quick proof of irrationality: Encyclopedia II - Golden ratio - A startlingly quick proof of irrationality | | 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 ...
See also:Golden ratio, Golden ratio - Definition, Golden ratio - History, Golden ratio - A startlingly quick proof of irrationality, Golden ratio - Alternate forms, Golden ratio - Mathematical uses, Golden ratio - Aesthetic uses, Golden ratio - Decimal expansion | | | Golden ratio, Golden ratio - A startlingly quick proof of irrationality, Golden ratio - Aesthetic uses, Golden ratio - Alternate forms, Golden ratio - Decimal expansion, Golden ratio - Definition, Golden ratio - History, Golden ratio - Mathematical uses, Golden angle, Golden function, Golden rectangle, Golden section search, Golden section (page proportion), Logarithmic spiral, Fibonacci number, Modulor, Sacred geometry, The Roses of Heliogabalus, Plastic number, Penrose tiles, Dynamic symmetry, Golden ratio base, Vitruvian man | | |
| | | Golden ratio: Encyclopedia II - Golden ratio - A startlingly quick proof of irrationality
Golden ratio - A startlingly quick proof of irrationality
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 contradiction. Thus this number cannot be so written, and it is therefore irrational.
Other related archivesA4, Acropolis, Architect, Architecture, Béla Bartók, Dynamic symmetry, Egyptians, Erik Satie, Euclid, Fibonacci number, Fibonacci numbers, Fibonacci sequence, For Ann (rising), Golden angle, Golden function, Golden ratio base, Golden rectangle, Golden section (page proportion), Golden section search, Greek letter, Hippasus of Metapontum, James Tenney, Lagrange, Le Corbusier, Leonardo Da Vinci, Logarithmic spiral, Luca Pacioli, Modulor, OEIS, Parthenon, Penrose tiles, Phidias, Pi, Pisot-Vijayaraghavan number, Plastic number, Pyramids of Giza, Pythagoras, Pythagoreans, Sacred geometry, Shepard tone, The Roses of Heliogabalus, Theodorus, Venus, Vitruvian man, acoustic scale, aesthetically, algebraic number field, ancient Egyptians, ancient Greeks, approximation theorem, beauty, continued fraction, equal tempered, fifth, geometry, glissandoing, golden mean base, golden rectangles, icosahedron, integers, irrational number, justly tuned, limit, lowest terms, major sixth, minor, numeral system, octave, paper sizes, pentagon, pentagram, phi, phyllotaxis, quadratic equation, ratio, reciprocal, recurrence, rule of thirds, rule of thumb, sixths, square root, symmetry, tau, woodwork
 Adapted from the Wikipedia article "A startlingly quick proof of irrationality", under the G.N U Free Docmentation License. Please also see http://en.wikipedia.org/wiki |
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