Contents:

"If equations are trains threading the landscape of numbers, then no train stops at Pi"

1.1. What is Pi

1.2. How is Pi Useful and Equally Important in Our Lives

1.3. Applications of Pi in Our Daily Life

"Pi is not just a collection of random digits. Pi is a journey; an experience; unless you try to see the natural poetry that exist in Pi, you will find it very difficult to learn"

2.1. Basic Calculation Method

2.2. History Behind Measuring Circles

2.3. Calculating the Value of Pi Yourself

2.3. Calculating Pi Using the Infinite Series

1. Understanding the Magical Constant

Pi or π is a Greek letter for "p" and is the ratio of the circumference of any circle to the diameter of that circle. Pi secretly stands as a constant to any circle regardless of its size. Changing the diameter or the circumference of a circle does not make it independent of Pi, which is what Pi is known for, for its secrecy. Over the years of history, finding the value of this secret constant has become a thing and to what Pi is known as today, it stands for being one among the most irrational, natural and infinite constant at all times. Pi has no standard value of itself, it keeps on going forever, without ever repeating itself, and that is the natural beauty which is to be vowed.

The saying below represents how important Pi is to our lives

"Anything with curvature has Pi," the professor said, "A straight line does not have Pi."

source: Pi common in everyday life, not just dessert

Our life is filled with a series of curves, there are ups and downs everyday and that is equally how Pi is important to us.

• Biochemists see pi when trying to understand the structure/function of DNA

  • To understand the relation and interaction of the DNA-protein, biologists and biochemists uses the π-π interaction through quantum chemical calculations for a clearer data.

• Navigations and the Global Positioning System

  • Aircrafts and aeroplane generally fly in path which makes an arc of circle, length of this path must be calculated in order to determine the amount of fuel left and fuel required.

  • The modern and advanced GPS systems use π as a method to locate and track the accurate positions.

• Architecture: Geometrical designs & Construction

  • Calculation of surface area and volume of curved objects like cylinder, cone, sphere, etc., like calculation of area & volume of machine parts requires Pi.

  • Designing of bridges and highway curves requires π to determine its accuracy and curvature.

• Signals and communication: The Sine wave

  • These waves have the fundamental period of 2π motion. Sine waves make communication much easier which makes π an important part of its play.

• Engineering

  • In almost all fields of Engineering, π plays a significant role in determining different accurate values. It is used in Electromagnetics, Sine wave, Machine Designing, Civil Engineering and much more.

2. Value of Pi

The most basic calculation method of Pi is to accurately measure the circumference of the circle and hence divide it by the diameter of that circle,

\[π = \frac{Circumference}{Diameter}\]

Because Pi (π) has so many important uses, then we need to be able to start to calculate it, at least to several decimal places accuracy. Someone had to come up with the approximate value for Pi (π) which appears on your calculator – it didn’t get there by magic!

Ancient Greek mathematician Archimedes came up with an ingenious method for calculating an approximation of π. Archimedes began by inscribing a regular hexagon inside a circle and then circumscribing another regular hexagon outside the same circle. He was then able to calculate the exact circumferences and diameters of the hexagons and could therefore obtain a rough approximation of π by dividing the circumference by the diameter. Archimedes then found a way to double the number of sides of his hexagons. He could then find a more accurate approximation of π by using polygons with more sides, which were closer to the circle. He did this four times until he was using 96 sided polygons. Archimedes calculated the circumference and diameter exactly and therefore could approximate π to being between 223/7 and 22/7. From then on, this fraction has remained as one of the most popular and memorable approximations of π ever since.
Around 600 years after Archimedes, the Chinese mathematician Zu Chongzhi used a similar method to inscribe a regular polygon with 12,288 sides. This produced an approximation of Pi (π) as 355/113 which is correct to six decimal places. It was nearly 600 more years until a totally new method was devised that improved upon this approximation.

Let's now do a fun activity of calculating the value of Pi.

Aim:

To calculate the value of Pi

Materials Required:

Compass, Blank Paper, Scale

Procedure:

Step 1:

Draw a circle on a blank paper using a compass and measure around the edges, i.e., its circumference. Note its value (for example: Let C= 82 cm).

Step 2:

Now draw the diameter of the circle and measure it using a regular plastic/steel scale. (Diameter (D) = 26 cm for the example taken).

Step 3:

Now use the formula:

\[∵ Circumference (C) = Pi (π) * Diameter (D)\] \[∴ π = \frac{C}{D} = \frac{82}{26} = 3.14159…\]

Conclusion:

Hence the value of π is 3.14159...

Mathematicians eventually discovered that there are in fact exact formulas for calculating Pi (π). The only catch is that each formula requires you to do something an infinite number of times which is understandable due to its never-ending value.
One of the most well known and beautiful ways to calculate Pi (π) is to use the Gregory-Leibniz Series:

\[\frac{π}{4} = 1 − \frac{1}{3} + \frac{1}{5} − \frac{1}{7} + \frac{1}{9}\]

If you continued this pattern forever you would be able to calculate exactly π/4 and then just multiply it by 4 in order to get π. However, if you start to add up the first few terms, you will begin to get an approximation for Pi. The problem with the series above is that you need to add up a lot of terms in order to get an accurate approximation of π. You need to add up more than 300 terms in order to produce Pi accurate to two decimal places!

Another series which converges more quickly is the Nilakantha Series which was developed in the 15th century. It converges more quickly, which means that you need to work out fewer terms for your answer to become closer to Pi.
Nilakantha Series:

\[π = 3 + \frac{4}{2*3*4} - \frac{4}{4*5*6} + \frac{4}{6*7*8} - \frac{4}{8*9*10} + ...\]

Mathematicians found more and more infinite series which brings the value of pi. One of the amazing things which interests people about Pi (π) is that there isn’t just one formula, but a large number of different ones for people to study.

3. FAQs on the Value of Pi