# 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{align}
\pi&=3.141592653589793238462643383279502884197\ldots\\ \\
\frac{22}{7}&=3.1428571428571428571428571428571428571\ldots\\

\frac{22}{7}-\pi&=0.0012644892673496\ldots.
\end{align}\]

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/