The Magic and Mystery of Pi (π)

Dr.R.Paul

Academic Dean

Pi is fundamental in any field that involves circles, waves, rotations, or periodicity,

making it a cornerstone of many mathematical and scientific disciplines.

March 14 as per the American convention, is celebrated as Pi Day worldwide as an ode to

the most well-known approximation (3.14) of the mathematical constant Pi.

The tradition was started by physicist Larry Shaw of the Exploratorium museum in

San Francisco in 1988, and has since seen global popularity. On the day, mathematicians try to

raise awareness on their subject among lay persons, through lectures, museum exhibitions and

pie (sic) eating competitions.

In 2019, UNESCO’s 40th General Conference designated Pi Day as the International Day

of Mathematics.

What is Pi?

Pi, often represented by the Greek letter π, is the most famous of all mathematical

constants. It represents the ratio of a circle’s circumference (boundary) to its diameter (a straight

line between two points on the circle’s boundary, passing through its centre). Regardless of the

circle’s size, this ratio always remains constant.

Pi is an irrational number -  it is a decimal with no end and no repeating pattern — which is most

often approximated to the 3.14, or the fraction 22/7.

How is Pi calculated?

The importance of Pi has been recognized for at least 4,000 years. Petr Beckman in his

classic, A History of Pi (1970), wrote that “by 2,000 BC, men had grasped the significance of the

constant that is today denoted by π, and that they had found a rough approximation of its value.”

Both ancient Babylonians and ancient Egyptians came up with their own measurements,

probably by drawing a circle of some diameter, and then measuring its circumference using a

rope of said diameter in length. Babylonians settled at 25/8 (3.125) as the value of Pi, while

ancient Egyptians settled at (16/9)2 (approximately 3.16).

It was Greek polymath Archimedes (circa 287-212 BCE) who came up with the method

to calculate Pi that remained in use till the 17th century. He realised that the perimeter of a

regular polygon of ‘n’ sides inscribed in a circle is smaller than the circumference of the circle,

whereas the perimeter of a similar polygon circumscribed around the circle is greater than its

circumference. He used this to calculate the limits within which the value of Pi must lie.

Now, as one keeps adding more and more sides to this polygon, it gets closer and closer

to the shape of a circle. Having reached 96-sided polygons, Archimedes proved that 223/71 < Pi

< 22/7 (in decimal notation, this is 3.14084 < π < 3.142858).

Following Archimedes, mathematicians constantly increased the number of sides of the polygon

to calculate Pi to ever greater decimal places. By 1630, Austrian astronomer Christoph

Grienberger calculated 38 digits of Pi using polygons with 1040 sides.

The problem with this method, however, is that it is extremely labour intensive. For instance, it

took Dutch mathematician Ludolph van Ceulen (1540-1610) a staggering three decades to

calculate Pi to 35 decimal points.

It would be Isaac Newton (1643-1727) who significantly simplified the process of

calculating Pi. In 1666, he calculated Pi up to 16 decimal places using calculus, which he

discovered along with mathematician Gottfried Wilhelm Leibniz (1646-1713). What had taken

previous mathematicians years to calculate now could be done in a matter of days.

By 1719, French mathematician Thomas Fantet de Lagny (1660-1734) had already calculated Pi

up to 112 correct decimal places. Today, with the help of modern computers, this method has

calculated the value of Pi up to 31 trillion (1012) decimal places.

But why make all these effort?

Circles are everywhere in the world. So are three-dimensional shapes like cylinders,

spheres, and cones, all of which carry the proportion of Pi. Knowing Pi’s value, thus, has some

crucial practical benefits in the fields of architecture, design, and engineering. From constructing

water storage tanks to fashioning hi-tech equipment for satellites, the value of Pi is

indispensable in all sorts of areas.

Pi (π) is a mathematical constant that represents the ratio of the circumference of a circle

to its diameter, approximately equal to 3.14159. Pi has a variety of important uses in

mathematics, science, and engineering:

1. Geometry of Circles:

Pi is essential in calculating lengths, areas, and volumes of circular and spherical shapes.

Eg.,

Circumference: 2 π r 

Area of a circle :  π  r2   

(Where r is the radius of the circle).

2. Trigonometry:

Pi is a key component of trigonometric functions such as sine, cosine, and tangent, which

are crucial in the study of waves, oscillations, and circular motion.

In trigonometry, angles are often measured in radians, where radians are equivalent to

180 degrees. This helps simplify many formulas and calculations.

3. Engineering and Physics:

Pi appears in formulas for calculating the motion of waves (sound waves, light waves,

etc.) and in analyzing periodic phenomena like pendulum motion, electrical circuits, and

vibrations.

In fluid dynamics, heat transfer, and other areas of physics, pi is used to model circular or

rotational motion, such as the movement of planets or particles.

4. Complex Numbers:

In complex analysis, Euler's formula, which relates pi to the imaginary number, the

number 1, and the base of natural logarithms, is one of the most famous equations in

mathematics. This formula is crucial in fields such as quantum mechanics and signal processing.

5. Probability and Statistics:

Pi appears in probability distributions, such as the normal distribution, where it helps

define the shape of the curve. The normal distribution, or "bell curve," is critical in data analysis

and inferential statistics.

6. Fourier Transforms:

Pi plays a role in Fourier analysis, which is used to break down complex waveforms into

simpler sine and cosine components. This is widely applied in signal processing,

telecommunications, and acoustics.

Moreover, Pi seems to be woven into descriptions of the very deepest workings of the

universe from calculating the vastness of space or understanding the spiral of DNA. “Pi is often a

key ingredient in the solution of a great many problems inspired by real-world phenomena… will

only increase its relevance as we continue to further our understanding of the world we live in,”

Prof Dorina Mitrea, chair of the Department of Mathematics at Baylor University, Texas,

told Newswise in 2023. However, the calculation of Pi 31 trillion digits is less obviously

“useful”. While Archimedes’ calculation was fairly adequate for all practical purposes that Pi

was used for in his time, today, Pi needs to be calculated to about 39 decimal places in order to

perform all calculations in the observable universe with virtually no error. Why then are

mathematicians so fixated on the number?

(The writer is the Academic Dean of Public College, Dimapur.)