Difference between revisions of "Phi"
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| − | Phi (<math>\phi</math>) is | + | '''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== | |
| + | <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>. | ||
| − | + | <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]]. | |
| − | + | 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. | |
| + | |||
| + | ==Golden ratio== | ||
| + | <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> | ||
| + | |||
| + | ==Other Usages== | ||
| + | * <math>\phi</math> is also commonly used to represent [[Euler's totient function]]. | ||
==See also== | ==See also== | ||
| − | * [[ | + | * [[Irrational number]] |
| − | * [[ | + | * [[Geometry]] |
| − | |||
| − | |||
* [[Zeckendorf representation]] | * [[Zeckendorf representation]] | ||
| − | + | [[Category:Constants]] | |
Revision as of 13:04, 1 January 2024
Phi (in lowercase, either 
or 
; capitalized, 
) is the 21st letter in the Greek alphabet. It is used frequently in mathematical writing, often to represent the constant 
. (The Greek letter Tau (
) was also used for this purpose in pre-Renaissance times.)
Contents
Use
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 
.
is also equal to the continued fraction 
and the continued radical 
. It is the only positive real number that is one more than its square and one less than its reciprocal.
It is also 
where 
is the nth number in the Fibonacci sequence. In other words, if you divide the 
th term of the Fibonacci series over the 
th term, the result approaches as increases.
Golden ratio
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 ; it has a number of interesting properties.
The first fifteen digits of in decimal representation are 
Other Usages
is also commonly used to represent Euler's totient function.
