The History of π
Written by Dr. David Murphy
Each year on March 14, people around the world celebrate a number. That’s strange! But March 14, when written as 3/14, gives the first few digits of the famous mathematical constant π, which is approximately equal to 3.14159265358979…. What is π and why is it so special that we honor it every March 14? To understand some reasons, let’s start at the beginning, or as close to the beginning as we can find, for the history of π spans nearly 4000 years and it “has been known for so long that [its origin] is quite untraceable” (MacTutor, A history of Pi).
First of all, π is defined as the ratio of the circumference to the diameter of a circle.[1] For this definition to make any sense, we need to know that this ratio is the same for all circles, not something that varies from one to another. Pause for a moment and reflect on this idea. All circles, no matter how large or how small, no matter where they are, have this constant somehow embedded in them. Their diameter may be large, but then so will be the circumference so the ratio is the same as when both are small. Though we don’t know who first proved this fact, it remains as true today as it was then.
The Babylonians and Egyptians knew this, as did the Hebrews and Chinese. All gave approximations of this ratio. In the Egyptian Rhind Papyrus (ca. 1650 BC), π is approximated by (16/9)2 = 3.16049…, a value that differs from π by about 0.0189 (so the relative error is only 0.6%). Other Egyptian and Mesopotamian approximations include 25/8 = 3.125, whose relative error is only 0.5%, and √10 = 3.16227766…, whose relative error is also less than 1%. Some argue that the Bible records the value of π as 3, for in I Kings 7:23 and II Chronicles 4:2 a round brass tub for Solomon’s temple is described to be “ten cubits from the one brim to the other … and a line of thirty cubits did compass it round about.” Yet, others contend that these measurements refer to the outer diameter of this thick-sided tub (to be a hand-breadth in thickness, II Chronicles 4:5) and its inner circumference, which yields the approximation of π as 355/113 = 3.14159292, far better than 3! This same approximation was given by Tsu Ch’ung Chi (429-501 AD).
Many of these approximations were likely obtained from measuring circular objects. If you try to check by measuring the circumferences and diameters of a drinking glass and a round table, you might find the values differ. So are mathematicians wrong? The answer is no, for the objects you measured aren’t perfect circles, but the mathematician (and all math students) study the idea of a circle, its form, rather than its manifestations. The transition from empirical to abstract mathematics based on deductive reasoning, which is necessary in order to know π and other mathematical results properly, was made by the ancient Greeks. While Euclid (around 300 BC), whose Elements is one of the greatest books of all time, never explicitly proves that the ratio of circumference to diameter is constant, in Book XII he does prove
Proposition 2: Circles are to one another as the squares on their diameters
and
Proposition 18: Spheres are to one another in triplicate ratio of their respective diameters.
Neither of these specifies the value of the common constant of area to squared-diameter for all circles or the constant that is the common ratio of volume to cubed-diameter for spheres, nor does it suggest there should be any relationship between them. Yet, they proved to be key steps toward this discovery.
Archimedes of Syracuse (287-212 BC), the greatest scientist of antiquity, in the third proposition of his Measurement of a Circle proved “The ratio of the circumference of any circle to its diameter is less than 22/7 but greater than 223/71.” That is, 223/71 < π < 22/7, or 3.140845… < π < 3.142857143…. In this, he pinned π down to two decimal places of accuracy as 3.14, the value we use to get March 14 as the date to celebrate. But why do we celebrate this number?
First of all, π is all around us. Those results of Euclid, while he didn’t specify the value of the ratios, all point to π. It is the ratio of the circumference to the diameter of any circle, but also the ratio of the area to the square of the radius of each circle. Moreover, for a sphere, the ratio of volume to the cube of the radius is always the same, and this is 4π/3, while the ratio of the surface area to the square of the radius is always 4π. “Okay,” you might say, “circles and circular things all involve π. That’s nice, but why does it matter? I don’t deal with circles every day, so why set aside one day each year to celebrate this circular constant?” It turns out that π appears in far more than just geometry. We see it again and again in calculus, especially when dealing with infinite series. Leonhard Euler (1707-1783), the most prolific mathematician of all time, proved that 1 + 1/4 + 1/9 + 1/16 + … + 1/n2 + … is equal to π2/6. More surprisingly, the bell curve that is so important in probability and statistics, which can be used to describe the distributions of the heights of men and their IQs, involves this same constant, π. That’s amazing! It’s for reasons like this one, how π shows up seemingly out of nowhere to be just the right thing for our problems, that we celebrate Pi Day each year. And this year especially, for recall those extra digits of π = 3.1415926…. This year, we have the once-in-a-lifetime opportunity to observe 3/14/15 (at 9:26 if you want more precision), an opportunity to surpass Archimedes’ approximation.
Yet, as precise as this is, we can never pin π down entirely. Mathematicians have proven that π is irrational (i.e., it cannot be expressed exactly as a fraction). Its digits continue, therefore, without any repeating pattern to be observed. One after another they continue mysteriously, just as π appears again and again mysteriously in so many applications. The mystery of π is our cause for celebration!
Sources and Resources
http://www-history.mcs.st-and.ac.uk/HistTopics/Pi_through_the_ages.html
[1] Recall that the circumference of a circle is the length around it while its diameter is the distance across it.
Dr. David Murphy is the Associate Professor of Mathematics at Hillsdale College.