π

How can you talk about interesting numbers without bringing up π?
History
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The oldest value we know for π comes from the Babylonians. (Man, but those guys were impressive mathematicians; almost any time you look at the history of fundamental numbers and math, you find the Babylonians in the roots.) They tended to work in ratios, and the approximation that they used 25/8s (3.125), which is not a terribly bad approximation. Especially when you realize *when* they came up with this approximation: 1900BC!
The next best approximation came from Egypt, around the time of Pharaoh Amenemhat in the mid 17th century BC, where it had been refined to 256/81 (3.1605). Which isn’t such a great step forward; it’s actually a hair *farther* from the true value of π than the Babylonian approximation.
We don’t see any real progress until we get to the Greek. Archimedes (yes, *that* Archimedes) worked out a better approximation. He used a *really* neat trick. He worked out how to compute the perimeter of a 96-sided polygon; and then worked out the perimeter of the largest 96-gon that could be drawn inside the circle; and the smallest 96-gon that the circle could be drawn inside. Here’s a quick diagram using octagons to give you a clearer idea of what he did:

circle-inscribe.jpg

That gives you an approximation of π as the average of 223/71 and 22/7 – or 3.14185.
And next, we find progress in India, where the mathematician Madhava worked out a power series definition of π, which allowed him to compute π to *13* decimal places. *13* decimal places, computing a power series completely by hand! Astounding!
Even better, during the same century, when this work made its way to the great Persian Arabic mathematicians, they worked it out to 9 digits in base-60 (base-60 was in inheritance from the Babylonians). 9 digits in base 60 is roughly 16 digits in decimal!
And finally, we get back to Europe; in the 17th century, van Ceulen used the power series to work out 35 decimal places of π. Alas, the publication of it was on his tombstone.
Then we get to the 19th century, when William Rutherford calculuated *208* decimal places of π. The real pity of that is that he made an error in the 153rd digit, and so only the first 152 digits were correct. (Can you imagine the amount of time he wasted?)
That was pretty much it until the first computers came along, and once that happened, the fun went out of trying to calculate it, since any bozo could write a program to do it. There’s a website that will let you look at its computation of [the first *2 hundred million* digits of π][digits].
The *name* of π came from Euler (he of the great equation, e + 1 = 0). It’s an abbreviation for *perimeter* in Greek.
There’s also one bit of urban myth about π that is, alas, not true. The story goes that some state in the American midwest (Indiana, Iowa, Ohio, Illinois in various versions) passed a law that π=3. Didn’t happen.
What is π?
—————–
Pretty much everyone is familiar with what π is. Take a circle on a plane. Measure the distance around the outside of it, which is called the circumference. Divide that by the diameter of the circle. That’s π.

circle-ratio.jpg

It also shows up in almost anything else involving measurements of circles and angles, from things like the sin function to the area of a circle to the volume of a sphere.
Where it gets interesting is when you start to ask about how to compute it. You get the relatively obvious things – like equations based on integrals to calculate the area of a circle. But then, you get the surprising ones. After all, π is a fundamental geometric number; it comes from the circumference of a circle.
So why in the world is the radius of a circle related to an infinite sum of the reciprocals of odd numbers? It is.
π/4 is the sum of the infinite series 1/1 – 1/3 + 1/5 – 1/7 + 1/9 – 1/11 + 1/13 …., or more formally:
pi-sum-eq.gif
Or how about this? π/2 =
pi-prod-eq.gif
What about a bit of probability? Pick any integer at random. What’s the probability that neither it nor any of its factors is a perfect square? 6/π2.
How about a a connection between circles and prime numbers? Take the set of all of the prime numbers P, then the *product* of all factors (1-1/p2) is.. 6/π2.
What’s the average number of ways to write an integer as the sum of two perfect squares? π/4.
There’s also a funny little trick for memorizing π, called a piem (argh!). A piem is a little poem where the length of each word is a digit of π.
How I need a drink, (3 1 4 1 5)
alcoholic of course, (9 2 6)
after the heavy lectures (5 3 5 8)
involving quantum mechanics. (9 7 9)
It’s a strange, fascinating little number.
[digits]: http://www.angio.net/pi/piquery

22 thoughts on “π

  1. MattF

    The whole story about the attempt, in 1897, to set a value of pi that was legal (and copyrighted!) in Indiana is told in Berggren, Borwein, & Borwein ‘Pi: A Source Book’– which, by the way, is a terrific reference. Anyhow, the bill was passed in the Indiana House, but fortunately didn’t get beyond the Indiana Senate.

