Happy Pi Day − March 14th!
Hi, everyone! Thought I'll write an article for this otherwise rather barren blog. Seeing that the next date of interest is the 14th of March and the day references a cryptic number of practical and transcendental applications, I have chosen to write a mélange about pi and associated matters.
First of all − why this particular date? Well, the number pi (in decimal format) is 3.14… and this can be written as 3/14 if we take the liberty of replacing the decimal point by the slash (/). Now, 3/14 looks very like a date written in American style i.e. with the month number appearing before the date number and the two separated by the slash. So, 3/14 translates to the third month, 14th day, and, voilà, there we have it folks, March the 14th!!!
So now you know why Pi Day is celebrated on the 14th of March. Hmm, hope that helps someone score some marks in an interview or a GK test.
Update: I have just found out that this is Einstein's birthday (Mar. 14, 1879 - Apr. 18, 1955)! What a great coincidence!
HC (March 28, 2014)
Just to put things in context, this article is strictly limited to the mathematical usage of pi. As such, I think it's worthwhile to define the mathematical usage first, so that the rest of the article can be understood within this context. Pi is the literal symbol that represents the ratio of the circumference of a circle to its diameter. This ratio is a constant − meaning that the ratio remains the same for all circles, doesn't matter if we take a small circle or a really big one. The ratio when expressed numerically is a real number, an irrational number and also a transcendental number. Since it is an irrational number its exact value cannot be computed so we only have an approximation for this ratio. This approximation is the standard textbook 22/7 that we have all studied in school or 3.14… in its decimal representation. It is this value that is represented by the literal symbol pi, so much so that we commonly say that "pi is 22/7", and almost always take it that pi is a synonym for 22/7.
But what is "pi" itself? Pi happens to be the 16th letter of the Greek alphabet. It is written (in uppercase) as Π − yes, like putting a roof over two vertical pillars! The lowercase pi is written in Greek as π and that is the symbol used to denote the number (whereas the uppercase pi is used as a symbol for another mathematical element). The Greek letter pi corresponds to the letter P of the Roman (Latin-English) alphabet. So pi can be written in English as p but I don't think anyone else (apart from me, that is) would consider that quite correct (Traditionalists! Oh, well…).
But why do we choose to denote the number with the Greek letter pi? Why not alpha? Or omega? Why pi? The answer to that question is murky and unclear. Just for your information, the number denoted by pi was known to the ancients of many nations many thousands of years ago, and very obviously they would not have referred to the number as pi! The association of the letter pi with the number 3.14… came about during the Renaissance Period in Europe when there was a tradition in mathematics and science to use Greek letters to represent well-known elements, values, and related concepts (such as the Greek letter theta (Θ uppercase; θ lowercase) being used to represent the measure of an angle in degrees). Wikipedia informs us that the Swiss mathematician Euler was responsible for popularizing the usage of pi by writing: "for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1" (1748) but pi (for the number 3.14…) was first published in a book by William Jones (1706) who credited John Machin for originating the symbolic usage of pi for 3.14…. Today the symbol pi (for the number 3.14…) is used universally in mathematics and science barring few exceptions.
Alright, so now we know why Pi Day is celebrated on the 14th of March, and what pi is, and who started the tradition of denoting the ratio with the symbol.
But what is this number that we're talking about so far? The number denoted by pi is defined as the ratio of the circumference of a circle to its diameter. This ratio happens to be fixed ("constant") for all circles regardless of size. So the ratio of the circumference of a circle with a diameter of 10 cm to its diameter is exactly the same as the ratio of the circumference of a circle with a diameter of 100 cm to its diameter. And what exactly is the value of this ratio? The value of the number denoted by pi is approximately 3.1415926535. Please note: approximately, not exactly. Mathematicians have concluded that the ratio is irrational and there is no end to the number. Another point to note is that there is no sequence of recurring digits in the value as known to date.
I just looked up Wikipedia and the current record for computing the value of pi stands at over 10 trillion digits! That's 1 followed by 13 zeroes! The value was computed using a supercomputer and some fancy algorithms. I'll just give you the value to 100 decimal places:
3.14159 26535 89793 23846 26433 83279 50288 41971 69399 37510 58209 74944 59230 78164 06286 20899 86280 34825 34211 70679
Phew! And to imagine that some people can actually (and correctly) recite the value of pi from memory to 67,000 digits! (Boy, he sure must have had plenty of time on his hands!)
