Happy pi day!

Happy \pi day! Today, March 14, is celebrated as \pi day, since when today’s date is written in American format (3/14) it corresponds with a common approximation of \pi (3.14). (I don’t know when Europeans celebrate \pi day. [Edit: According to Jester (see below), Euopeans celebrate it on July 22 -- 22/7 is another common approximation of \pi.]) \pi, of course, is the ratio between any circle’s circumference and diameter, and shows up all over the place in mathematics (not only in geometry). As you probably know, it isn’t equal to 3.14, or 22/7; it’s actually irrational, which means that it can’t be expressed as a fraction, and its decimal representation continues on forever without repeating.

\pi \approx 3.1415926535 8979323846 2643383279 5028841971 6939937510 5820974944 5923078164 0628620899 8628034825 3421170679 8214808651 3282306647 0938446095 5058223172 5359408128 4811174502 8410270193 8521105559 6446229489 5493038196 4428810975 6659334461 2847564823 3786783165 2712019091 4564856692 …

In celebration of this amazing number, here are some fun \pi facts and links for you!

\displaystyle \frac{\pi}{4} = 1 - \frac{1}{3} + \frac{1}{5} - \frac{1}{7} + \frac{1}{9} - \dots

\displaystyle \frac{\pi^2}{6} = 1 + \frac{1}{4} + \frac{1}{9} + \frac{1}{16} + \frac{1}{25} + \dots

\displaystyle \frac{\pi}{2} = \frac{2 \cdot 2}{1 \cdot 3}\cdot \frac{4 \cdot 4}{3 \cdot 5} \cdot \frac{6 \cdot 6}{5 \cdot 7} \cdot \frac{8 \cdot 8}{7 \cdot 9} \dotsb

\displaystyle \pi = \int_{-1}^{1} \frac{2}{1 + x^2}

\displaystyle \frac{2}{\pi} = \sqrt{\frac{1}{2}} \cdot \sqrt{\frac{1}{2} + \frac{1}{2} \sqrt{\frac{1}{2}}} \cdot \sqrt{\frac{1}{2} + \frac{1}{2} \sqrt{\frac{1}{2} + \frac{1}{2} \sqrt{\frac{1}{2}}}}\dotsb

Edit: Denise of Let’s play math! has a few more interesting pi-related links and quotes, including a few explaining how to calculate pi by throwing hot dogs on your kitchen floor. I’m not kidding, go check it out.

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6 Responses to Happy pi day!

  1. Pseudonym says:

    Tomorrow is the Ides of March, so Beware!

  2. Brent says:

    Heh, true, thanks for the warning! =)

  3. Jester says:

    We, Europeans, celebrate it on 22/7 (european format) :D

  4. Brent says:

    Ah, I see!

  5. Steve Gilberg says:

    If I try to figure out how people discovered all those pi equations, I will sprain my brain.

    What’s about equally fascinating to me is the sharp probability dropoff in the “find your name in pi” link. If you’re looking for a set of six characters, you have a 97.6% chance of finding it in the first 4 billion digits; if it’s a set of seven, your chance is only 11%!

  6. Brent says:

    Most of those pi equations emerged from various areas of mathematics, some fairly deep, some not as deep, but the point is that I don’t think anyone ever really set out explicitly to discover an equation for pi. They were studying something else and all of a sudden (perhaps unexpectedly), pi showed up. Bingo, a formula for pi! And since pi is so intimately connected with so many different areas of mathematics, it comes up a lot.

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