# Beautiful Formulas for pi=3.14…

The number $\pi$ is defined a s the ratio of a circle’s circumference $C$ to its diameter $d$:
$\pi=\frac{C}{d}.$

$\pi$ in decimal starts with 3.14… and never end.

I will show you several beautiful formulas for $\pi$.

## Art Museum of formulas for $\pi$

### Beautiful formula of $\pi$ (continued fraction).

$\pi$ is an irrational number. This means that $\pi$ can not be written as a ratio of two integers:$\pi \neq \frac{n}{m}$ for any integers $n, m$.

However, $\pi$ can be written as an infinite series of nested fractions, known as continued fraction.
There are several known continued fractions that are equal to $\pi$.
\begin{align*}
\pi&= 3 + \cfrac{1}{7
+ \cfrac{1}{15
+ \cfrac{1}{1 + \cfrac{1}{292 + \cdots}}}}
\20pt] \pi&= \cfrac{4}{1 + \cfrac{1^2}{2 + \cfrac{3^2}{2 + \cfrac{5^2}{2 +\cfrac{7^2}{2 + \cdots}}}}} \\[20pt] \frac{4}{\pi}&= 1+\cfrac{1}{3 + \cfrac{2^2}{5 + \cfrac{3^2}{7 + \cfrac{4^2}{9 +\cfrac{5^2}{11 + \cdots}}}}} \end{align*} It is mysterious that \pi in decimal shows no pattern but the expressions of \pi in continued fractions have simple patterns. ### Beautiful formula of \pi. \begin{align*} \frac{\pi}{4}&=1-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots= \sum_{n=0}^{\infty}\frac{(-1)^n}{2n+1} &&\text{Leibniz formula for \pi}\\[12pt] \frac{2}{\pi}&=\frac{\sqrt{2}}{2}\cdot \frac{\sqrt{2+\sqrt{2}}}{2}\cdot \frac{\sqrt{2+\sqrt{2+\sqrt{2}}}}{2}\cdots &&\text{Franciscus Vieta}\\[12pt] \frac{\pi}{2}&=\frac{2}{1}\cdot \frac{2}{3}\cdot \frac{4}{3} \cdot \frac{4}{5} \cdot \frac{6}{5}\cdot \frac{6}{7}\cdot \frac{8}{7} \cdot \frac{8}{9}\cdots &&\text{John Wallis}\\[12pt] \frac{\pi^2}{6}&=\frac{1}{1^2}+\frac{1}{2^2}+\frac{1}{3^2}+\frac{1}{4^2}\cdots && \substack{\text{Leonhard Euler}\\ \text{ the value of Riemann zeta function \zeta(2)}}\\[12pt] \sqrt{\pi}&=\int_{-\infty}^{\infty}e^{-x^2}\, dx &&\text{Gaussian integral}\\[12pt] \frac{\pi}{4}&=4\arctan{\frac{1}{5}}-\arctan{\frac{1}{239}} && \text{John Machin}\\[12pt] e^{2\pi i}&=-1 && \text{Euler’s formula} \end{align*} ### Miscellaneous • March 14th (3/14) is Pi Day. • The mirror reflection of the English alphabet letters PIE is looks like 314. (See the above picture.) • \pi contains 2017. If you want to know more about the fun fact about the number 2017, check out the post Mathematics about the number 2017 • ## Reference Wikipedia article about \pi contains more extensive facts about \pi. ### More from my site • Even Perfect Numbers and Mersenne Prime Numbers Prove that if 2^n-1 is a Mersenne prime number, then \[N=2^{n-1}(2^n-1) is a perfect number. On the other hand, prove that every even perfect number $N$ can be written as $N=2^{n-1}(2^n-1)$ for some Mersenne prime number $2^n-1$.   Definitions. In this post, a […]
• How to Prove a Matrix is Nonsingular in 10 Seconds Using the numbers appearing in $\pi=3.1415926535897932384626433832795028841971693993751058209749\dots$ we construct the matrix $A=\begin{bmatrix} 3 & 14 &1592& 65358\\ 97932& 38462643& 38& 32\\ 7950& 2& 8841& 9716\\ 939937510& 5820& 974& […] • Find the Largest Prime Number Less than One Million. Find the largest prime number less than one million. What is a prime number? A natural number is called a "prime number" if it is only divisible by 1 and itself. For example, 2, 3, 5, 7 are prime numbers, although the numbers 4,6,9 are not. The prime numbers have always […] • Mathematics About the Number 2018 Happy New Year 2018!! Here are several mathematical facts about the number 2018. Is 2018 a Prime Number? The number 2018 is an even number, so in particular 2018 is not a prime number. The prime factorization of 2018 is \[2018=2\cdot 1009.$ Here $2$ and $1009$ are […]
• A One-Line Proof that there are Infinitely Many Prime Numbers Prove that there are infinitely many prime numbers in ONE-LINE.   Background There are several proofs of the fact that there are infinitely many prime numbers. Proofs by Euclid and Euler are very popular. In this post, I would like to introduce an elegant one-line […]
• Top 10 Popular Math Problems in 2016-2017 It's been a year since I started this math blog!! More than 500 problems were posted during a year (July 19th 2016-July 19th 2017). I made a list of the 10 math problems on this blog that have the most views. Can you solve all of them? The level of difficulty among the top […]
• Mathematics About the Number 2017 Happy New Year 2017!! Here is the list of mathematical facts about the number 2017 that you can brag about to your friends or family as a math geek. 2017 is a prime number Of course, I start with the fact that the number 2017 is a prime number. The previous prime year was […]
• Infinite Cyclic Groups Do Not Have Composition Series Let $G$ be an infinite cyclic group. Then show that $G$ does not have a composition series.   Proof. Let $G=\langle a \rangle$ and suppose that $G$ has a composition series $G=G_0\rhd G_1 \rhd \cdots G_{m-1} \rhd G_m=\{e\},$ where $e$ is the identity element of […]

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1. 07/13/2017

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##### Mathematics About the Number 2017

Happy New Year 2017!! Here is the list of mathematical facts about the number 2017 that you can brag about...

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