 Photo from Gwen Adams at King Arthur Flour

Today, Pi Day, celehbrates the mathematical constant π. Most first encountered π in geometry class when calculating the perimeter and area of a circle, where P=2πr and A=πr. When solving to a decimal value, students might use the approximation 3.14 – hence, March 14 being Pi Day.

π falls in with the imaginary number i, Euler’s constant e, and the numbers 1 and 0 as one of the five most fundamental numbers in mathematics. This elusive number has appeared in records as far back as ancient Egypt, but not until the early 18th century did it have the name it does today. In 1706, self-taught mathematician William Jones (1675-1749) gave the constant its Greek symbol π, replacing the old Latin mouthful quantitas in quam cum multiflicetur diameter, proveniet circumferencia. (Fun fact: Though from humble backgrounds, Jones had connections to several eminent scientists of his day, including Sir Isaac Newton and John Machin.)

Through history, mathematicians have worked to calculate π to more and more precise terms though, as an irrational number, it never really ends. Computers have optimized this process and calculated π out to billions of degrees. In the days before computers, the mathematicians of old resorted to good old geometry to approximate the value of π.

In this post I briefly discussed Archimedes’ (d. 212 B.C.) method of exhaustion, by which he approximated the circumference of a circle by circumscribing and inscribing regular polygons on the circle until, theoretically, he had a regular polygon with an infinite number of sides, at which point he would have a circle. He concluded, using a 96-gon, that 3 $\frac{10}{71}$ < π  < 3 $\frac{1}{7}$. Image from Spreadshirt Media

The ancient Greeks played with this method of exhaustion business for some time before dropping the idea. Then, in the first few centuries A.D., Chinese mathematicians estimated the value of π to an even more precise degree using the methods of the ancient Greeks. Zu Chongzhi (祖冲之, 429-500) calculated π to be $\frac{355}{113}$, between 3.1415926 and 3.1415927, using a 12,288-gon. Mathematicians in India, such as Brahmagupta (598-670), also explored π. The Jaina school of mathematics gave π as the $\sqrt{10}$, which, although inaccurate (the first three digits of $\sqrt{10}$ are 3.16), was a tidy number to use.

A  new way of calculating π emerged in the late 17th and early 18th centuries with Maclaurin and Taylor series of the arctangent, the inverse of the tangent. (For more on that calculation, see this article.) With this innovation, mathematicians calculated π to hundreds of decimal places. Proofs in the 18th and 19th centuries affirmed in the irrationality and transcendence of π.

Though 3.14 may have worked well enough for a π approximation in geometry class, in many physical calculations today people scientists and engineers need a more exact value. When one studies trigonometry, one discovers that π is inherently linked to the trigonometric functions by the unit circle. Thus, any phenomenon involving rotations or waves invites π to the party. This includes the rotation of parts in a jet engine, scanning devices in MRIs and CATs, and high-frequency electronics, which may require precision of π between 1/10,000 and 1/1,000,000,000,000,000!