Golden ratio

:This article is about the mathematical ratio. For the Aristotelian concept of "golden mean" see Nicomachean Ethics.

History

The golden ratio was first studied by ancient mathematicians due to its frequent appearance in geometry. The golden ratio may have been understood and used by the Egyptians. The discovery of irrational numbers, numbers that cannot be represented as an exact ratio of two integers, is usually attributed to Pythagoras (or to the Pythagoreans, notably Theodorus) or to Hippasus of Metapontum. Euclid spoke of the "golden mean" this way, "A straight line is said to have been cut in extreme and mean ratio when, as the whole line is to the greater segment, so is the greater to the lesser".

Related Topics:
Geometry - Integers - Pythagoras - Theodorus - Hippasus of Metapontum - Euclid

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The golden ratio is symbolized by the Greek letter φ (phi) or less commonly by τ (tau).

Related Topics:
Greek letter - Phi - Tau

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Two quantities are said to be in the golden ratio, if "the whole is to the larger as the larger is to the smaller", i.e. if

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:rac{a+b}{a} = rac{a}{b}.

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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:

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:rac{a}{b} = rac{b}{a-b}.

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After multiplying the first equation with a/b or the second equation with (a − b)/b, both of these equations are seen to be equivalent to

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:left(rac{a}{b} ight)^2 = rac{a}{b} + 1

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and hence

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:rac{a}{b} = arphi.

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This definition gives the value of arphi stated above. Alternatively, some define the number of the golden ratio to be the so-called golden ratio conjugate (also erroneously called the silver ratio or silver mean),

Related Topics:
Conjugate - Silver ratio

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hatarphi = rac{1}{arphi} = arphi-1. The ratios arphi:1 and 1:hatarphi are equivalent. Also, some use the symbols au or arphi to designate the number called hatarphi here.

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The fact that a length is divided into two parts of lengths a and b which stand in the golden ratio is also (in older texts) expressed as "the length is cut in extreme and mean ratio". This can be easily visualized using a line that is divided into two segments, as in the diagram.

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For those who struggle with algebra but can at least handle the idea of fractions of equal value, the inquiry that leads to the golden number is expressed this way: Is there a number that compares to 1 as 1 compares to the same number minus one? The answer is yes, but there is only one number, and it is the golden number x in this example.

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:rac{x}{1} = rac{1}{x-1},

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or, equivalently, the quadratic equation

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:x^2-x-1=0.,

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This quadratic equation has two roots:

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:{1+sqrt{5} over 2} pprox 1.618034, mathrm{and} {1-sqrt{5} over 2} pprox -0.618034.

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If a house has a rectangular "golden window" with a length of 1 unit of measurement, then its width is the golden number minus 1, about 0.618 of the unit of measurement. If the shorter side of the window is instead determined to have a width of 1 unit of measurement, then its length is the golden number, about 1.618 units of measurement.

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The golden ratio value pprox 1.618034 is the only positive number that is exactly 1 less than its own square.

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:x^2=x+1,!

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