This blog post is inspired by another blog post – Number Devil.
I really enjoyed reading the book “The Number Devil”. A lot of interesting mathematical concepts are explained in very friendly way – a boy named Robert who always dreams of fishes and ants and falling into a hole suddenly starts dreaming of numbers and there he meets the number devil. But he is not a devil at all, he shows him some of the beautiful things that numbers can do. And here I am sharing some interesting stuffs I found while reading the book.
1. In the very first dream of numbers that Robert gets, the number devil shows him the following mathematical trick.
The pattern seems to go on forever, but it doesn’t.
2. Take any number larger than and multiply it by . Let’s take . . Now between and there are three prime numbers , and . Similarly there will always be at least one prime number between a number ‘’ which is larger than and another number ‘’ that you get by multiplying ‘’ by .
3. Take any even number greater than , you can always find two prime numbers which add up to that number (Goldbach Conjecture). For example take . The two prime numbers that add up to are and .
4. In one of his dreams the number devil talks about his friend Bonacci and the Bonacci numbers. A sequence of numbers that starts from 1. The sequence appears when you count the growth of rabbits and also when you count the number of branches in a tree!
Fibonacci devised a thought experiment to figure out the growth of rabbits in a year. So he assumed that no rabbit dies in one year span and a rabbit takes a month to grow up from its soft white fur to brown. These assumptions are not realistic but it made the problem easier.
So according to his assumption next month two rabbits are born. Then in a month they also grow up and the first mother rabbit gives birth to another pair of rabbits which add up to 3 pairs and so on.
So the sequence that we get is 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89… The numbers follow a simple pattern. Every number can be obtained by adding two previous numbers. So the next Fibonacci number will be .
5. In another of Robert’s dream he and the number devil build a large triangle by piling small cubes. The cubes are arranged as the bricks are arranged in a wall. On the very first cube number devil writes 1. And then in another row two 1’s. The number in each cube is the sum of the numbers written in the cubes above it.
When they completely fill the triangle with numbers, the number devil lights up all the even numbers in the triangle and there emerges a beautiful pattern that Robert watches in amazement.
He then shows some other patterns made by numbers when he lights up all odd numbers and then all the numbers divisible by 5. He tells Robert that you can also find natural numbers, triangle numbers, Fibonacci numbers etc. in the triangle. Another interesting things that you can do is if you have to add all the numbers, say from 1 to 9 then you just have to check what’s the 9th number in the row of triangle numbers which is the third row in Pascal’s triangle. And the number is 45. So all the numbers from 1 to 9 add up to 45 and you get the answer without actually adding every single number!