Pi DAY
1. Context
March 14, or 3/14 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
In 2019, UNESCO’s 40th General Conference designated Pi Day as the International Day of Mathematics
2. What is Pi?
Pi, commonly denoted by the Greek symbol π, stands out as the most renowned among mathematical constants. It signifies the relationship between a circle’s circumference (its outer boundary) and its diameter (a straight line connecting two points on the circle’s perimeter and passing through its center). Irrespective of the circle's dimensions, this ratio remains consistent. Pi is categorized as an irrational number, meaning it is a decimal that continues indefinitely without repetition. It is typically approximated as 3.14 or expressed as the fraction 22/7
3.How is Pi calculated?
- Pi (π) is calculated using various mathematical methods. One of the earliest methods is to measure the circumference and diameter of a circle and divide the circumference by the diameter. However, this method is not very accurate and can only provide an approximation of pi.
- Another common method is through geometric approaches, such as the method of inscribed and circumscribed polygons. This involves inscribing regular polygons (polygons with equal sides and angles) inside and around a circle and calculating their perimeters. As the number of sides of the polygons increases, their perimeters approach the circumference of the circle, yielding a more accurate value for pi.
- There are also infinite series and calculus-based methods for calculating pi, such as the Leibniz formula, the Gregory-Leibniz series, and various integral representations.
- Modern computers and algorithms have been used to calculate pi to trillions of digits using sophisticated mathematical algorithms and supercomputers. However, for most practical purposes, using approximations like 3.14 or 22/7 is sufficient
4.Evolution of Pi Calculation
- 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 10^40 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
5. Significance
- Circles abound in the world around us, as do three-dimensional shapes like cylinders, spheres, and cones, all of which embody the constant proportion of Pi.
- Understanding the value of Pi holds significant practical advantages across various domains such as architecture, design, and engineering.
- Whether it's constructing water storage tanks or designing sophisticated satellite equipment, Pi's value is essential in numerous applications.
- Furthermore, Pi appears to be intricately intertwined with explanations of the fundamental workings of the universe, from assessing the vast expanses of space to unraveling the helical structure of DNA.
- Professor Dorina Mitrea, chair of the Department of Mathematics at Baylor University, Texas, highlighted in a 2023 interview with Newswise that Pi serves as a crucial element in solving a multitude of problems inspired by real-world phenomena. As our comprehension of the world advances, Pi's relevance is only expected to grow.
- Nevertheless, the computation of Pi to 31 trillion digits may not seem immediately practical. While Archimedes' calculation sufficed for the practical purposes of his time, today, achieving a precision of about 39 decimal places is necessary to conduct calculations in the observable universe with virtually zero error
6. Way Forward
The digits of Pi never end and never show a pattern. They go on forever, seemingly at random except that they can’t possibly be random, because they embody the order inherent in a perfect circle
Source: Indianexpress
