Reading this article can help you become a Pi Day expert — at least, relative to many of your colleagues.

Related: Infographic: Irrational numbers to celebrate Pi Day

Circle back to the beginning

Human use of circles dates at least to the second Millennium BCE — long before the start of the Greek alphabet in the eighth century BCE. The letter pi (π) is the 16th letter of this 24 letter alphabet. But the naming of the number π is a long story with several steps.

First, published measurement of the relation of the circumference of a circle to its diameter was about a 500-year wait. “Measurement of a Circle,” an approximately 250 BC treatise by Archimedes of Syracuse (est 287 BCE – 212 BCE), specifies this relationship in its Proposition 3: “The ratio of the circumference of any circle to its diameter is greater than 3¹⁰/₇₁ but less than 3¹/₇.”

Second and third, the ratio had to be more accurately estimated and a symbol assigned. After roughly 2,000 years, mathematicians could write:

π = 3.14159265358979323846264338327950288419716939937510…

The key actor was William Jones (1675 – 1749), a Welsh mathematician from Anglesey and the author credited with first identifying π with 3.14159(…). His “Synopsis Palmariorum Matheseos” was published in 1706 as an introduction to mathematics. In Art 76 on page 263, π is introduced with the following sentences.

There are various other ways of finding the Lengths, or Areas of particular Curve Lines, or Planes, which may very much facilitate the Practice; as for Instance, in the Circle, the Diameters is to the Circumference as 1 to

(16/5-4/239)-(1/3)(16/5³-4/239³)+(1/5)(16/5⁵-4/239⁵)-etc. = 3.14159 etc ≡ π

This series (among others for the same purpose, and drawn from the same Principle) I receiv'd from the Excellent Analyst, and my much esteem'd Friend Mr. John Machin; and by means thereof, Van Ceulen's Number, or that in Art. 64.38* may be Examin'd with all desireable Ease and Dispatch.

* Correct reference is 65.38

Without apology, this introduction presents a humble π. There is no announcement of new notation, but merely the equating of a number and π as if the symbol π were already in common use.

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Mathematicians led the way

Credit to Jones for introducing the symbol can be found on the slate plaque erected by Anglesey County Council at the primary school in Llanfechell, Anglesey. The Welsh wording on the plaque celebrates the introduction of the mathematical symbol π by Jones, who grew up and received his early education in the area.

If the number describes the ratio of the circumference to the diameter, why is the symbol π instead of γ (gamma) corresponding to the “c” in circumference? A plausible reason is that common 1706 usage for “circumference” was “perimeter” or “periphery,” each of which argues for π corresponding to “p.” Other reasons are also possible.

John Machin's calculation stopped with the hundredth digit in the decimal expansion for π. Writing a single symbol greatly simplifies using numbers with infinitely many digits in equations. In this way, the symbol π facilitated advancement in mathematics in general.

For example, the fourth step to π Day is an extension of π from the real numbers to the complex numbers. Real numbers correspond to points on an infinite line with two defining points – zero and one.

Complex numbers were discovered in the 16th century by the Italian mathematician Gerolamo Cardano. By setting i² = -1 and drawing a line perpendicular to the real line through the point 0 with a corresponding point 1 above zero, the two lines define “real” and “imaginary” axes for the complex plane. Complex numbers are written as “a+bi” where a is the real part of the number and b is the imaginary part in deference to the imaginary number i. Leonhard Euler (1707 – 1783) was a Swiss mathematician who helped extend π to complex numbers through an equation known as Euler's Identity. Euler's identity is: e^πi + 1= 0

This single equation describes the relationship among five key mathematical constants (e, π, i, 1, and 0) and, for our celebratory purposes, demonstrates the importance of π.

Euler's identity holds for complex numbers. Similar identities for quaternions (a number system introduced by William Rowan Hamilton in 1843) and for surreal numbers (a number system introduced by John Horton Conway in 1969) might also exist.

Incidentally, to my knowledge, Euler is the only human to have two mathematical constants named for him:

Euler's constant = e = 2.71828 …

Euler-Mascheroni constant = γ =0.57721…

Neither constant has a day. Euler's constant e would require February to have 71 days, and γ would need months with 57 days. π is clearly more compatible with the calendar.

Earlier I mentioned that π permeates contemporary business. As the economy grows and pursues new directions, such as early diagnosis and management action, no doubt the new approaches will in some sense be as easy as π.

π Day celebrates a history leading to crucial considerations in contemporary insurance and risk management programs. It celebrates things you do every day. Enjoy it.

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