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Pi

What is pi?

In Euclidean geometry, pi is defined as the ratio of a circle’s circumference to its diameter with an approximate value of 3.14. Pi is considered as an irrational number and a mathematical constant that is represented using the symbol π. Since it is a mathematical constant, its value cannot change. 

Pi (π) is also known as a transcendental number since it is not a root of any polynomial with rational coefficients – which also means that it is impossible to solve ancient problems of squaring the circle with a compass and straightedge. 

The use of the Greek letter π was first used to indicate the circumference to diameter ratio of a circle by Welsh mathematician William Jones in 1706. It is also known as Archimedes’ constant.

What is the value of pi? 

The approximate value of pi can be expressed as either decimals or fractions.

Decimals 

We often use the value of 3.14 to represent π but since it is an irrational number, its decimal representation is a combination of infinitely many digits and not in a repeating pattern. Below is the approximate value of pi with different numbers of decimal places.

Approximation value of pi in 5 decimal places3 . 1 4 1 5 9
Approximation value of pi in 10 decimal places3 . 1 4 1 5 9 2 6 5 3 5
Approximation value of pi in 20 decimal places3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6
Approximation value of pi in 50 decimal places3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0
Approximation value of pi in 100 decimal places3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9 2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 1 1 7 0 6 7 9
Approximation value of pi in 200 decimal places3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9 2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 1 1 7 0 6 7 9 8 2 1 4 8 0 8 6 5 1 3 2 8 2 3 0 6 6 4 7 0 9 3 8 4 4 6 0 9 5 5 0 5 8 2 2 3 1 7 2 5 3 5 9 4 0 8 1 2 8 4 8 1 1 1 7 4 5 0 2 8 4 1 0 2 7 0 1 9 3 8 5 2 1 1 0 5 5 5 9 6 4 4 6 2 2 9 4 8 9 5 4 9 3 0 3 8 1 9 6
Approximation value of pi in 500 decimal places3 . 1 4 1 5 9 2 6 5 3 5 8 9 7 9 3 2 3 8 4 6 2 6 4 3 3 8 3 2 7 9 5 0 2 8 8 4 1 9 7 1 6 9 3 9 9 3 7 5 1 0 5 8 2 0 9 7 4 9 4 4 5 9 2 3 0 7 8 1 6 4 0 6 2 8 6 2 0 8 9 9 8 6 2 8 0 3 4 8 2 5 3 4 2 1 1 7 0 6 7 9 8 2 1 4 8 0 8 6 5 1 3 2 8 2 3 0 6 6 4 7 0 9 3 8 4 4 6 0 9 5 5 0 5 8 2 2 3 1 7 2 5 3 5 9 4 0 8 1 2 8 4 8 1 1 1 7 4 5 0 2 8 4 1 0 2 7 0 1 9 3 8 5 2 1 1 0 5 5 5 9 6 4 4 6 2 2 9 4 8 9 5 4 9 3 0 3 8 1 9 6 4 4 2 8 8 1 0 9 7 5 6 6 5 9 3 3 4 4 6 1 2 8 4 7 5 6 4 8 2 3 3 7 8 6 7 8 3 1 6 5 2 7 1 2 0 1 9 0 9 1 4 5 6 4 8 5 6 6 9 2 3 4 6 0 3 4 8 6 1 0 4 5 4 3 2 6 6 4 8 2 1 3 3 9 3 6 0 7 2 6 0 2 4 9 1 4 1 2 7 3 7 2 4 5 8 7 0 0 6 6 0 6 3 1 5 5 8 8 1 7 4 8 8 1 5 2 0 9 2 0 9 6 2 8 2 9 2 5 4 0 9 1 7 1 5 3 6 4 3 6 7 8 9 2 5 9 0 3 6 0 0 1 1 3 3 0 5 3 0 5 4 8 8 2 0 4 6 6 5 2 1 3 8 4 1 4 6 9 5 1 9 4 1 5 1 1 6 0 9 4 3 3 0 5 7 2 7 0 3 6 5 7 5 9 5 9 1 9 5 3 0 9 2 1 8 6 1 1 7 3 8 1 9 3 2 6 1 1 7 9 3 1 0 5 1 1 8 5 4 8 0 7 4 4 6 2 3 7 9 9 6 2 7 4 9 5 6 7 3 5 1 8 8 5 7 5 2 7 2 4 8 9 1 2 2 7 9 3 8 1 8 3 0 1 1 9 4 9 1 2

As stated, no matter how many decimal places occur, irrational numbers can’t have repeating digits or patterns. Hence, for easier calculations and computations, we use the approximation of 3.14 to denote the value of pi

Fractions

Irrational numbers cannot be expressed as a ratio of any two integers. However, we use the ratio of 22/7 or 355/13 to approximate the value of pi. More so, no common fractions can be the exact value of pi

What is the origin of pi?

Approximations for the value of π started way back in the Common Era with only two decimal places. During the mid-first millennium, Chinese mathematicians improved its approximation by seven decimal places. However, progress was not made not until the late medieval period. 

The earliest record of written approximations can be found in Babylon and Egypt. According to some Egyptologists, ancient Egyptians used the approximation 22/7 for π which is based on the measurements of the Great Pyramid of Giza (c. 2560 BC) as early as the Old Kingdom which asserts doubt to some. The Rhind Papyrus in Egypt, which dated around 1650 BC but was transcribed from a document from 1850 BC revealed the use of π as ($\frac{16}{9}$)2 which is approximately 3.16 to solve for the area of a circle. Meanwhile, a clay tablet from 1900 – 1600 BC was found in Babylon that contains a geometrical statement where it uses the value of $\frac{25}{8}$ or exactly 3.125. 

