Archimedes’ Constant

The number pi (π) can be simply stated: it is the ratio of a circle’s circumference (C) to its diameter (d).

\pi = \frac{C}{d}

The Greek mathematician Archimedes (c. 287 BC – c. 212 BC) was the first person to come up with a computational method of calculating π. He inscribed and circumscribed polygons with the same number of sides inside and outside of a circle. The value of π is between the perimeter of the inscribed polygon and the perimeter of the circumscribed polygon as shown in the diagrams below. By increasing the number of sides of the inscribed and circumscribed polygons, the value of π can be estimated more closely. The number π is thus sometimes called Archimedes’ Constant.

Archimedes’ method of calculating π

Archimedes’ Constant was not called π until Welsh mathematician William Jones (1675-1749) began using it in 1706. π is the first letter of the Greek word for periphery (περιφέρεια).

The number π (3.1415926535897932384626433…) has some remarkable properties, a few of which are

  • π cannot be expressed as a ratio of two integers (it is an irrational number).
  • The exact decimal representation of π has an infinite number of digits.
  • The decimal digits of π never exhibit a repeating pattern.
  • The decimal digits of π are randomly distributed, but this has not yet been proven.
  • π cannot be a solution of any equation involving only sums, products, exponents, and integers (it is a transcendental number).

It is worth noting that there are an infinite number of transcendental numbers (and, therefore, at least an infinite number of irrational numbers). But π is remarkable in that it pervades both mathematics and physics, often in ways that appears to have nothing to do with circles, spheres, or even geometry.

The value of π has now been calculated out to 100 trillion decimal places (1014) by Japanese computer scientist Emma Haruka Iwao. Like other recent attempts to calculate the most digits of π, Iwao used the Chudnovsky algorithm. Her record-breaking calculation took nearly 158 days using cloud computing between October 14, 2021 and March 21, 2o22.

Interestingly, the value of π can be calculated using a couple of simple infinite series.

The great Swiss mathematician Leonhard Euler (1707-1783) obtained the following:

\pi = \sqrt{6\left ( \frac{1}{1^{2}}+\frac{1}{2^{2}}+\frac{1}{3^{2}}+\frac{1}{4^{2}}+\cdots \right )}

And earlier, German mathematician Gottfried Wilhelm Leibniz (1646-1716) and Scottish mathematician David Gregory (1659-1708) independently arrived at an even simpler infinite series to generate π, though it converges so slowly that it is of little practical use.

\pi = 4\left ( \frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\cdots \right )