Pi Machine Week 1: Introduction

Activity info on ESCOT web site.

Have you ever wondered what the value of pi is, or where this number came from?

By definition, Pi is a ratio that compares the circumference of a circle to its diameter. One very interesting fact about this ratio is that its value is always the same number, no matter which circle you use to compute it! Let's take a look at some examples of different circles:

Another interesting thing about pi is that it is an irrational number. This means that it cannot be written as a fraction (note: 22/7 is just an approximation of the value), and it cannot be written as a repeating or terminating decimal. Therefore, we can only estimate its value, we could never really write all the digits of pi down even if we tried!

In this investigation, we will attempt to estimate the value of pi using a new method. Instead of measuring and creating ratios, we are going to use the computer. Because the computer can generate random numbers very quickly, we can use it to create a simulation, which is a re-enactment of an actual experiment. The experiment that we are simulating works like this: pretend you had some rice in your hands, and you drop all the grains onto a dart-board which is lying on the floor. We can simulate the way that the rice falls by generating random points on a circular board.

Activity

Imagine a circle of radius =1 that is drawn inside a square so that it touches all four sides of the square like the picture on the left. Now imagine that points were generated randomly anywhere in the square. Which of the following four choices do you think best estimates what percentage of the points would fall on the circle?

1. 25% (0.25)
2. 50% (0.5)
3. 75% (0.75)
4. 95% (0.95)

Record your prediction in the answer section below.



Part B: Collect data to investigate your prediction.

Run the simulation provided in this page and collect data to support your answers. This simulation generates random points in the board. If the point is in the circle, it appears with a red color. If the point is outside the circle and in the square, it appears with a blue color. Please note that the more points are generated in the same position, the darker the color becomes.

To run the simulation for your experiments, follow these steps:
1. Set the Number of Points that you want the simulation to generate in the Simulation Properties window.
2. Click on the "Run" button.
3. When the simulation is done (the Number of Points will count down to 0).
4. Repeat the experiment for a different number of points. Before running the simulation again, you need hit the "Reset" button. Wait for the picture in the simulation window to completely reload. It may take some time to reload as there are thousands of agents, so please be patient! Enter another number of points and run again.

We suggest that you experiment with 100, 1000, 10000, and 100000 points. Please note that the more points you use the more accurate and consistent the results.



Part C: Reflections

Notice that the ProbabilityInCircle simulation property tells you the percentage of points that fall in the circle (expressed as a decimal number). Are numbers close to the estimate you chose in part A above? Can you explain why or why not? To get full credit for this question, write a mathematical explanation of what happened. To do this, you want to think about comparing the areas of each of the figures. Hint: recall that the radius of the circle is 1.



Part D: Conclusion and Extension

There is something very interesting about these results. If you multiply the ProbabilityInCircle simulation property by 4, you get an approximation for pi! Can you give a mathematical explanation for why this happens? Write your answer in the answer area below.