Archimedes, a Greek mathematician, devised the first reported procedure for rigorously computing the value of π around 250 BC using a geometrical approach using polygons. Because of this discovery that prevailed for about 1000 years – π has been called as “Archimedes constant.” Using a regular hexagon inside and outside a circle, he computed for the upper and lower bounds of π – which continued until he reached a 96-sided regular polygon. By simply calculating the perimeter of these regular polygons, he was able to prove that $\frac{223}{71}$ < π < $\frac{22}{7}$ or 3.1408 < π < 3.1429. 

Ptolemy wrote the value of π as 3.1416 in his book Almagest around 150 AD Others believed that Ptolemy’s value for π is obtained from either Archimedes or Apollonius of Perga. In his Āryabhaṭīya (499 AD), the Indian astronomer Aryabhata used the value of 3.1416 for π. Additionally, around 1220, Fibonacci calculated π having the value of 3.1418 using a polygonal approach which was not the same as Archimedes. 

There have been different approaches in finding the approximate value of pi such as using polygon approximation and infinite series. 

Where does the symbol of pi come from?

The Greek letter π was originally used as an abbreviation of the Greek word periphery (περιφέρεια) and was combined in ratios with δ (for diameter) or ρ (for radius) to create circle constants. In 1647, it was first recorded in Oughtred’s “δ.π” in order to express the ratio of periphery and diameter. 

Welsh mathematician William Jones used the Greek letter alone to represent the circumference to diameter ratio of a circle in his 1706 work Synopsis Palmariorum Matheseos, or a New Introduction to the Mathematics.

Euler’s book of Essay Explaining the Properties of Air in 1727, he started using the single-letter π with the value of pi being equal to 6.28. Nevertheless, his first use of π = 3.14 can be dated back to 1736 in his Mechanica in 1736. In 1748, he wrote “for the sake of brevity we will write this number as π; thus π is equal to half the circumference of a circle of radius 1″ in his book Introductio in analysis infinitorum. Because of Euler’s connections with other mathematicians in Europe, the usage of the Greek letter π spread quickly. This practice also made an impact in the Western world as it was adopted and practiced. 

Why the need to explore more digits of pi?

One of the reasons why mathematicians and researchers continue to solve for the digits of π is because it provides a more precise answer to some numerical calculations. For most cosmological calculations, thirty-nine digits of π are necessary for the accurate calculation of the circumference of the observable universe.

People have worked tirelessly to compute thousand and millions of digits of pi. This endeavor can be linked to the human impulse to break records, and such accomplishments frequently make international news. Furthermore, these calculations are also being used to test the functions of supercomputers and high-precision algorithms. 

What is the importance of pi?

Since π is always associated and related to a circle, it is used for various formulae in geometry and trigonometry. Hence, a variety of mathematical problems requires the use of pi like:

  • solving arc lengths or any other curves; 
  • areas of circle, ellipses, sectors, and other curved surfaces; and
  • volumes of any solids. 

The table below shows some formulae in geometry that involve π. 

FormulaMathematical Use
C = 2πr or C = πdComputing the circumference of a circle where r is the radius of the circle and d is the diameter of a circle.
A = πr2Computing the area of a circle where r is the radius of the circle.
A = πabComputing the area of an ellipse where a and b are the semi-major axis and semi-minor axis, respectively. 
V = $\frac{4}{3}$πr3Computing the volume of a sphere where r is the radius.
SA = 4πr2Computing the surface area of a sphere where r is the radius. 
H = $\frac{1}{2}$π2r4Computing for the hypervolume of a 3-sphere where r is the radius. 
S = (n – 2)(π)Computing the sum of internal angles of a regular convex polygon with n sides. 
A = $\frac{ns^2}{4}$ cot $\frac{\pi}{n}$Computing the area of a regular convex polygon with n sides and side measure of s
r = $\frac{s}{2}$ cot $\frac{\pi}{n}$Computing the inradius r of a regular convex polygon with n sides and side measures of s
R = $\frac{s}{2}$ csc $\frac{\pi}{n}$Computing for the circumradius R of a regular convex polygon with n side and side measure of s

Angles are used in trigonometric functions, and radians are commonly used as measuring units by mathematicians. Since angles are measured by radians, π plays a vital role in measuring angles since a complete revolution of angle in a circle is equivalent to 2π. 

Furthermore, it is used in physics and engineering formulas to explain the periodic phenomena in:

  • pendulum motions;
  • string vibrations; and 
  • alternating electric currents. 

Other disciplines of research, such as statistics, physics, Fourier analysis, and number theory, use π in some of their key formulas. 

What is a pi day?

Pi Day is a yearly commemoration of the mathematical constant (pi) – which is celebrated every 14th of March. This date came from the fact that the first three significant digits of π are 3, 1, and 4. Larry Shaw staged the first and large-scale known celebration of Pi Day in 1988 at the San Francisco Exploratorium, where he works as a physicist. 

On March 12, 2009, the House of Represantatives in the United States passed a nonbinding resolution (111 H. Res. 224) declaring March 14, 2009, as National Pi Day. In 2020, Google created a Google Doodle where circles and pi symbols are all over the word Google. In 2015, March 14 was referred as “Super Pi Day” since when written in a month/day/year format, it will be written as 3/14/15 – which is the first 5 digits of pi. More so, 3/14/15 at 9:26:53 represents the first 10 digits of π. 

Due to a pun based on the words “pi” and “pie” being having the same pronunciation in English, and the coincidental circular shape of many pies, Pi Day is being celebrated by eating and throwing pies, and discussing the significance of the number. Additionally, some schools even throw competitions for the most students who can memorize the most number of decimal places of pi.

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