In honor of Pi

Today’s the day that some people are calling the Pi Day of the Century. You’ve probably heard that $\pi$ is an irrational number, which means its decimal expansion never ends and never repeats. Some of you saw a really cool talk in the Math Club on Thursday that showed why. The decimal expansion begins 3.14159265359….Today’s date (in MM/DD/YY format) is 3/14/15, so $\pi$ day is celebrated today at 9:26:53. The traditional way to celebrate is to eat pie…too much pie.

The constant $\pi$ arises in many ways in math and in nature (not just from circles). Since we just finished our section on infinite series, I thought I’d mention one here. Since we know that $\tan(\frac{\pi}{4})=1$, we also know that $\arctan(1) = \frac{\pi}{4}$. Therefore, $4 \arctan(1)$ is equal to $\pi$. Now, if you were asked to find the Taylor series for the function $f(x) = \arctan(x)$ centered at $x=0$, you’d eventually answer T(x)= x​1/3x^3+1/5x^5-1/7x^7+…. This means that the Taylor series for $4f(x) = \arctan(x)$ centered at $x=0$ is 4T(x) =4(x​1/3x^3+​1/5x^5−1/7x^7+…). Since $T(x)$ converges to $\arctan(x)$, $4T(1)$ converges to $4 \arctan(1)$, which is $\pi$. But 4T(1) = 4(1-1/3+1/5-1/7+…). So here we have an infinite series that converges (which test would you use?) and it converges to $\pi$.

Here are some links about $\pi$ and about $\pi$ day.

Why Pi Matters by Steven Strogatz

Mathematics Much More than Pi Day by Darren Glass

Happy Pi Day 2015 by Hojae Lee

Neil deGrasse Tyson’s Twitter…containing a fun tweet storm about $\pi$ day (among other nerdy things)

Clip from Star Trek Spock knows what’s up: $\pi$ isn’t just irrational, it’s also transcendental. This means that $\pi$ can’t be the solution to any polynomial equation.

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