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Difference between revisions of "Phi"
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'''Phi''' (in lowercase, either <math>\phi</math> or <math>\varphi</math>; capitalized, <math>\Phi</math>) is the 21st letter in the Greek alphabet.  It is used frequently in mathematical writing, often to represent the constant <math>\frac{1+\sqrt{5}}{2}</math>. (The Greek letter tau (<math>\tau</math>) was also used for this purpose in pre-Renaissance times.)
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'''Phi''' (in lowercase, either <math>\phi</math> or <math>\varphi</math>; capitalized, <math>\Phi</math>) is the 21st letter in the Greek alphabet.  It is used frequently in mathematical writing, often to represent the constant <math>\frac{1+\sqrt{5}}{2}</math>. (The Greek letter [[Tau]] (<math>\tau</math>) was also used for this purpose in pre-Renaissance times.)
  
 
==Use==
 
==Use==
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<math>\phi</math> appears in a variety of different mathematical contexts: it is the limit of the ratio of successive terms of the [[Fibonacci sequence]], as well as the positive solution of the [[quadratic equation]] <math>x^2-x-1=0</math>.
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<math>\phi</math> appears in a variety of different mathematical contexts: it is the limit of the ratio of successive terms of the [[Fibonacci sequence]], as well as the positive solution of the [[quadratic equation]] <math>x^2-x-1=0</math>.
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<math>\phi</math> is also equal to the [[continued fraction]] <math>1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}}}</math> and the [[nested radical|continued radical]] <math>\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}</math>. It is the only positive real number that is one more than its [[perfect square| square]] and one less than its [[reciprocal]].
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It is also <math>{\lim_{x \to \infty}} \frac{F_{x+1}}{F_x}</math> where <math>F_n</math> is the nth number in the [[Fibonacci sequence]]. In other words, if you divide the <math>n</math><sup>th</sup> term of the Fibonacci series over the <math>(n-1)</math><sup>th</sup> term, the result approaches <math>\phi</math> as <math>n</math> increases.
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==Golden ratio==
 
==Golden ratio==
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<math>\phi</math> is also known as the Golden Ratio. It was commonly believed by the Greeks to be the most aesthetically pleasing ratio between side lengths in a [[rectangle]].  The [[Golden Rectangle]] is a rectangle with side lengths of 1 and <math>\phi</math>; it has a number of interesting properties.
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<math>\phi</math> is also known as the golden ratio. It was commonly believed by the Greeks to be the most aesthetically pleasing ratio between side lengths in a [[rectangle]].  The [[golden rectangle]] is a rectangle with side lengths of 1 and <math>\phi</math>; it has a number of interesting properties.
  
 
The first fifteen digits of <math>\phi</math> in decimal representation are <math>1.61803398874989</math>
 
The first fifteen digits of <math>\phi</math> in decimal representation are <math>1.61803398874989</math>
  
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==Other useages==
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==Other Usages==
 
* <math>\phi</math> is also commonly used to represent [[Euler's totient function]].
 
* <math>\phi</math> is also commonly used to represent [[Euler's totient function]].
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==See also==
 
==See also==
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[[Category:Constants]]
 
[[Category:Constants]]
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Latest revision as of 19:38, 6 October 2024

Phi (in lowercase, either $\phi$ or $\varphi$; capitalized, $\Phi$) is the 21st letter in the Greek alphabet. It is used frequently in mathematical writing, often to represent the constant $\frac{1+\sqrt{5}}{2}$. (The Greek letter Tau ($\tau$) was also used for this purpose in pre-Renaissance times.)

Use

$\phi$ appears in a variety of different mathematical contexts: it is the limit of the ratio of successive terms of the Fibonacci sequence, as well as the positive solution of the quadratic equation $x^2-x-1=0$.

$\phi$ is also equal to the continued fraction $1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\frac{1}{1+\cdots}}}}$ and the continued radical $\sqrt{1+\sqrt{1+\sqrt{1+\sqrt{1+\cdots}}}}$. It is the only positive real number that is one more than its square and one less than its reciprocal.

It is also ${\lim_{x \to \infty}} \frac{F_{x+1}}{F_x}$ where $F_n$ is the nth number in the Fibonacci sequence. In other words, if you divide the $n$th term of the Fibonacci series over the $(n-1)$th term, the result approaches $\phi$ as $n$ increases.

Golden ratio

$\phi$ is also known as the golden ratio. It was commonly believed by the Greeks to be the most aesthetically pleasing ratio between side lengths in a rectangle. The golden rectangle is a rectangle with side lengths of 1 and $\phi$; it has a number of interesting properties.

The first fifteen digits of $\phi$ in decimal representation are $1.61803398874989$

Other Usages

See also

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