    Reply
  2. James

    My favourite pi mnemonic is the pi song from Songs To Wear Pants To. STWPT is a website where the site owner will write one-minute songs on topics suggested by readers. One request was a song that was a mnemonic for the first fifty digits of pi:
    http://www.archivestowearpantsto.com/tracks/0052_i_am_the_first_fifty_digits_of_pi.mp3
    Man, I can’t, I shan’t
    Formulate an anthem
    Where the words comprise mnemonics
    Dreaded mnemonics for pi.
    The numerals just bother me, always,
    Even the dry anterior.
    Try to request something lower (zero)
    In numerary aptitude.
    Even I, pantaloon galant,
    I cannot actuallize
    The requested mnemonics
    Leading fifty, I…
    He does have a mistake in this — it the last two lines should have 3,9,9,3,7,5,1, but only have 3,9,9,7,5,1 letters. But then, he did say that he couldn’t do it…

    Reply
  3. malpollyon

    Fifty digits of pi is nothing, “Poe, E.: Near a Raven” (google it) is a mnemonic for the first SEVEN HUNDRED AND FORTY digits.

    Reply
  4. malpollyon

    Of course “Cadaeic Cadenza” is rumored to contain the first 3835 digits, but every time I try to check, I faint in sheer awe.

    Reply
  5. Chris Nelson

    Fascinating as always.

    worked out the perimeter of the largest 96-gon that could be drawn inside the circle; and the smallest 96-gon that the circle could be drawn inside

    Shouldn’t the second “inside” be “outside”?

    Reply
  6. slawekk

    >Pick any integer at random
    What distribution are we using here? Typically, when we don’t specify a distribution we mean the uniform one. But there is no uniform distribution on (all of) integers…

    Reply
  7. Mark C. Chu-Carroll

    Slawekk:
    Technically, it’s a limit.
    If you take the limit of (sum from 1 to n : probability(n has no perfect square factors))/n as n approaches infinity, the value approaches 6/π2.

    Reply
  8. 胖猫

    Dunno if it’s just my browser messing up, but don’t you have Wallis’ infinite product shown twice? Shouldn’t the first one be Leibniz’ infinite sum?

    Reply
  9. Jawad Shuaib

    I memorize 10 digits per day to keep my memory sharp. By now, I have pi memorized to several 100s. The way I memorize the number is to divide a long section of numbers into groups of 10, like:
    1415926535 8979323846 2643383279
    Then, I further sub divide a group of 10 digits into two numbers:
    14 15 92 65 35 | 89 79 32 38 46 | 26 43….
    By storing these numbers in my brain this way, I can recall them by easily saying “fourteen fifteen ninety two…etc”, instead of saying “hundred and fourty one, five hundred and ninety two…” which is some what more difficult.
    Remember, the key to memorizing anything is organizing information in the brain.

    Reply
  10. parik

    Facinating,
    the most amazing thing is to realize the ratio of circumference to diameter is always constant, this can practically concluded easily , but to arrive and this conclusion theoritcally might have been difficult , especially in ealier days when there no concept of limits , i am still looking for geometrical theorm which says pi = circuference/diameter

    Reply
  11. livetoad

    A minor quibble on place names. You follow the well-established practice of referring to ancient Iraq as “Babylon”. Everyone does this. Actually much of the math that gets credited to Babylon comes from Sumer and other ancient locales. (Babylon may seem ancient, but consider how old the culture was when Hammurabi established his empire there!) You could refer to “Mesopotamian math”, but then we don’t normally say “Nilotic” or “Gangetic” or “Yaluvian” [?!?] math (although some do). Normally we refer to the math of ancient “Egypt” or “India” or “China”, even though those names are anachronistic — and not even the currently correct native names! (India is “Bharat”, etc.) Why not just be consistent and refer “ancient Iraq” rather than “Babylon”?

    Reply
  12. Matt

    parik–
    There’s a ‘similarity’ argument that proves what you are looking for. Since all the properties of a circle have to be functions functions of a single parameter (the radius), one concludes right away that any two ‘one-dimensional length’ properties (such as the diameter and the circumference) have to be proportional.

    Reply
  13. Krishna

    Hi,
    Can you provide some usage of pi in real world where I can explaing to students and public in general.
    thanks
    Krishna

    Reply
  14. Pingback: Infinite and Non-Repeating Does Not Mean Unstructured | Good Math/Bad Math

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