.: The Red Brick – Dubai College's Student Newspaper :.

“Infinity is a floorless room, without walls and ceilings.”

This description of the vast expanse that is infinity would leave most of us clueless, if not a little frightened. How does one begin to deal with something that never ends – infinity seems to defy all our logic and attempts to tame it. A few, however, saw in infinity not a vast expanse of frightening proportions, but a world of endless possibilities, free to be explored and delved into. The realms of infinity represented a world where anything and everything could happen, and a few of the greatest minds reveled in this freedom.

One such mind was Georg Cantor (although he was eventually placed in an asylum). He was the inventor of set theory, a mathematical way of expressing collections of objects. His most important work came with the definition and classification of infinite collections of objects. He defined infinity, and proved several results, including the fact that there are an infinite number of infinites.

The easiest way of thinking about infinity is to think of the largest number you can think of, then add one, and repeat the process. What Cantor aimed to do was to define infinity mathematically, and prove some of the counter-intuitive aspects of it. We can explain some of these aspects using the idea of a hotel with an infinite number of rooms – an analogy developed by David Hilbert, the great German mathematician, in his attempt to explain Cantor’s work.

Hilbert asks us to imagine a hotel with an infinite number of rooms, and that each room is occupied. The rooms are numbered 1, 2, 3, 4… and so on. Remember, there is no room labelled infinity, since there will always be another room. He tells us that every single room in the hotel is occupied – not one room is unoccupied. He then asks what happens if another customer comes along asking for a stay at the hotel? Can we accommodate them?

In fact we can. All we do is ask the person in room 1 to move to room 2, and the person in room 2 to move to room 3 and so on. You might ask where the person in room infinity moves to – there is no room infinity! And so, the first room is left unoccupied and we are able to accommodate the extra customer. And yet, at the start, every single room in the hotel was occupied. We can extend this argument to any number of customers that arrive wanting to stay at the hotel. If two customers arrive, we ask the person in room 1 to move to room 3, the person in room 2 to move to room 4, and so on. If n customers arrive, we ask the person in room 1 to move to room n+1, the person in room 2 to move to room n+2, and so on, freeing up n rooms. This defines the operation infinity+n = infinity, which is rather counter-intuitive.

Hilbert then asks what happens if an infinite number of new customers arrive. Is it still possible to accommodate the all? Incredibly, it is. Before you look at how it can be done, think about it for a few minutes. What we do is move the person in room 1 to room 2, the person in room 2 to room 4, the person in room 3 to room 6, and so on (move the person in room n to room 2n). Thus, all the odd numbered rooms are left unoccupied, and since there is an infinite number of odd numbers, we can accommodate the infinite number of new customers – again quite paradoxical. The operation is thus defined as infinity+infinity = infinity. What happens if two buses carrying an infinite number of people each arrive – we can use a similar argument, and accommodate all the new customers. Thus, we can extend the operation to infinity+ n infinities = infinity.

What happens then if an infinite number of buses come carrying an infinite number of people arrive? The mathematics here is a little complex – it is however, still possible to accommodate all the customers! Cantor, using a method known as the diagonalisation argument, shows that this incredible notion is indeed true. He was, however, stumped when asked whether it would be possible to accommodate all the rational numbers, another infinite set. Surely this was possible? Through some clever logic, Cantor established that it was not, and herein lays his greatest and most paradoxical of achievements. He had proven that infinity had size – that some infinites were larger than others. Two sets that were infinite did not necessarily have to be the same size as each other.

As paradoxical as it seems, it is indeed true – the mathematical proof is absolute. It is little wonder that Cantor was fascinated by the world of infinity, with its incredible and surprising quirks. Some might ask what is all the use of this, where can any of this be put to practical use. What is the point of it all? The point is, does it have to have a point?

Raaghav Ramani

Differentiation is perhaps the most useful aspect of mathematics, with applications in economics, physics and, in general, almost all practical fields. Though GH Hardy claimed that ‘useful mathematics’ is often the least interesting, differentiation, with its thousands of applications, is one of the most interesting ideas in mathematics – it deals with the wonderful concept of the infinitesimally small.

Differentiation is the idea of finding the rate of change of one variable with respect to another. This rate of change can be approximated by drawing a tangent to the curve and then working out what the gradient approximately is. But this is a mere approximation; as far back as 300 BC, mathematicians asked what the exact rate of change was. They thus introduced the idea of limits; letting the length of the tangent become smaller and smaller, using points that are closer and closer together, until you can work out the exact gradient at the point.

