(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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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

Read more here: » Golden ratio: Encyclopedia II - Golden ratio - Decimal expansion

Almost all irrational numbers are transcendental and all transcendental numbers are irrational: the article on transcendental numbers lists several examples. er and πr are irrational if r ≠ 0 is rational; eπ is also irrational. Another way to construct irrational numbers is as irrational algebraic numbers, i.e. as zeros of polynomials with integer coefficients: start with a polynomial equation p(x) = an xn + an-1 xn−1 ...

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Irrational number, Irrational number - History, Irrational number - The square root of 2, Irrational number - Another proof, Irrational number - The golden ratio, Irrational number - Transcendental and algebraic irrationals, Irrational number - Logarithms, Irrational number - Decimal expansions, Irrational number - Open questions, Irrational number - The set of all irrationals, Irrational number - Another irrational number

Read more here: » Irrational number: Encyclopedia II - Irrational number - Transcendental and algebraic irrationals

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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Irrational number, Irrational number - History, Irrational number - The square root of 2, Irrational number - Another proof, Irrational number - The golden ratio, Irrational number - Transcendental and algebraic irrationals, Irrational number - Logarithms, Irrational number - Decimal expansions, Irrational number - Open questions, Irrational number - The set of all irrationals, Irrational number - Another irrational number

Read more here: » Irrational number: Encyclopedia II - Irrational number - Decimal expansions

One proof of the irrationality of the square root of 2 is the following reductio ad absurdum. The proposition is proved by assuming the negation and showing that that leads to a contradiction, which means that the proposition must be true. Assume that √2 is a rational number. This would mean that there exist integers a and b such that a / b = √2. Then √2< ...

See also:

Irrational number, Irrational number - History, Irrational number - The square root of 2, Irrational number - Another proof, Irrational number - The golden ratio, Irrational number - Transcendental and algebraic irrationals, Irrational number - Logarithms, Irrational number - Decimal expansions, Irrational number - Open questions, Irrational number - The set of all irrationals, Irrational number - Another irrational number

Read more here: » Irrational number: Encyclopedia II - Irrational number - The square root of 2

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 ...

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

Read more here: » Golden ratio: Encyclopedia II - Golden ratio - Aesthetic uses

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