# Why is $\pi$ not equal to $\frac{22}{7}?$

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#### Contents

## Pi

We all know that $\pi$ is the ratio of the circumference of a circle to the diameter. But why is $\pi$ not equal to $\frac{22}{7}?$

$\pi$ is never equal to $\frac{22}{7}$. To prove this statement, you can try to solve this integral

$\int_{0}^{1} \frac{x^4(1-x)^4}{1+x^2} \, dx.$

The result would be $\frac{22}{7}-\pi,$ and since area cannot be negative, that means $\frac{22}{7}$ must be greater than $\pi$.

Below is the graph of the function

True or False?$\large e^{\frac{22i}{7}} + 1 =0$

Clarification: $i = \sqrt{-1}$

Here, the answer is obviously false. $_\square$

## Continued Fraction

$\frac{22}{7}$ is the second convergent of continued fraction of $\pi$, which means that $\frac{22}{7}$ is only an approximation of $\pi$ but not an exact value.

$\pi=[3;7,15,1,292,1,1,1,2,1,3,1,14,2,1,1,2,2,2,2,\ldots]$ (OEIS A001203)

The second convergent $=[3;7]=3+\frac{1}{7}=\frac{22}{7}.$

## Decimal Expansions

We have

$\begin{aligned} \pi&=3.141592653589793238462643383279502884197\ldots\\ \\ \frac{22}{7}&=3.1428571428571428571428571428571428571\ldots\\ \frac{22}{7}-\pi&=0.0012644892673496\ldots. \end{aligned}$

So they are not equal.

**Cite as:**Why is $\pi$ not equal to $\frac{22}{7}?$.

*Brilliant.org*. Retrieved from https://brilliant.org/wiki/why-pi-is-not-equal-to-frac227/