The Mathematical Art of Infinity

 by Sawsene B

The concept of infinity is one that has captured the minds of mathematicians for centuries. We are all familiar with the concept that numbers go on and on, represented by the familiar symbol ∞ one that has blurred the lines of mathematics, philosophy and even spirituality.

In the book How to Count to Infinity by Marcus du Sautoy- a quick I strongly recommend for anyone interested in the art of mathematics- we are introduced to many interesting concepts including the theories surrounding counting to infinity, a seemingly impossible task. The writer suggests the following: if you were to count from 1-10 in 8 seconds, then 11-20 in 4 seconds, 21-30 in 2 seconds, 31-40 in 1 second, 41-50 in ½ second and continuing this on well it would take 16 seconds to reach infinity. 

Take N to be the total number of seconds

N = (8 + 4 + 2+1+½+¼ +1/8 …)

2N = 2(8 + 4 + 2+1+½+¼ +1/8 …)

2N = (16 + 8 + 4 + 2+1+½+¼ +1/8 …)

 

To simplify N we can subtract it from 2N

2N - N = (16 + 8 + 4 + 2+1+½+¼ +1/8 …)-(8 + 4 + 2+1+½+¼ +1/8 …)

Now all the numbers from 8 onwards would cancel out leaving us with 16. NOw as Du Sautoy explained, this would be impossible in practice, requiring us to be able to speak faster than the speed of light which is impossible and would require an infinite amount of energy.

In 1845 was born the father of infinity Georg Cantor, a German mathematician who devoted much of his studies towards the concept of infinity. He suggested the idea that some infinities are greater than others. For example the quantity of natural numbers( integers) is infinite, and yet its size is equal to the multiples of 2 to infinity. Using this Cantor introduces his paradox of infinity hotels; proving these are countable infinities.

However as Cantor soon discovered this is not always the case as irrational numbers come into play. Take pi or the square root of two, these are just 2 of an infinite number of infinite decimal numbers. This adds a layer of complexity to the theory of infinity itself. This caused Cantor to have an extraordinary revelation that changed how we view this concept of counting- that this infinity out of all the infinite sets of number of integers and fractions, is infinitely larger.

This leads us to a mind-blowing realisation, that infinity is not simply a singularity, but that there are an infinite number of infinities, each larger than the previous, perhaps one of the most prevalent theories in mathematics.

Cantor’s revolutionary discoveries of the infinite have not only expanded our knowledge on mathematics but also provided us with the potential of using infinity in our day-to-day algorithms. Marcus du Sautoy even suggested Cantor's diagonalisation argument could one day help us find our perfect match.

Credit: How to count to infinity- Marcus Du Sautoy