What is the mathematical symbol Pi mean ? How did it come into being ? What does it stand for ? This is something that I have always wondered for quite a while ?
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There is a great deal of interest nowadays in the book, Life of Pi. The concept of Pi has fascinated people through the ages, from unknown Egyptian and Babylonian architects who must first have encountered its challenge to modern mathematicians and scientists aided by supercomputers who have now gauged its value to a trillion decimal places.
Pi simply is the constant, denoted ∏, which when multiplied with the diameter of a circle, will give its circumference. A simple approximation that many of us use is 3.142 but architects, engineers and scientists require a more precise definition and hence the search for a precise answer. Archimedes is known to have estimated the value of Pi by applying the common sense logic that a circle must be bounded internally and externally by a regular polygon of large enough number of sides. He is said to have calculated the dimension of a 96 sided polygon that bounded a circle and thereafter gave up attempting any more precise a definition of the elusive Pi.
We now know that Pi is an irrational number, which means that it cannot be expressed as a ratio of any two integers. There is archaeological evidence that the ancient Egyptians knew something about this fraction, and were taken up enough by its unfathomable nature, that they enshrined it in their monuments. The Great Pyramid at Giza constructed in 2500 BC, was built with a perimeter of 1760 cubits and a height of 280 cubits which gives a ratio equivalent to 2 ∏. It cannot be pure chance that this great monument was built to this ratio. The equivalent value of ∏ in this ratio comes to be 3 + 1/7 or 22/7 which is what students in schools still commonly use in its place.
Around 1400 AD, an Indian, Madhava of Sangamagrama estimated Pi to eleven decimal places by equating Pi to an infinite series. The development of infinite series to estimate mathematical values was a great innovation that helped achieve greater precision in calculations. Srinivasa Ramanujam is known to have expressed Pi in the form of several such series, each of them giving another way of reaching closer to the value of Pi.
A German, Ludolph van Ceulen, devoted the greater part of his life to estimating the value of Pi, and he managed to solve value Pi to 35 decimal places using geometrical methods in the sixteenth century. He was so proud of his achievement, that he had these decimals inscribed on his tombstone. Ever since, the constant is often known as Ludolph’s Constant. It is also better known as Archimedes’ Constant.
It was only afterwards, in the 18th century that the nature of Pi was understood as irrational, and it took another century to understand that it is a transcendental number, which means that there is no polynomial with rational coefficients for which Pi is the root. That simply means that Pi is not constructible with compass and straight edge.
Teachers of mathematics have naturally tended to use Pi, as a symbol for mathematical enquiry. March 14 is celebrated as Pi Day in some parts of the world. The date relates closely to 3.14 which is an approximation of the value of Pi. Schools and colleges take the opportunity to hold competitions and displays related to maths on this day. A popular contest is often around how many digits of Pi can be memorised and retold accurately. People have managed to recite over 10000 digits from memory.
There are many websites dedicated to Pi, and its teaching. We have even come across verses and lyrics composed around the inscrutable Pi. Here is one from a website dedicated to the teaching of Pi:
“Oh, number Pi, Oh, number Pi
You’re truly transcendental.
Oh, number Pi, Oh, number Pi
You’re physical and mental.
You stretch the bounds…of all we know,
And tell our circles where to go
Oh, number Pi…..”
Roby John 115 Q, 2 A, 0 C |
By definition, pi is the ratio of the circumference of a circle to its diameter.
Pi is always the same number, no matter which circle you use to compute it. For the sake of usefulness people often need to approximate pi. For many purposes you can use 3.14159. Here’s pi to many more digits: 3.14159265358979323846.
Pi is an infinite decimal. Unlike numbers such as 3, 9.876, and 4.5, which have finitely many nonzero numbers to the right of the decimal place, pi has infinitely many numbers to the right of the decimal point.
If you write pi down in decimal form, the numbers to the right of the 0 never repeat in a pattern. Some infinite decimals do have patterns - for instance, the infinite decimal .3333333… has all 3′s to the right of the decimal point, and in the number .123456789123456789123456789… the sequence 123456789 is repeated. However, although many mathematicians have tried to find it, no repeating pattern for pi has been discovered - in fact, in 1768 Johann Lambert proved that there cannot be any such repeating pattern.
As a number that cannot be written as a repeating decimal or a finite decimal (you can never get to the end of it) pi is irrational: it cannot be written as a fraction (the ratio of two integers).
The area of a circle is pi times the square of the length of the radius, or “pi r squared”:
A = pi*r^2
Roby John 115 Q, 2 A, 0 C |