A circumference
divide by diameter
irrational pi
– a "pi-ku" by Paul Doherty
CUTTING π
Materials
circular object
string
scissors
tape
To Do and Notice
Carefully wrap string around the circumference of your circular object. Cut the string when it is exactly the same length as the circumference. Now take your “string circumference” and stretch it across the diameter of your circular object. Cut as many “string diameters” from your “string circumference” as you can. How many diameters could you cut? Compare your data with that of others. What do you notice?
What’s Going On?
This is a hands-on way to divide a circle’s circumference by its diameter. No matter what circle you use, you’ll be able to cut 3 complete diameters and have a small bit of string left over. Estimate what fraction of the diameter this small piece could be (about 1/7). You have “cut pi,” about 3 and 1/7 pieces of string, by determining how many diameters can be cut from the circumference. Tape the 3 + pieces of string onto paper and explain their significance.
DISCOVERING π
Materials
variety of circular or cylindrical objects (paper plates, jar lids, cans, etc.)
measuring tape (or meter stick and string)
basic function calculator (optional)
To Do and Notice
Recall that diameter is the distance across a circle and that the circumference is the distance around the circle. Let’s go hunting: look for circles. Use centimeters as your units to measure the diameter and the circumference of each circle you find (remember that the base of a cylinder is also a circle!), and record your data on the following chart. To complete the chart, use your circumference and diameter measurements from each circle and find the sum, difference, product, and quotient of each set of data. Make sure you include your units.
What’s Going On?
You probably noticed that one of the columns above gives you close to the same value for every circle. Your neighbor’s chart will show you the same. This value is usually a little more than 3, close to the constant ratio pi. The more carefully you make your measurements, the closer this value comes to pi. Pi is the constant ratio of any circle’s circumference to its diameter. Pi is an irrational number, which means that it cannot be written as a ratio of two integers, and that its decimal expansion goes on forever and is non-repeating. If we stop the decimal expansion of pi at a certain place, we get an approximation to the number: the more decimal places we retain, the better the approximation we get.
pi = 3.141592653589793......, and a common approximation is pi ≈ 3.14
Notice that the linear centimeter units remain as linear measurements when you add or subtract the circumference and diameter. The product of two linear measurements gives you square units. The quotient has no units, as centimeters/centimeters “cancel”. Pi is unit-less.
TOSSING π
Materials
large sheet of drawing paper or cardboard
meterstick
pen
toothpicks (30 or more)
calculator
To Do and Notice
Draw a series of parallel lines on the paper or cardboard, as many as will fit, making sure that the distance between each line is exactly equal to the length of your toothpicks. Now, one by one, randomly toss toothpicks onto the lined paper. Keep tossing until you’re out of toothpicks—or tired of tossing.
It’s time to count. First, remove any toothpicks that missed the paper or poke out beyond the paper’s edge. Then count up the total number of remaining toothpicks. Also count the number of toothpicks that cross one of your lines.
Now use this formula to calculate an approximation of pi:
Pi = 2 × (total number of toothpicks) / (number of line-crossing toothpicks)
What’s Going On?
This surprising method of calculating pi, known as Buffon’s Needles, was first discovered in the late eighteenth century by French naturalist Count Buffon. Buffon was inspired by a then-popular game of chance that involved tossing a coin onto a tiled floor and betting on whether it would land entirely within one of the tiles.
Increasing the number of tosses improves the approximation, but only to a point. This experimental approach to geometric probability is an example of a Monte Carlo method, in which random sampling of a system yields an approximate solution. You can play with a simulation of this activity on this page.
SEARCHING π
Materials
Internet access
To Do and Notice
Pick a number sequence that’s special to you—perhaps your birth date.
Go to the Pi-Search Page and type your sequence in the search box at the top of the page. This web site will search the first 200 million digits of pi in a fraction of a second. (See “How it works” on the Pi-Search Page to find out how this is accomplished.) If it finds your sequence, it will tell you at what position in pi your sequence begins and will display your sequence along with surrounding digits.
No result? Try another sequence. The shorter the sequence, the better the odds of finding it.
What’s Going On?
Pi is an irrational number, which means that its digits never end and that it doesn’t contain repeating sequences of any length. If Pi-Search didn’t find your sequence of numbers, that’s probably because the sequence occurs somewhere past the first 200 million digits. Note the qualification “probably”: Mathematicians can’t say with absolute certainty that pi contains every possible finite number sequence—but they strongly suspect that this is the case.
As of 2011, pi has been calculated to 10 trillion decimal places. When mathematicians study any sample of this huge number, they find that each digit, 0–9, occurs as often as any other, and that the occurrence of any digit seems unrelated to the preceding digit. This makes pi appear to be statistically random. If this statistical randomness is unending, then pi must contain all finite sequences of digits, including the birth dates of everyone ever born and yet to be born. It would also contain every winning lottery number—too bad we don’t know how to identify them.
