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 )

A New Infinite Series, Convergent and Irrational

Infinite series are a log of func. All kidding aside, you may have heard of the sum of reciprocal squares.

\sum\limits_{n=1}^{\infty}\frac{1}{{n}^2}=\frac{1}{{1}^2}+\frac{1}{{2}^2}+\frac{1}{{3}^2}+\frac{1}{{4}^2}+\frac{1}{{5}^2}+\frac{1}{{6}^2}+\frac{1}{{7}^2}+\cdots

The sum of this slowly convergent series is approximately equal to 1.644934. Is there anything special about this number? Italian mathematician Pietro Mengoli (1626-1686) first posed the question in 1644 (published 1650), what is the exact sum of this infinite series? This problem was not solved until 90 years later by Swiss mathematician Leonhard Euler (1707-1783) in 1734 (published 1735). Euler proved that the exact sum is

\sum\limits_{n=1}^{\infty}\frac{1}{{n}^2}=\frac{{\pi}^2}{6}

There’s that number pi, the most famous of the irrational numbers, showing up once again in mathematics, ostensibly having nothing at all to do with circles. What’s an irrational number? It is any real number that cannot be expressed as a ratio of two integers. The decimal digits of an irrational number neither terminate nor end in a repeating sequence (e.g. 1/3 = 0.3333… or 9/11 = 0.81818181…).

Determining the exact value of the sum of reciprocal squares infinite series is known as the Basel Problem, named after the hometown of Euler (who solved it) and the Bernoulli family of mathematicians (who were not able to solve it).

My admired colleague in England, Abdul Ahad, has come up with a variant of the sum of reciprocal squares where every third term starting with n = 4 is subtracted rather than added.

\frac{{\pi}^2}{6} -2\sum\limits_{n=1}^{\infty}\frac{1}{{\left (3n+1  \right )}^2}=\frac{1}{{1}^2}+\frac{1}{{2}^2}+\frac{1}{{3}^2}-\frac{1}{{4}^2}+\frac{1}{{5}^2}+\frac{1}{{6}^2}-\frac{1}{{7}^2}+\cdots

Ahad has shown that this new series is convergent and sums to an irrational number, approximately equal to 1.40146804. The infinite sum portion of the above expression is approximately equal to 0.12173301 and is also an irrational number. Multiplying by 2 gives us an irrational number, and subtracting from π2/6, which is itself irrational, results in our final result being irrational.

Interestingly, almost all real numbers are irrational, strange as they are.

References

Ahad, Abdul. “An interesting series.” M500 Magazine 278, 8-9 (2017). http://m500.org.uk/wp-content/uploads/2018/11/M278WEB.pdf.

Ahad, Abdul. “A New Infinite Series with Proof of Convergence and Irrational Sum.” Res Rev J Statistics Math Sci, Volume 4, Issue 1 (2018). https://www.rroij.com/open-access/a-new-infinite-series-with-proof-of-convergence-and-irrational-sum.pdf.