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.) | '''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.) | ||
| − | == | + | == Uses == |
| − | <math>\phi</math> | + | * <math>\phi</math> is also commonly used to represent [[Euler's totient function]]. |
| − | + | * <math>\phi</math> is used to express the golden ratio. | |
| − | <math>\phi</math> is | ||
| − | |||
| − | |||
| − | + | <math>\phi</math> appears in a variety of different mathematical contexts: | |
| − | <math>\phi</math> | ||
| − | The first fifteen digits of <math>\phi</math> in decimal representation are <math>1.61803398874989</math> | + | === 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 golden ratio is also equal to the positive solution of the [[quadratic equation]] <math>x^2-x-1=0</math> as well as 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 more 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^th</math> term of the Fibonacci series over the <math>(n-1)^th</math> term, the result approaches <math>\phi</math> as <math>n</math> increases. The first fifteen digits of <math>\phi</math> in decimal representation are <math>1.61803398874989</math> | ||
| − | == | + | == See also == |
| − | |||
| − | |||
* [[Irrational number]] | * [[Irrational number]] | ||
* [[Geometry]] | * [[Geometry]] | ||
Latest revision as of 20:12, 14 February 2025
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.)
Uses
is also commonly used to represent Euler's totient function.- is used to express the golden ratio.
appears in a variety of different mathematical contexts:
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 golden ratio is also equal to the positive solution of the quadratic equation 
as well as the continued fraction 
and the continued radical 
. It is the only positive real number that is one more than its square and one more than its reciprocal. It is also 
where 
is the nth number in the Fibonacci sequence. In other words, if you divide the 
term of the Fibonacci series over the 
term, the result approaches as 
increases. The first fifteen digits of in decimal representation are 
See also
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