All About π : Mystery, History, and Epiphany

π – vector version*

The recent date 3-14-15 generated a few news stories about National Pi Day, but outside of a math class, most people rarely think about this fascinating irrational number. Here’s how I teach students about π.

I begin by explaining the basic difference between rational and irrational numbers. Don’t panic, stay with me. It will be okay. All rational numbers can be expressed as the ratio of two integers. The following numbers are examples of rational numbers:  0 = 0/2, 1.5 = 3/2, 53 = 53/1, 2.6666 . . . = 8/3 and 4.3187187187 . . . = 43,144/9,990.

At some point in the decimal expansion of any rational number, the digits start to repeat in a pattern. If a number is irrational, there is no point in the decimal expansion where the digits start to repeat in a pattern. In grad school, an epiphany came when I learned that there are infinitely many more irrational numbers than rational numbers and the number of rational numbers is an infinitely small fraction of the irrational numbers. The first 51 digits of the irrational number π are:

π = 3.14159  26535  89793  23846  26433  83279  50288  41971   69399  37510  . . .

Starting with the 762nd digit and ending with the 767th digit, the decimal expansion of π = 999999. This sequence of digits is called the Feynman Point, named after the brilliant physicist Richard Feynman. In one of this lectures, Feynman said he would like to memorize π to the 762nd digit so he could recite the digits to that point and then quip, “nine nine nine nine nine nine and so on.”

When I ask people the definition of π, most of them, including some college grads, can’t give me the correct definition. The constant π is defined to be the ratio of the circumference of a circle to the diameter of the circle. As I understand it, Euclid’s Elements did not explicitly state that the ratio of the circumference of a circle to the diameter of a circle is always the same number for any circle. The intuitive reason that this ratio is the same for all circles is that all circles are similar figures, and corresponding parts of similar figures are proportional. (There are some interesting discussions on the web about why this ratio is the same for any two circles.)

In the Bible, I Kings 7:23 indirectly gives π a value of 3. “Then he made the molten sea; it was round, ten cubits from brim to brim, and five cubits high, and a line of thirty cubits measured its circumference.” Apparently some people have found this approximation of 3 for π to be alarming, because it might give people the idea that certain passages in the Bible are not to be taken literally. The creationist author Theodore Rybka did a creative piece of mathematics to show that the Bible actually implies π = 3.14 which agrees with the modern version of π to two decimal places.

William Shanks (1812–1882), a British amateur mathematician, over a 15 year period calculated π to 707 decimal places. Shanks also calculated the digits of other mathematical constants such as e and the natural logarithm of 2, 3, 5 and 10 to 137 decimal places, and he published a table of the prime numbers up to 60,000. In 1944, D. F. Ferguson, using a mechanical calculator, showed that Shanks’s calculation was correct only up to the first 527 places.

Modern computers continue to set records for calculating digits of π. On January 7, 2010, using only a personal computer, Fabrice Bellard, a French computer scientist, set a new world record by calculating almost 2.7 trillion digits of π. The last time I checked, over 10 trillion digits of π have been calculated.

What about memorizing the digits of π? It’s not as difficult as you might think. When my son was in fifth grade, a friend of his came to our house to play. One of the boys asked me a question about π. I just couldn’t help myself. Within one hour, both boys could recite the first 50 digits of π. They didn’t know any better; to them it was just great fun. The last year I taught high school math, I recited 70 digits of π to the students in one of my classes. (The first 70 were posted on the wall above the chalkboard, and a student had challenged me to memorize them. Of course, I could not resist the challenge.)

Recently, one of my wife’s friends asked me if there will another Pi day. I said that if she lives to 3-14-2115, she’ll see another one. Then I added that there should be a Pi Day next year, because π rounded to 4 decimal places = 3.1416. This put a smile on her face and gave her enough reason to schedule a Pi party for next year.

No doubt about it, π is a fantastic number. Some people feel that it has mystical properties. How π is used to describe relationships in mathematics and nature is unbounded. Go to the web and find out the world record for memorizing the digits of π—you will be blown away. You can also explore the vast properties, mysteries, and rich history of π. Who knows? You may start a new hobby.

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*Vector version of w:Image:Pi.eq.C.over.d.png from the English Wikipedia, Public Domain

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