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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