Pi with infinite connections
View(s):The approximate value of Pi (π) is 3.14 in mathematics. Therefore, March 14 is celebrated as Pi Day. In 2019, UNESCO proclaimed March 14 as the International Day of Mathematics.
Ancient Babylonians and Egyptians figured out that the circumference of a circle is slightly more than three times its diameter. But they could not come up with an exact value for this ratio although they knew that it is a constant. This constant was later named as π which is a letter in the Greek alphabet.
It was the Greek mathematician Archimedes (250 BC) who was able to find an upper bound and a lower bound for this constant. He drew a circle of diameter one unit and drew hexagons inside and outside the circle such that the sides of each hexagon touch the sides of the circle. In mathematics the circle passing through all vertices of a polygon is called a ‘circumcircle’ and the largest circle that fits inside a polygon tangent to all its sides is called an ‘incircle’. The total length of the smaller hexagon then becomes the lower bound of π and the length of the hexagon outside the circle is the upper bound. He realized that by increasing the number of sides of the polygon can make the bounds get closer to the value of Pi and increased the number of sides to 12,24,48 and 60. He argued that by increasing the number of sides will ultimately result in obtaining the original circle, thereby laying the foundation for the theory of limits. He ended up with the lower bound as 22/7 and the upper bound 223/71. He could not continue his research as his hometown Syracuse was invaded by Romans and was killed by one of the soldiers. His last words were ‘do not disturb my circles’, perhaps a reference to his continuing efforts to find the value of π to a greater accuracy.
Archimedes can be considered as the father of geometry. His contributions revolutionized geometry and his methods anticipated integral calculus. He invented the pulley and the hydraulic screw for drawing water from a well. He also discovered the law of hydrostatics. He formulated the law of levers which states that a smaller weight placed farther from a pivot can balance a much heavier weight closer to it. He famously said “Give me a lever long enough and a place to stand and I will move the earth”.
Mathematicians have found many expressions for π as a sum of infinite series that converge to its value. One such famous series is the Leibniz Series found in 1674 by the German mathematician Gottfried Leibniz, which is given below.
π = 4 ( 1 – 1/3 + 1/5 – 1/7 + 1/9 – ………….)
The Indian mathematical genius Ramanujan came up with a magnificent formula in 1910. The short form of the formula is as follows.
π = 9801/(1103 √8)
For practical applications an approximation is sufficient. Even NASA uses only the approximation 3.141592653589793 for its interplanetary navigation calculations.
It is not just an interesting and curious number. It is used for calculations in navigation, encryption, space exploration, video game development and even in medicine. As π is fundamental to spherical geometry, it is at the heart of positioning systems in GPS navigations. It also contributes significantly to cybersecurity. As it is an irrational number it is an excellent foundation for generating randomness required in encryption and securing communications. In the medical field, it helps to calculate blood flow rates and pressure differentials. In diagnostic tools such as CT scans and MRI, pi is an important component in mathematical algorithms and signal processing techniques.
This elegant, never-ending number demonstrates how mathematics transforms into practical applications that shape our world. The possibilities of what it can do are infinite as the number itself. It has become a symbol of beauty and complexity in mathematics. “It matters little who first arrives at an idea, rather what is significant is how far that idea can go.” said Sophie Germain.
Mathematics fans are intrigued by this irrational number and attempt to calculate it as far as they can. In March 2022, Emma Haruka Iwao of Japan calculated it to 100 trillion decimal places in Google Cloud. It had taken 157 days. The Guinness World Record for reciting the number from memory is held by Rajveer Meena of India for 70000 decimal places over 10 hours.
Let us celebrate Pi for opening doors to infinite applications.
R N A de Silva
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