# The Search for Pi: 3.14 and Beyond

Started in 1988 at San Francisco’s Exploratorium museum, March 14, or Pi Day, has become a classroom tradition. But what is pi? Why do we celebrate it by slicing up circular junk food? And why are mathematicians still calculating digits of pi into the trillions of decimal places?

Pi is a mathematical constant, meaning its value never changes. It describes the relationship between a circle’s circumference (the distance around a circle) and its diameter (the distance across a circle at its widest point). Today, we also describe pi as an irrational, transcendental number. In other words, a real thinker—let’s break down this description to understand why it’s possible that we’re still finding new digits of pi.

First we have to start with real numbers. Real numbers make up pretty much all numbers that are not purely conceptual (like infinity, ∞). However, real numbers are not always tidy and neat. The “tidiest” real number is a whole number (0, 1, 2, 3, etc.). Slightly messier are the fractions (¼, ½, ¾, etc.) but these are still simple enough: divide 7 by 8 and you can express it as 0.875. Whole numbers and fractions are both rational numbers because they can be expressed as one particular number divided by another. Then we have the irrational numbers like pi. Pi is real, but there is no fraction that accurately describes it because after its decimal point is a never-ending but distinct row of numbers without a repeating pattern. Some numbers like 1/3 end in a repeating pattern, as in 0.333…, making them never-ending but still rational. Even calculating out to trillions of decimal places, mathematicians have yet to find a discernible pattern to pi.

Whew! Ok, now we’ve reached the transcendental part. Real numbers are not just rational or irrational, they’re also algebraic or transcendental. Basically (very basically), pi cannot be solved using an algebraic equation, which uses whole numbers to solve for x. As a quick refresher, a super simple algebraic equation looks something like this: x = 3–2x where x = 1.

Pi is used in equations that are mapping the shape of the universe, searching for planets outside our galaxy, discovering how DNA folds, simulating atoms and molecules in computer models, making predictions in particle physics, and much more, so it’s not just a cool parlor trick or a way to get in The Guinness Book of World Records. It’s used in science and engineering every day.

But pi wasn’t always known as an irrational, transcendental number. It has taken mathematicians 4,000 years to uncover the mysteries of pi, and they may not be done yet.

The Babylonians and the Egyptians were the first to identify pi as the relationship between the circumference and diameter of a circle. Instead of using a mathematical system based on tens like the metric system many of us are familiar with, they used a sexagesimal system based on 60 (1). To approximate the value of pi, they inscribed a hexagon within a circle. The sides of the hexagon were equivalent to the circle’s radius, so the outer shape of the hexagon roughly correlated to the circumference of the circle, enabling them to estimate pi at 3.125 (the first five decimal places of pi are 3.14159 so they were pretty close). The Rhind Papyrus written by the Egyptian scribe Ahmes around 1,650 BCE recorded pi as 3.16049 (1).

Eudoxus of Cnidus (400–350 BCE) was a Greek astronomer and mathematician. Famous for his geometric, celestial model that interlaced the motions of the Sun, moon, and the five planets known at the time, Eudoxus was particularly interested in the study of proportions and he demonstrated that the area of a circle is proportional to the square of its radius.

While we often define pi as the proportion of its circumference to its diameter, A= πr2 is perhaps the first use of pi many of us memorize in school, and the premise of this equation led mathematicians on the long path of trying to solve the constant.

One of Greece’s most famous mathematicians—if not the most famous—Archimedes of Syracuse (287-212 BC)

took the calculation of pi to new heights by using multiple polygons (with five sides) inscribed in a circle, enabling increasingly accurate calculations. Imagine laying polygon cutouts into tight, circular rotations within a circle; the more you lay down, the more sides this composite polygon has and the closer it gets to representing the circumference of the circle. While this inscribed and circumscribed polygon method has some physical limitations (Archimedes used only two regular polygons in his calculations), it was accurate enough to give him a value for pi in the range between 3.143 and 3.141 (1), closer than the Babylonians’ 3.125 estimate. Polygons can be used to approximate the area radius of a circle. The more sides a polygon has, the closer it comes to matching the dimensions of a circle.

Archimedes was not the only mathematician to use this method. About 400 years later in China, Liu Hui would include 192 sides of inscribed polygons to get a value of 3.14159, which is 100 percent accurate—if you’re only going to the first five decimal places. In another 200 years, the father-son mathematician duo of Tsu Ch’ung-chih and Tsu Keng-chih calculated more than 24,000 polygon sides. Their results added another accurate decimal place to the growing number.

