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The Ratio of the Circumference of a circle to its Diameter
The most interesting number that students come across in the Primary curriculum is π . They get to know π as the ratio between the circumference & diameter of a circle. There are many interesting facts about π.
It is also usually said that its value is equal to 3.14. But this is just an approximate value which is sufficient for our daily needs.
We can think of π as a number which relates the length of a straight line (diameter of a circle) to the length of a curve (circumference) related to that straight line.
π was the first mystery number known to all civilizations because the circle was a common figure which had to be drawn for religious and commercial reasons. They know that the ratio between the circumference & the diameter of a circle was a constant. It must have taken a lot of time and many approximations of π in many different sizes of circles, before all civilizations came to the conclusion that the relation between the circumference & diameter was a fixed number for any circle.
After this realization, the most logical step for all civilizations was to find values of π sufficient enough for their practical needs. 3 and √10 were some of the approximations discovered more than 2000 years ago.
Calculating Digits of π
Archimedes is supposed to have made the first attempt. He realised that a circle can both be inscribed and circumscribed with polygons, whose perimeters could be calculated. By increasing the number of sides of the polygon, he could get a fairly accurate value.
He arrived at a value of around between 223/71 & 22/7 by using a polygon with 96 sides!
As knowledge of mathematics evolved, attempts were made to arrive at more accurate approximate values. 22/7 and 355/113 were some of these values. 355/113 is more accurate than 22/7.
Ludolph van Ceulen (1540 - 1610), who spent a major part of his life calculating values of π. He developed on Archimedes' work and discovered 35 digits in pi by supposing a polygon with literally billions and billions of sides. He got these digits inscribed on his tombstone.
But until 16thcentury, no more understanding was achieved, except that is it an interesting number. π was proved to be an irrational number in 1761 meaning whereby that it cannot be expressed in rational form i.e as a/b where a & b are integers. Irrational numbers turned out to be numbers which could only be expressed as the sum of an infinite series of fractions.
In 1658 it was expressed as a sum of infinite continued fractions. In 1674, Leibniz succeeded in expressing it as the limit of an infinite series with simple fractions and pattern. Then many infinite series were found out whose sum (limit) was π.
Finding as many digits as possible of π became almost an industry. But scientists have estimated that if we calculate any distance over the curved surface of the Earth, using the value of π up to 15 decimal places, i.e 3.141592653589793, it would be off by a gap 10,000 times thinner than a hair!
And it would only take the first 39 digits to calculate the circumference of the known universe to the width of a hydrogen atom!
Computing Digits of π
Before the invention of modern mathematics, the approach to calculating the value of π was to approximate the circle with a polygon of a huge number of sides.
The calculation of values of π became much easier after the invention of calculus and infinite series.
It became faster with the advent of computer software.
Possibly the speed of calculation of the digits tests the power of computer software.
In March 2019, Emma Haruka Iwao, an employee at Google, computed 31.4 trillion digits of pi using y-cruncher and Google Cloud machines. This took 121 days to complete.
In January 2020, Timothy Mullican announced the computation of 50 trillion digits over 303 days.
In August 2021, a Swiss team calculated 𝛑 to 62.8 trillion digits, about 25% further than the previous record. They accomplished this while testing a new supercomputer at the Competence Center for Data Analysis, Visualization and Simulation (DAViS).
They have also announced that the last ten digits in their calculation of 𝛑 was 7817924264.
Calculating more digits of π has become an intellectual as well as a technological challenge. So we can expect more digits of π discovered in the years to come!
π Through String Theory
Two physicists of the Indian Institute of Science (IISc), Aninda Sinha ) along with Arnab Priya Saha, have accidently arrived at a new series for π in the course of their study of particle collisions in high-energy physics in quantum theory.
If you are interested in this series and a brief history finding digits of π you can watch this video from Presh Talwalkar's Mind Your Decisions - https://www.youtube.com/watch?v=t1ZnptSEPI8
The Symbol π
π which is pronounced as “pie” was originally used to represent perimeter. In 1706 it was used by William James, a Welsh mathematicians and an FRS, to represent the ratio of the circumference to the diameter of a circle. Since then it has become one of the most recognized mathematical symbols.
Before being represented by the Greek symbol π it was called Archimedes Constant. According to available evidence, Archimedes was the first to calculate the value of π.
π is an Irrational Number
Expressed in decimal numbers, an irrational number has the following property. The digits in its decimal part never end and do not show any pattern or repetition.
The irrationality of π, meaning it cannot be expressed as a simple fraction, was proven in 1768 by Johann Lambert.
This was a significant milestone in the understanding of π, as it ruled out the possibility of expressing π as a ratio of two integers. This irrationality also implies an infinite, non-repeating decimal expansion, making π a subject of study in the field of number theory.
π is a Transcendental Number
German mathematician Ferdinand Lindemann proved in 1882 that π couldn’t be produced by a combination of the five operations addition, subtraction, multiplication, division, and square root extraction. It is not a root of any non-zero polynomial equation with rational coefficients.
