Difference between revisions of "Phi"
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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>. | <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 [[ | + | <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 [[square number| 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 n<sup>th</sup> term of the Fibonacci series over the (n-1)<sup>th</sup> term, the result approaches <math>\phi</math> as n increases. | 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 n<sup>th</sup> term of the Fibonacci series over the (n-1)<sup>th</sup> term, the result approaches <math>\phi</math> as n increases. | ||
Revision as of 22:00, 27 February 2020
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 nth term of the Fibonacci series over the (n-1)th term, the result approaches as n 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.
