## What is your favourite number?

*“Oh, I’d never really thought about it” *

* “What a strange question” “I don’t have one”*

*“They are all the same really” “Do you actually have one?”*

*“Numbers are dull” “I don’t like numbers”*

*“7, I can’t explain why, I just love it”*

Above are just some of the responses I have had over the years to this question. Overwhelming, the question is met with surprise and sometimes scorn and I think this is a shame because numbers are a huge part of my life. I do have a favourite number. In fact, I have a *number* of favourite numbers. There are so many to choose from! I don’t like to restrict myself, so I have a favourite prime and a favourite composite, a favourite rational number a favourite irrational number. And sometimes they change depending on my mood. I know that, at this point, some will be thinking that this is a bit bizarre. But is it really that odd to have a favourite or several favourites and it is really that odd to talk about it or them? Samuel Taylor Coleridge once said:

*“Nothing is so contagious as enthusiasm”*

I think numbers could do with a little of this – they seem too often to get a bad rap. Few of us would be phased by the following questions:

What is your favourite colour?

What is your favourite word?

What is your favourite character from Harry Potter?

What is your favourite song?

So what is odd about numbers? I would argue zero, zilch, zip. In fact, numbers provide a veritable feast of interest and excitement and I hope that, by giving you an insight into some of my favourites, I might be able to convince you to take another look. The first number that we will consider went against the very core beliefs of the Pythagorean Brotherhood and ultimately lead to its destruction. Hippasus, who saw fit to utter this unutterable number to the outside world, was allegedly murder for his troubles. Fly back in time to around 500BC and you will find The Pythagorean Brotherhood happily ensconced in their world of numbers busily exploring a few triples:

One fateful day, someone, maybe it was Hippasus himself, saw fit to pose the following question:

Imagine a square with sides of length 1. What is the length of the diagonal of this square?

Of course, they were able to work out the answer because they knew Pythagoras’ theorem:

*“The square of the hypotenuse of a right-angled triangle is equal*

* to the sum of the squares of the other two sides”*

So, what harm can a unit square pose I hear you question? Well, let’s take a look. By the Pythagorean theorem we can work out the length of the diagonal as follows:

So, the length of the diagonal is equal to √2 , i.e., the number which, when multiplied by itself, produces 2. And what is so very dreadful about that I hear you say? It is perhaps important to mention here that the Pythagoreans loved integers or whole numbers. In fact, they believe that all numbers could be expressed as a ratio of integers and all of their teaching and beliefs were build upon this fact. There is an infinite supply of such numbers and today we know them as the rational numbers. The Pythagoreans would have known immediately that the answer to √2 was not an integer: 1×1=1 which is less than 2 and 2×2=4 which is greater than 2 and there are no integers between 1 and 2. So they would have begun to look for a ratio of two integers that when multiplied together produced 2.

Try as they might, they simply couldn’t find the answer. This terr ible discovery had to be kept secret. Such heresy couldn’t be uttered beyond the confines of the brotherhood. Everyone was sworn to secrecy and the “number” would remain unutterable. Today we call numbers such as √2 irrational. They are numbers whose decimal expansion do not terminate, nor end with a repeating sequence. √2 begins with the digits but no list of digits can represent √2 exactly and there is no repeating pattern within these digits. The problem for the Pythagoreans was not just that the discovery of √2 went against the very core of their beliefs but, once discovered, other such unutterable numbers kept on popping up. Afterall, there is an infinite supply of irrational numbers too! One day Hippasus decided to utter and, whether it was at the hands of the gods or the brotherhood, he was drowned for his unacceptable outburst.

Now for another of my favourites. We all know of banned books but what about banned numbers? There is evidence of counting dating back 35,000 years ago, yet for centuries, this number was either completely absent from number systems or was only given place holder status. Although it was first accepted as a proper number in India in the 7^{th} Century, it took another 1000 years before it was widely accepted across the world. In fact, this number was banned in Florence in 1299 because it was believed to encourage fraud. The number I am talking about is, of course, the one, and only, 0.

Zero, I hear you say, but it’s not worth anything. How can it possibly be a favourite? My love of zero comes as a result of the many juxta positions associated with it. It is nothing and yet central to mathematics: consider for a moment arithmetic without it – think addition, subtraction, or multiplication with Roman Numerals – not an appealing prospect. Now consider the symbol for zero. It is an empty hole and, at the same time, an unending circle with no beginning and no end. You might think that two such different interpretations would be unable to fit the same number and yet both are appropriate for zero. The former signifies zeros nothingness and the latter is indicative of the infinite loop in which you would find yourself if you attempted to divide by zero! Finally, and perhaps my favourite point about zero, is that it is the bringer of order and of chaos. Zero paved the way for Newton and Leibniz, in the 17^{th} Century, to develop a brand-new branch of mathematics – calculus. Today, calculus allows us to predict the future – the stuff of dreams! It is behind the growth models for covid 19 – models that provided understanding to help fight against this deadly virus. Calculus essentially boils down to division by zero and yet, as already mentioned, division by zero, in the traditional sense, will land you in an infinite loop. The result of such division can be found in the function of – it is both simple and elegant. Look at how the y values shoot off upwards unendingly as x drops below 1^{1} and approaches zero from the positives and then how they shoot downwards unendingly as x approaches zero from the negatives. For me, it evokes the image of Shakespeare’s star-crossed lovers. The hopelessness of their situation is epitomised by the infinite loop in which the function finds itself.

I have long sought to associate such narratives to functions but that is a story for another day.

My final choice feels simpler, but it is no less beautiful in my opinion. It is the number 6. The reason I love 6 is because it is a perfect number. Perfect numbers are positive integers that are equal to the sum of their proper factors. For example, the proper factors^{2} of 6 are 1, 2 and 3 and 1+2+3=6. To me this is a truly beautiful result. So simple and yet poetic. If you want further evidence of the reason behind my choice, it may help you to know that perfect numbers relate to Mersenne primes. I love prime numbers too and particularly love Mersenne primes because I think their form is beautiful and though prime, they feel tantalisingly close to being composite but that is a story for another day. Today we know of 51 perfect numbers, all of which are even, so why have a chosen this particular one? 28 is the next perfect number and for some reason it just doesn’t resonate with me. Though 496 as the 3^{rd} perfect number has beauty, 6 has an additional quality that I believe makes it doubly perfect. Not only do the sum of the proper factors of 6 produce 6 but the product does too:

1+2+3=6 and 1×2×3=6

^{1} When we divide by a positive number less than 1, you get a bigger number, i.e. 1÷½=2, 1÷¼=4

^{2} proper factors are the factors of a number other than the number itself