This idea is more clearly explained through a graphical method:

The gradient is given by the change in the ‘y-length’ divided by the change in the ‘x-length’. However, as said before, this will be an approximation. The brilliant mathematics lies behind the idea of letting the value of h become smaller and smaller, so that the distance between the two points is less and less, so a better approximation is reached. The best approximation is reached when the the distance between the two points becomes zero; when h is zero. Thus, the general formula for differentiation is given by:

Using this idea, can you work out what the gradient will be for the following scenarios?

1) f(x) = x

2) f(x) = x ²

3) f(x) = x ⁿ

For more information on differentiation:

http://www.khanacademy.org/video/calculus–derivatives-1–new-hd-version?playlist=Calculus

The Menger Sponge: A shape with an infinite surface area - but zero volume

The Mandelbrot Set: A fractal created using imaginary numbers

The Koch Snowflake: Another fractal; created using triangles. The shape repeats itself no matter how far one zooms in

A fractal created using Newton's iterative method for solving a quartic equation

A Hypercube: a four-dimensional cube

 BUT – my favourite shape of all:

The triangle is the fundamental shape in geometry – any regular polygon can be constructed using triangles. But how can a circle be constructed out of triangles – I’ll leave you to figure this one out!

Teacher: What is 2k + k?
Student: 3000!

Q: Why do you divide Sin over Tan?
A: Just cos!

Q: Did you hear the one about the statistician?
A: Probably….

Norbert Wiener, a prominent mathematician and professor, was quite the celebrity at MIT where he taught – students were generally in awe of his immense mathematical ability, but more so of his bewildering antics. An MIT student once saw him in a post-office, and wanted to introduce himself to the great man, for few had the opportunity to meet him. He was, however, in deep thought, pacing around with brow furrowed; the student imagined he was likely running through complex problems in that great mind of his, and that any interruption would be most unwelcome. Still, the student tentatively walked forward and introduced himself, saying: “Good morning Professor Wiener”. The professor stopped in his tracks, looked up and smacked his forehead, shouting: “Wiener! That’s the one!”

This rather fantastically titled article looks at what, in my humble opinion, brings mathematicians, and such personalities, together to work in a discipline that is the foundation for so many different fields. The word ‘discipline’ here is key; mathematics, above all, is a discipline – one fused with rigour and cemented with precision, and yet laced with elegance and delicate intricacies. As the mathematician G.H. Hardy claimed: “Beauty is the first test: there is no permanent place in this world for ugly mathematics”.

The base of mathematics is proof – without rigorous proof, mathematics becomes useless. For the scientist, it might be adequate to prove that a theorem holds true by showing that it does for hundreds, or even thousands, of values and assuming it does for others. For the mathematician, this means nothing. A proof in mathematics must encompass all the values, all the possibilities. In their book ‘Principia Mathematica’, Bertrand Russel and Alfred North Whitehead famously provided a 378 page proof of the fact that 1+1=2. After proving this, they stated: “The above result can be quite useful”.

But while proof is this rock upon which mathematics is built, the most intriguing and interesting aspects of mathematics arise from the exploration of different ideas and methods. To explore mathematics and the curious properties of numbers and shapes requires insight and intuition – the ability to observe and deduce details. Few could claim to have more insight into mathematics than Srinivasa Ramanujan, a shy clerk from Madras, who independently rediscovered hundreds of theorems in mathematics and proposed thousands more – all without any formal education. His wonderful (perhaps naive, but wonderful all the same) belief that everything had a formula, combined with G.H. Hardy’s immovable insistence upon rigorous proof, produced one of the most fruitful collaborations in mathematics. Together, Ramanujan and Hardy revolutionised number theory, discovering and proving many curious properties about numbers. One such property was regarding the partition numbers – the number of ways you can write a number as a sum of other numbers. For example, the number 3 has 3 partitions; it can be written as 3+0, or 1+2, or 1+1+1. Hardy and Ramanujan asked how many partitions the nth number had. They came up with a function:

One has to imagine from where these two mathematicians evoked such a formula. Contrary to popular thought, mathematics is perhaps more an art than a science – creativity and originality are the hallmarks of great mathematicians, as well as precision and detail. G.H. Hardy claimed that he pursued mathematics “only as a creative art”, and this is crucial. Mathematicians do not, and in my humble opinion should not, pursue mathematics for its practical applications, or for its functional value, or for its all-pervading nature. People should pursue mathematics for mathematics.

The beauty of mathematics lies primarily in the fact that there is so much left to be explored and proved. New developments and discoveries invariably throw up fresh new questions, and mathematics is constantly expanding. As Freeman Dyson claimed about Ramanujan’s mathematics:

“That was the wonderful thing about Ramanujan… He discovered so much, and yet he left so much more in his garden for other people to discover… I have intermittently been coming back to Ramanujan’s garden. Every time when I come back, I find fresh flowers blooming.”

By Raaghav Ramani