I would like to mention here few of the interesting things that I found in the book.
Before reading this book I was not aware of how to calculate π (please do not tell me that we can find by dividing circumference of a circle by its diameter, it will not give you accurate result). People always talk about calculating the value of π up to 1000 decimal places or even more. But what is the correct way of finding the value up to higher decimal places? First person to calculate the approximate value of π was Archimedes. He used polygons to find the value of π. This method is called Method of Exhaustion. He found that the value of π lies between 22/7 and 223/71 and for next five hundred years nobody provided a better approximation. Later a German mathematician Van Ceulen used a polygon of 4,611,686,018,427,387,904 sides and figured out the value of π up to 35 decimal places.
3.1415926535897932384626433832795029
Then I searched internet and found many more interesting facts about π such as the proof that π is irrational and equation to find the value of π up to any number of decimal places.
More about π can be found at http://en.wikipedia.org/wiki/Pi
Next constant discussed in the book is Phi (uppercase Φ, lowercase φ or ϕ) rhymes with "eye". An absolutely incredible number. It is also known as Divine Proportion, Golden Ratio, etc. People consider it as Gods favorite number because it is most widely found in nature. Let me give you some examples.In humans, ratio of leg length to the whole body, ratio of length of thigh and length from knee to foot, etc. Sea shell is considered as one of the best example of Golden Ratio. One more fascinating fact is relation of Fibonacci Sequence with the Divine Proportion. If you calculate the ratio of any number in the sequence to the number before this number, you will notice that it gives you approximate value of Golden Ratio. The more you move towards higher values the better you will get the approximation. For example in the following Fibonacci Sequence:
1 1 2 3 5 8 13 21 34 55 89 144 233 ....
3/2 -- 1.5
5/3 -- 1.66
8/5 -- 1.6
13/8 -- 1.625
21/13 -- 1.61538
and so on.
The formula for finding most approximate value of Golden Ratio is :
Another number which caught my attention is e. Its value is 2.71828 18284 59045 23536 (truncated to 20 decimal places)
And the general formula is :
e = 1/0! + 1/1! + 1/2! + 1/3! + 1/4! + 1/5! + 1/6! + ....
But in this book I found how Jacob Bernoulli found this number. He was calculating the amount of money required to be paid on a principal of Re 1 at the interest of 100%. He found that when compounded annually the amount is Rs 2, and when compounded biannually amount is 2.25,... similarly when compounded daily it becomes Rs 2.71. So he found that we are moving towards a constant 2.718. Which was later named as e because of mathematician Euler.
Apart from these constants I got to know one amazing fact about perfect numbers. I knew that a perfect number is defined as a positive integer which is the sum of its proper positive divisors. But the interesting thing is perfect numbers are related to Prime Numbers. For any prime number P which can be written in the form for 2n − 1, there exists a perfect number which is equal to 2n−1(2n − 1). Try putting n=2,3,5,7 you will get 6, 28, 496 and 8128 respectively.
I will not write more about the book. It has numerous fascinating stories about Scientists and there discoveries.You must read the book and satisfaction is Guaranteed :).
Agry
Note: This book has very few mistakes for which you can find the amendments at http://www.peterjbentley.com/amendments.pdf.