High school mathematics usually gets by with a precision of 4 decimal places (i.e. 3.1416) but engineering requires higher precision and accuracy. As far as the maximum accuracy and precision required is concerned I'll quote from Wikipedia:
"According to Jörg Arndt and Christoph Haenel, thirty-nine digits are sufficient to perform most cosmological calculations, because that is the accuracy necessary to calculate the volume of the known universe with a precision of one atom." (Emphasis mine)
(I'm not complaining, 39 digits are fine by me!)
Apart from the decimal fraction representation, the approximate value of pi can also be expressed as a common fraction as 22/7 which gives a value of 3.142857 (that is, correct to 2 decimal places) or the less common 355/113 which gives a value of 3.14159292 (that is, correct to 6 decimal places).
OK, so we know that pi represents the ratio of the circumference of a circle to its diameter, and that the value of this ratio is an irrational constant number approximating 3.14159…. So?? There are many, surely thousands of, values that are used in science and mathematics. These others are given short shrift in comparison to the fame of pi. Why?
The reason for pi's popularity is that it is Nature's pet! Scientists and mathematicians are amazed at the regular appearance of the value of pi in almost every aspect of the natural world. All measurements associated with circles (and derivatives thereof) require the involvement of this number. It is quite magical in the manner in which it pops up in the most unexpected of contexts. It is this magicality (nay, mysticality is a better way of expressing it) that has attracted such phenomenal recognition of the value denoted by pi.
And yet it is simplicity itself…
Do you want to find out the value of pi yourself, firsthand? Absolutely simple! All you need is a round can, a piece of string, and a measuring tape. Wrap the string around the can just once (as exactly as you can) − the length of the string that completes one circuit of the can is the circumference of the can. Measure it with the measuring tape. Next measure the diameter of the can by spanning the top (or bottom) of the can with the measuring tape. Armed with the circumference and the diameter whip out your handy-dandy pocket calculator (any self-respecting cellphone has a calculator utility built-in!) and do the division. Repeat the experiment about ten times and compute the average of all the results. The answer should be in the region of 3 point something. There! That's how you get the value of pi!
Want to refine the value of pi? Just repeat the steps above with a simple variation − wrap the string around the can 10 times to get the circumference magnified by a factor of 10. Divide this magnified circumference by the diameter and then divide the answer by 10. This should produce a better approximation of pi. Taking a larger can helps − try one with a diameter of 4" or more.
Remember when you were given your first geometry set (box)? One of the first things a kid does is to draw a "flower" using the pair of compasses that comes with the set. Simply open up the compass to a suitable "radius" and draw a circle. Then (with the same radius) place the point of the compass on the circumference of the circle and draw an arc between any two arbitrary points on the circumference of the circle. This arc will intersect the circle at two new points. Using any one of the "new" points as the center, repeat the process of drawing an arc as before. This will produce a "flower" inside the original circle. Count the number of "petals" of this "flower" − six!
This means that the circumference of the circle is 6 times the radius of the circle or, in other words, the circumference is 3 times the diameter (since the radius is half the diameter). But 3 is not pi! Or, is it? Well, actually if you proceed carefully you will notice that when you draw the arcs progressively the final intersection will not align exactly with the starting point − it will be a little short. This means that the circumference is a tad more than 6 times the radius and hence a tad more than 3 times the diameter. That difference is what makes pi irrational! I leave it up to you to find that difference!
Archimedes (that's the 3rd century BC!) used a similar method to get the ratio. He drew a regular polygon and summed the lengths of all the sides to get the circumference approximating the circle around the polygon. He found that the value of the ratio lay between 223/71 (3.14084) and 22/7 (3.14285) with a 96-sided polygon. An excellent approximation! A Chinese mathematician of roughly the same era determined that the ratio lay between 3.141024 and 3.142708 with a 192-sided polygon. Phew!
A 12-sided polygon. The circle is in black, the polygon is in red.
A 24-sided polygon. The circle is in black, the polygon is in red. Note how the 24-sided polygon approximates the circle better than the 12-sided polygon above.