They were called “transcendental,” possibly because of their capacity to transcend the realm of algebraic numbers.
π was found to belong to a special class of irrational numbers, which were called transcendental numbers. Transcendental numbers cannot be expressed as solutions of an algebraic equation.
This places π in the same category as Euler’s number e, another transcendental constant.
We know that √2 is an irrational number. But it can be expressed as the solution to the equation x2 = 4 which is an algebraic equation. Hence √2 is called an algebraic irrational number.
Squaring the Circle
Squaring a circle was an attempt to construct a square whose area was the same as a given circle, in a finite number of steps, using only a straight edge and a compass. This is a problem that dates back to ancient Greek mathematics.
Mathematically it assumed that π could be derived through algebraic methods.
Lindemann's proof that π was not an algebraic irrational, effectively put a closure to the problem of squaring a circle.
π in Science & Nature
π occurs in unexpected expressions & equations which describe physical phenomena.
Pi, the symbol π, is more than just a number used to calculate the dimensions of circles. It is a constant that pervades the universe, linking seemingly disparate fields and phenomena in a tapestry of mathematical beauty.
We can think of π as a quantity which relates curved lines to straight lines.
Today, we need it to determine area and circumference of circles. It's critical to accurate computation of angles, and angles are critical to navigation, building, surveying, engineering and more. Radio frequency communication is dependent on sines and cosines which involve pi.
Why do we need to compute the value of π to many decimal places? This effort was complemented by the invention of computers. The speed of calculating the number of digits of π was used as a measure of the speed of a computer.
An outcome of this effort was the discovery that digits in an irrational number can be used for coding messages so that they are not “understandable” to anyone who does not have the key. Today messages sent through the internet are scrambled using many techniques which were the result of study of irrational numbers.
Leonard Euler discovered an equation which connects π & e in an equation which according to surveys, is considered the most beautiful equation in all of science.
e iπ= -1
Pi Day
Larry Shaw, an American physicist, organised the first Pi Day in 1988 at the San Francisco Exploratorium. He and his wife celebrated the first π Day by handing out slices of fruit pie and tea at 1:59 p.m., which is the three digits after 3.14. The π Day quickly became popular in US schools.
In its 40th General Conference in November 2019, the United Nations Educational, Scientific, and Cultural Organization (UNESCO) decided to declare Pi Day (March 14) as the International Day of Mathematics.
Most countries celebrate 14thMarch every year as Pi Day, because 14thMarch is written (in many countries) as 3.14. May be India should celebrate 22nd July (written as 22/7) as the Indian π Day!
The birth days of Albert Einstein and Stephen Hawking also fall on the π Day.
Remembering Digits of π
The following sentences have been constructed to help us remember π to a few decimals. Counting the letters in each word, you can write the value of π.
May I have a large container of coffee?
How I wish I could enumerate?
The English physicist James Jeans wrote this piem: How I want a drink, alcoholic of course, after the heavy lectures involving quantum mechanics. Counting the letters in each word, you can write 3.1415 92653 58979, giving 15 digits of pi, more digits than you are ever likely to need.
Approximations of π
In arithmetic expressions, only rational approximations of π can be used. Here are some of them.
22/7 = 3.1428.....
355/113 = 3.14159292.....
104,348/33,215 = 3.14159265392....
I recently came across the interesting expression √(7+ √(6+ √5) ) as a fairly accurate approximation of π. The square roots are nested. The answer is 3.141632545.
π in Indian Mathematics
One of problems of ancient Hindu was the construction of yagnya altars. One of the requirements was the construction of a circular altar whose area was equal to that of a square altar. The early part of the vedic literature assumes a value of 3 for π. The later sulba sutras give the value accurate to many decimals.
Aryabhatta in the 5thcentury worked out a value which was accurate up to 4 decimal places! In the 14thcentury, Madhava worked out the infinite series for π/4 which was discovered by Liebnitz 300 years later. This is never taught in our schools, either in the math or the history class!
A Strange Coincidence
The value of π is closely related to that of the gravitational constant "g". π^2 is almost equal to g which 9.80665! There is no physical reason for this coincidence.
Because of this, there is a relation closely connecting the length & time measurement units. The time period of a pendulum which is 1m long is almost 2 seconds, for a complete swing. Such a pendulum is called a "Seconds Pendulum".
Self-Locating Numbers in π
The commonly used value of Pi is 3.14159. If we take the ordinal position of the digits, "1" comes in the "position 1"
Several other "number strings" have been identified which occur in the "position" specified by them!
"16470" occurs in the "16470th" position!. This is also true of 44899, 79873884, 711939213.
We can call them as "self-locating" numbers in π.
The PI Function in MS Excel
π occurs in so many scientific calculations that MS Excel has a PI() function which returns its value to 15 decimal places.
π is Not So Special
Though this article is about π, we also need to remember that there is nothing very special about it. It is important to us since it is related to the geometry of a circle.
Any other irrational number will also have all these properties!
And the Georg Cantor proved that the infinity of irrational numbers is more than the infinity of rational numbers. This means that there are more irrational numbers than rational numbers.
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