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at3rg0
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The golden ratio is irrational. Do you know any clever proofs for this fact? I put this here, because it's not homework--only more of a discussion.
matt grime said:Clever? Does the fact that it is an algebraic number that is not an integer count as clever?
at3rg0 said:The golden ratio is irrational. Do you know any clever proofs for this fact? I put this here, because it's not homework--only more of a discussion.
your rightDodo said:I don't get it. There has to be a gazillion rationals between those two fractions, no matter how big n is or how close to the limit you are.
al-mahed said:could someone prove that irrational OP integer = irrational, OP = operations +, -, / and *
ramsey2879 said:How about
[tex]\frac{F_{2n}}{F_{2n-1}} < Phi < \frac{F_{2n+1}}{F_{2n}}[/tex]
If you assume [tex]phi = a/b[/tex] then the above inequality conflicts with that.
Those are "two fractions". Those are two sequences of fraction. Phi is between every pair of corresponding numbers in those sequences.Dodo said:I don't get it. There has to be a gazillion rationals between those two fractions, no matter how big n is or how close to the limit you are.
ramsey2879 said:so we have phi is a root of x^2- x -1 but the discriminate is [tex]\sqrt{5}[/tex] so phi is irrational.
CRGreathouse said:Let z be an integer, n be a positive integer, and x be an irrational number.
x + z is irrational (else a/b - z = (a-bz)/b which is rational)
x - z is irrational by the above.
x * n is irrational (else a/b / n = a/(bn) which is rational)
x / n is irrational (else a/b * n = (an)/b which is rational)
x * 0 is rational
x / 0 is undefined
The Golden Ratio, also known as Phi, is a mathematical constant represented by the Greek letter phi (ϕ). It is approximately equal to 1.6180339887 and is considered to be the most aesthetically pleasing ratio in art and architecture.
The Golden Ratio can be calculated by dividing a line into two unequal parts, where the longer part divided by the shorter part is equal to the sum of the two parts divided by the longer part. This results in a ratio of approximately 1.6180339887.
The Golden Ratio is considered irrational because it cannot be expressed as a fraction of two integers. It is a non-repeating, non-terminating decimal and its digits go on infinitely without forming a pattern.
The proof of the irrationality of the Golden Ratio was first given by the ancient Greek mathematician Euclid. It involves assuming that the Golden Ratio is rational and then arriving at a contradiction, thus proving that it is in fact irrational.
The irrationality of the Golden Ratio has significant implications in mathematics, art, and architecture. It is considered to be a fundamental concept in nature and is found in various natural phenomena such as the structure of plants, the spiral shape of galaxies, and the growth patterns of certain animals. It is also used in various fields such as music, design, and even stock market analysis.