Please note that the diagrams shown above are meant to explain the basic concept behind the method. The accuracy can be enhanced by increasing the radius of the circle. However, it is impractical to draw really big circles on paper. Instead, the ancients used geometrical algorithms to perform the computations.
Approximations of pi through the ages
The ratio of the circumference to the diameter was known to the ancients, even as early as the 19th century BC, but this we surmise from the monuments that they built and from mostly indirect references. Wikipedia informs us that the Babylonians took the ratio to be 25/8 (3.12). The Egyptians used 256/81 − the square of (16/9), or (4/3) squared twice − approximating to 3.16. In the 9th century BC, the Indian mathematician Yajnavalkya used 339/108 (3.13888). Then came Archimedes (as described above) and Ptolemy (2nd century BC) with 377/120 (3.1416) − that's as close as they come!
In the 5th century AD it was Aryabhatta who calculated the ratio to 62832/20000 (3.1416). The Chinese used 22/7 and 355/113 for the ratio. Later on, in the 14th century AD, the Indian mathematician Madhava computed the ratio correct to 11 decimal places − 3.14159265359. Now that's good! Just about then a Persian mathematician hit 17 correct decimal places − 3.14159265368979324 using a regular polygon with more number of sides than I can count! Well, 268,435,456 sides to be precise! That's more than 268 million sides if you didn't notice!
Why do we say that pi is a transcendental number? A transcendental number cannot be computed from an algebraic formula; that is, we cannot construct an algebraic formula that evaluates to the subject number. Well, pi's like that! We cannot construct an algebraic statement that evaluates to the value of pi.
In A Lighter Vein…
"How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics."
How true! But what's that got to do with pi? Well, it just happens to be a pretty well-known mnemonic for pi, that's all! This particular mnemonic is attributed to the English scientist James Jeans (1877 - 1946). The number of letters in each word denotes the respective digit of pi.
How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics
3 1 4 1 5 9 2 6 5 3 5 8 9 7 9
People have come out with longer versions of text − both prose and poetry − as aides memoire for pi. Just as poetic licence permits a poet to transcend standard grammar for the sake of meter and rhyme, constrained writing allows the same "liberty" for compositions that represent the digits of pi. That's called Pilish, if you please! When the text takes the form of a poem it is often humorously referred to as a "piem". Mike Keith, an American mathematician, composed such a piem and I reproduce the first two stanzas (80 words = 80 digits!) for your perusal:
Midnights so dreary, tired and weary,
Silently pondering volumes extolling all by-now obsolete lore.
During my rather long nap - the weirdest tap!
An ominous vibrating sound disturbing my chamber's antedoor.
"This", I whispered quietly, "I ignore".
Perfectly, the intellect remembers: the ghostly fires, a glittering ember.
Inflamed by lightning's outbursts, windows cast penumbras upon this floor.
Sorrowful, as one mistreated, unhappy thoughts I heeded:
That inimitable lesson in elegance - Lenore -
Is delighting, exciting...nevermore.
The whole effort (3835 words = 3835 digits!!!) is named "Cadaeic Cadenza" and can be viewed in its entirety at http://www.cadaeic.net/cadenza.htm.
What does "cadeic" mean? If you analyse the word cadeic letter by letter you'll find that the ordinal of each letter's position in the Roman (Latin-English) alphabet spells out the corresponding digit of pi - "c" is the third letter (3), "a" is the first letter (1), "d" is the fourth letter (4)… and so on…
Mr. Keith has also written an entire book of 10,000 words representing 10,000 digits of pi! Wow! To find out more about the book, "Not A Wake", just visit http://www.cadaeic.net/notawake.htm.
What about you? Can you come up with something of your own? Of course it must be original! Do it, and send it to me and I'll post it here. Maybe I can even get someone to sponsor a prize for the best effort! What prize? Well, how about a pie?
I'll wind up this article with two popular misconceptions about pi:
First of all, pi is NOT a number, least of all 22/7 or 3.141… Pi is a Greek letter used as a symbol representing (among many other things) the ratio of the circumference of a circle to its diameter.
Second, 22/7 is not the ratio at all! 22/7 is simply a convenient approximation of the subject ratio. In fact, Archimedes set 22/7 as the upper bound for the approximation while 223/71 was the lower bound.
On that note, I'll just wave goodbye to y'all! Take care and have yourself a happy pi day!
12th March, 2014