Pi (pronounced like “pie”) is often written using the greek symbol “π”. The definition of π is: The Circumference divided by the Diameter of a Circle.
March 14th is Pi Day, July 22nd is Pi approximation day. This right here is an anomaly, as 22/7 is a more accurate approximation of Pi than is 3.14. We mathematicians can live with using the MM/DD date format as that is the only way we have a sniff at having a Pi-Day, but calling the more accurate number as the approximation day is a bit rich.
History of Pi
The Babylonians had a stab at it as early as 2000 BC coming as close as 3.125 or 25/8. Archimedes provided a breakthrough by inscribing and circumscribing polygons about a circle and generating lower and upper bounds.
Our own Ramanujan developed some juicy methods that were later incorporated into computer algorithms. Inevitably, machines entered the fray and we have apparently now computed up to 2.17 trillion digits.
The beauty of the 2.7 trillion-digit approximation is that from 2000 BC to 2100 AD we have moved by 0.5% – those Babylonians must have been pretty cool then, huh.
CAT Preparation Imitates Pi
When we are young we are told that Pi equals 22/7, we feel good about ourselves and solve a bunch of problems with it. Then after a while, we realize that Pi is not really 22/7 and has many more layers to it; on top of it, the damn thing is irrational. Kinda like the CAT-based selection process.
Like CAT preparation itself, it appears to be endless. But don’t let the shoulders drop, like PI itself CAT preparation is useful in a variety of contexts, is delightfully fascinating and is most definitely not pointless.
An Intriguing Geometry Question
Ratio of Areas of a Square in a Circle and a Circle in a Square:
Consider the following diagram:
Let the side of Square ABCD be x. The diameter of the circle circumscribing it will be equal to the diagonal of the square, which is equal to .
So, the area of the circle: .
Area of the square: x2.
The ratio of the area of the square ABCD to the circle circumscribing it: x2: = 2:
Now, consider the circle – EFGH. The diameter of this circle will be equal to the side of the square ABCD.
So, area of this circle:
Ratio of the square ABCD to the circle inscribed in it: x2:
Additionally, the side of square ABCD is the diagonal of the square IJKL, so, their areas will be in the ratio of 2:1
The ratio of areas of the circles ABCD and EFGH is also 2:1
As a general rule,
- The ratio of the area of a square to the circle circumscribing it: 2:
- The ratio of the square to the circle inscribed in it: 4:
- If the pattern of inscribing squares in circles and circles in squares is continued, areas of each smaller circle and smaller square will be half the area of the immediately bigger circle and square respectively.
Check out this wonderful article on preparing for Quant from July!
No discussion on Pi ends without a reference to the Circle of Life and a call-out to the utterly delightful phrase ‘Hakuna Matata’. Hakuna Matata means ‘No worries for the rest of your Days’.
We at 2IIM wish that you jump into this CAT journey, have plenty of fun preparing for it and subscribe to the carefree philosophy Hakuna Matata.
Happy CAT preparation to you all!
Rajesh Balasubramanian takes the CAT every year and is a 4-time CAT 100 percentiler. He likes few things more than teaching Math and insists to this day that he is a better teacher than exam-taker.