Today is Pi Day. You know, March 14. 3/14 is sort of like 3.14. Get it? OK, it's a bit of a stretch because 3/14 looks like a fraction and not Pi. Whatever. We still call it Pi Day.
Even if the date of Pi Day is a little weird, Pi is still pretty awesome. Here are some things you might not know about Pi.
If you have a circle, you can measure two things: the distance around the perimeter of the circle (circumference) and the distance across the widest part of the circle (diameter). No matter how big your circle, the ratio of circumference to diameter is the value of Pi. Pi is an irrational number---you can't write it down as a non-infinite decimal. This means you need an approximate value for Pi.
The simplest approximation for Pi is just 3. Yes, we all know that's incorrect, but it can at least get you started if you want to do something with circles. In the past, many math books listed Pi as 22/7. Again, this is just an approximation but it is better than the value of 3 (actually 22/7 is closer to Pi than just writing 3.14).
The early history of mathematics covers many approximations of the value of Pi. The most common method would be to construct a many-sided polygon and use this to calculate the perimeter and diameter as an estimate for Pi. Other cultures found ways to write Pi as an infinite series---but without a computer, this can be sort of difficult to calculate out very far.
There are many methods to calculate Pi but I will go over the simplest to understand. It starts with the inverse tangent function. We know that the inverse tangent of 1 is π/4 and we can use this to calculate Pi. No, you can't just plug it into your calculator and get Pi---that assumes you already know Pi. Instead, we need to do a Taylor Series expansion of the inverse tangent.
The basic idea behind the Taylor Series is that any function sort of looks like a power series if you just focus on one part of that function. Using this, I can represent the inverse tangent of some value (x) as an infinite series:
Expanding this function about the point x = 1 should be equal to π/4. This means we get the following for π: (note: fixed equation on 3/14/16)
That's it. Now you can just plug away at this formula for as long as you like---or you could have a computer do it. Here is a program that calculates the first 10,000 terms in the series (just press play to run it):
See, that's not so difficult for a computer. However, you can see that even after 10,000 terms the calculated value is still different than the accepted value. This isn't the best series to calculate Pi---but I said that earlier.
This is my favorite Pi activity. Here is the idea. Generate pairs of random numbers between 0 and 1 to create random x,y coordinates. Plot these points on a 1 by 1 grid and calculate their distance to the origin. Some of these will have a origin distance less than 1 and some will be greater than 1. The points with a distance of less than one are "inside a circle"---actually it's a quarter of a circle. So, by counting points inside the circle compare to the total points I get an estimate of the area of this circle which should be π/4. That's it.
OK, here is the program.
You really should play around with this (because it's fun). Try changing the number of points or something like that. I included a "rate(1000)" statement so you can see the points being added. Oh, run it more than once---each time you get a different result because of the random part.
Get out your calculator. Use 9.8 m/s2 for the local gravitational constant (g). Now try this:
That's pretty close to the accepted value of Pi---and it's not a coincidence. It comes from the original version of the meter as a unit of length. One way to define a meter is to create a pendulum that takes 1 second to make one swing (or 2 seconds for the period). If you remember, there is a relationship between period and length for a pendulum (with a small oscillation amplitude):
Put in 1 meter for the length and 2 seconds for the period and boom---there is your connection. Here is a more detailed explanation.
This is Euler's Identity.
If you don't think that equation is crazy and awesome, then you aren't paying attention. It makes a relationship between these five numbers:
- Pi: you know, circles and stuff.
- e: the natural number. This number is very important in calculus and other things (here is my explanation from before).
- i: the imaginary number. With this number (the square root of negative 1) we can write complex numbers (combination of real and imaginary).
- 1: the multiplicative identity. It may seem silly, but multiplying by one is very important---just take unit conversions as an example.
- 0: the additive identity. Without the number zero, you really can't have place value so you are stuck with a number system like the Roman Numerals.
But why does this equation work? That's not such a simple answer. Of course, you could use Euler's formula for exponentials:
However, that is sort of like explaining magic with more magic. For me, the problem is that we like to think of numbers as real countable things. But you can't count an imaginary number. You can say that 32 is like 3 groups of 3, but what about 31.32? Or what about 3-3.2i? Those are pretty tough to picture. If you still want to grok this Euler Identity, check out this site.
Imagine a large sphere. If you know the diameter of this large sphere, you can also find the circumference using the value of Pi. Now replace the sphere with the diameter of the observable universe at 93 billion light years (yes, I know this is bigger than 13 billion light years---it's complicated). If we don't know the exact value of Pi, but one 152 digits then we don't know the exact circumference. However, the uncertainty in the circumference is less than the Planck length---the smallest unit of distance measurement that has any meaning. You need even fewer digits of Pi to get a uncertainty in the circumference smaller than the size of an atom.
So, should we just stop looking for more and more digits of Pi? No, we need to continue the quest for a better appoximation of Pi. Anyway, who knows what we will find out there in the digits of Pi. There is already the Feynman point in which there is a sequence of six 9's in a row. And don't forget this classic comic from xkcd.
Do you want Pi Day Homework? OK, here are a few questions for you.
- Find a better numerical recipe for calculating the digits of Pi and do it (in Python or whatever). Warning, you maybe have to import something like the decimal module so that you can display many digits of number.
- Calculate (or estimate) how many digits of Pi you need to calculate the circumference of the universe to within the size of 1 atom.
- Assuming the digits of Pi are random, what is the probability of finding a series of seven 9's in a row? How many digits would you need to calculate to have a 50 percent chance of seeing these seven 9 nines?
- Go back to the random number calculation for Pi. Change the program so that it plots random points in three dimensions instead of just two.