A century later, another method for calculating pi would be developed in India in the sixteenth century by the Kerala school, which was responsible for several mathematical achievements that would be re-discovered in Europe a couple of centuries later. One such achievement was the representation of pi as a series notation (π/4 = 1 -1/3  + 1/5 – 1/7…).

Meanwhile in Europe, mathematicians were launching a rigorous search for the limits of pi. Using Archimedes’ method with 500 million polygon sides, the German mathematician Ludolph Van Ceulen found 35 digits of pi by the end of his life in 1610. (To really cement his achievement, he had the digits carved on his tombstone). The French mathematician and “father of modern algebraic notation” François Viète was likewise on the case. After using an inscribed polygon method similar to the Greeks’ to solve for a whopping 393,216 sides, Viète would use algebraic notation to write pi as an infinite product (2). Following Viete’s formula in 1579, the seventeenth and eighteenth centuries saw a slew of new formulas proposed by mathematicians including John Wallis, William Brouncker, James Gregory, Gottfried Leibniz, and (perhaps not surprisingly) Sir Isaac Newton.

Something we haven’t mentioned up until this point: when all these people were finding expressions for pi, how were they discussing it? Before the symbol π was first to describe the mathematical constant by Williams Jones in 1706, mathematicians were burdened with the Latin phrasequantitas in quam cum multiflicetur diameter, proveniet circumferencia (the quantity which, when the diameter is multiplied by it, yields the circumference)”. The symbol represents the never-ending value that relates the circumference and diameter of a circle and would become popularized by Leonhard Euler half-a-century later . Mathematicians finally arrived at our current description of pi (recall it’s a mathematical constant and irrational, transcendental number) in 1881 when Ferdinand von Lindemann showed that pi is transcendental, or unable to be solved using an algebraic equation with rational coefficients. Proving that pi is transcendental ends a somewhat mystical search by the Greeks to “square the circle” or convert the area of a circle to a square.

In the nineteenth century computers would change everything, including the race for pi’s furthest digits. Mathematician John Wrench, a researcher for the U.S. Navy in the 1940s and 50s, and Levi Smith calculated pi to 1,000 digits using a mechanical calculator in 1948. After a decade of rapid computer advances, in 1961, he and Daniel Shanks tried for more digits of pi on an IBM 7090—a “fully-transistorized system” that replaced the company’s vacuum-tube computer, the 790, and found more than 100,000 digits. Wrench would go on to be a pioneer in developing the high-speed numerical methods used on computers for difficult hydrodynamics and aerodynamics calculations.

Then in 1976 two mathematicians, Richard Brent and Eugene Salamin, independently found an algorithm (a set of calculations or steps used on a computer) that “doubles the number of accurate digits with each iteration.” Since then, trillions of digits of pi have been calculated on computers using a number of methods, not only the Brent–Salamin algorithm. You may question the usefulness of calculating so many digits of pi; after all, 15 or 16 digits are plenty accurate for technology like the navigation system of the International Space Station.

But some mathematicians are interested in the rules that govern the digits of pi, not the digits themselves. In fact, in 2010 a Yahoo researcher used cloud computing to break up the calculations for pi to “skip ahead,” so to speak, to the two-quadrillionth binary digit of pi (a quadrillion is 10 with 15 zeroes behind it). In addition to learning more about the mysteries of pi, exercises like this can help test the limits of computation. If you’re ever searching for pi achievements, you may find a range of numbers as pi is discussed in terms of the actual numbers behind the decimal points and the computer binary digits, or bits, calculated. For example, 2.7 trillion decimal points is as many as nine trillion bits.

Meanwhile, some of us may use our old-fashioned human brains to memorize a few digits of pi on Pi Day. The current world-record holder recited 70,030 digits of pi, but you can start a little smaller. Maybe try the first 10 digits? 3.141592653.

References

1. Beckmann, Petr, A History of Pi. St. Martin’s Griffin, 1975.
2. C. Huber, “Viete’s Method for Calculating Pi,” School Science and Mathematics Association, 1984. DOI: 10.1111/j.1949-8594

19. The World of Pi, http://www.pi314.net/eng/salamin.php.

20. Scientific American, “How Much Pi Do You Need?” http://blogs.scientificamerican.com/observations/how-much-pi-do-you-need/

21. BBC, “Pi record smashed as team finds two-quadrillionth digit,” http://www.bbc.com/news/technology-11313194

22. New Scientist, “New Pi record exploits Yahoo’s computers,” https://www.newscientist.com/article/dn19465-new-pi-record-exploits-yahoos-computers/.

23. Pi World Ranking List, http://www.pi-world-ranking-list.com/.

24. Pi Day.org, http://www.piday.org/million/.

Katie Jones is a science writer at Argonne National Laboratory. Follow Katie on Twitter @Kelyce_writer.