Pi Pops Up Where You Dont Expect It

Happy Pi Day, where we celebrate the worlds most famous number. The exact value of =3.14159 has fascinated people since ancient times, and mathematicians have computed trillions of digits. But why do we care? Would it actually matter if somebody got the 11,137,423,895,285th digit wrong?

Probably not. The world would keep on turning (with a circumference of 2r). What matters about isnt so much the actual value as the idea, and the fact that seems to crop up in lots of unexpected places.

Lets start with the expected places. If a circle has radius r, then the circumference is 2r. So if a circle has radius of one foot, and you walk around the circle in one-foot steps, then it will take you 2 = 6.28319 steps to go all the way around. Six steps isnt nearly enough, and after seven you will have overshot. And since the value of is irrational, no multiple of the circumference will be an even number of steps. No matter how many times you take a one-foot step, youll never come back exactly to your starting point.

^{Calculating the area of a circle with wedges.} Jim.belk

From the circumference of a circle we get the area. Cut a pizza into an even number of slices, alternately colored yellow and blue. Lay all the blue slices pointing up, and all the yellow slices pointing down. Since each color accounts for half the circumference of the circle, the result is approximately a strip of height r and width r, or area r². The more slices we have, the better the approximation is, so the exact area must be exactly r².

Pi in other places

You dont just get in circular motion. You get in any oscillation. When a mass bobs on a spring, or a pendulum swings back and forth, the position behaves just like one coordinate of a particle going around a circle.

^{Simple harmonic motion is another view of circular motion.}

If your maximum displacement is one meter and your maximum speed is one meter/second, its just like going around a circle of radius one meter at one meter/second, and your period of oscillation will be exactly 2 seconds.

^{The area of the space under the normal-distribution curve is the square root of pi.} Autopilot, CC BY-SA

Pi also crops up in probability. The function f(x)=e^-x, where e=2.71828 is Eulers number, describes the most common probability distribution seen in the real world, governing everything from SAT scores to locations of darts thrown at a target. The area under this curve is exactly the square root of .

How did get into it?! The two-dimensional function f(x)f(y) stays the same if you rotate the coordinate axes. Round things relate to circles, and circles involve .

Another place we see is in the calendar. A normal 365-day year is just over 10,000,000 seconds. Does that have something to do with the Earth going around the sun in a nearly circular orbit? Actually, no. Its just coincidence, thanks to our arbitrarily dividing each day into 24 hours, each hour into 60 minutes, and each minute into 60 seconds.

Whats not coincidence is how the length of the day varies with the seasons. If you plot the hours of daylight as a function of the date, starting at next weeks equinox, you get the same sine curve that describes the position of a pendulum or one coordinate of circular motion.

Advanced appearances of

More examples of come up in calculus, especially in infinite series like
1 – (¹₃) + (¹₅) – (¹₇) + (¹₉) + = /4
and
1² + (¹₂)² + (¹₃)² + (¹₄)² + (¹₅)² + = ²/6
(The first comes from the Taylor series of the arctangent of 1, and the second from the Fourier series of a sawtooth function.)

Also from calculus comes Eulers mysterious equation
eⁱ + 1 = 0
relating the five most important numbers in mathematics: 0, 1, i, , and e, where i is the (imaginary!) square root of -1.

^{A graph of the exponential function y=e^x.} Peter John Acklam, CC BY-SA

At first this looks like nonsense. How can you possibly take a number like e to an imaginary power?! Stay with me. The rate of change of the exponential function f(x)=e^x is equal to the value of the function itself. To the left of the figure, where the function is small, its barely changing. To the right, where the function is big, its changing rapidly. Likewise, the rate of change of any function of the form f(x)=e^ax is proportional to e^ax.

^{The relationship between an angle, its sine, cosine and a circle.} 345Kai, CC BY-SA

We can then define f(x)= e^ix to be a complex function whose rate of change is i times the function itself, and whose value at 0 is 1. This turns out to be a combination of the trigonometric functions that describe circular motion, namely cos(x) + i sin(x). Since going a distance takes you halfway around the unit circle, cos()=-1 and sin()=0, so eⁱ=-1.

Finally, some people prefer to work with =2=6.28 instead of . Since going a distance 2 takes you all the way around the circle, they would write that eⁱ = +1. If you find that confusing, take a few months to think about it. Then you can celebrate June 28 by baking two pies.

Lorenzo Sadun, Professor of Mathematics, University of Texas at Austin

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