What is Anti-Matter?

To start with, antimatter really isn't that different from ordinary matter, it is composed of particles (anti-particles, but still particles nonetheless) just like normal matter is. In fact the only real difference between matter and antimatter is that (generally) they have the opposite (but exact same magnitude) charge to their matter counterparts.

Antimatter is, however, a relatively new concept for science. In fact the first real, serious suggestion that antimatter could even exist was in 1928, by the English theoretical physicist Paul Dirac. He started out with pretty modest intentions, he wanted to discover how an electron spins, something that was not known at the time. And he managed it, the original equation he came up with was a slightly messy looking one, but still rather neat for a problem that had alluded physicists for so long.

But Dirac wasn't satisfied there, he felt that the equation could be simplified further and he felt that notation was the key, he worked and eventually, he managed to create an equation to explain the spin of an electron with just a few terms.

He then decided that this was ready for full publication, but when he did he encountered a few problems. The solutions to the equation implied that if it was how the electron behaves it must have a particle that is identical to it, exact for it having opposite charge. It wasn't generally accepted that this could be true, but Dirac believed that the equation was constructed so elegantly and beautifully that it must be true, and in fact this particle was discovered to exist in 1932 by Carl Anderson, this was the positron.

The main useful property on anti-matter is that when it collides with matter, it annihilates. This means that all of the mass of the two particles is converted into energy, and it is a lot of energy. Using Einstein's famous equation, E = mc², we can work out how much energy is produced when annihilating an electron and positron. They each have a mass of 9.11 × 10^-31 kilograms and c is the speed of light, 299,792,458 metres per second. So the energy produced in the annihilation of an electron and positron is, E = 2 × 9.11 × 10^-31 kg × 299,792,458² m/s = 1.64 × 10^-13 joules.

Now this may seem incredibly low, but this is for one pair of tiny particles. If we had a collection of electrons and positrons the mass of a grain of sand (about 2.3× 10^-5 kilograms) this energy level jumps to 4.13 × 10¹² joules. This is about 50 times more than the average energy a car uses in a year. From matter no greater than the size of a grain of sand.

The energy released is in the form of gamma radiation which although it can be harmful it has very useful properties too. It is only through annihilation that PET scans (positron emission tomography) are possible. Electrons is fired at where a scan is needed to be taken from, and with the aid of a radioactive isotope it collides with a positron and produces a gamma ray. The gamma radiation is picked up via detectors and the data that is received can be reconstructed to show a detailed 'slice' of the area.

Because of the massive amounts of energy that are produced from a relatively small mass it would seem reasonable to try and harness this energy to power, well, everything. But here lies the problem. It is very difficult to produce large amounts of antimatter artificially. CERN when fully operational can produce 10 million anti-matter particles a minute, but even at this rate it would take over 100 billion years to produce just 1 gram of anti-hydrogen.

As you might imagine it is also very difficult to store anti-matter, because when it interacts with matter it annihilates into energy. In fact anti-matter has only been stored for 16 minutes at most in the whole of history, this is yet another stumbling block as to why anti-matter is very hard to utilise.

Because of both of these factors it is estimated (by NASA in 1999) that one gram of anti-hydrogen would cost $62,500,000,000,000 (that is 62.5 trillion dollars!). Because of this massive sum of money for such small amounts of anti-matter, research into it is a big thing in modern physics.

Basic Introduction to Topology

Topology is the study of properties that are preserved within objects through twisting, stretching and deformations of that object. However you are not allowed to tear the object, this is key.


In more formal terms, topology is the study of shapes without reference to distance. Or in yet more formal terms it is the study of continuous functions. The study of continuous functions requires the knowledge of range and domain of a function, it studies functions with domains and ranges that make the function continuous. But if you knew that, it is unlikely you would be reading this blog post for absolute beginners as to what topology is!


Also a slight detour from the topic, a continuous function is one where a small change in the input results in a small change in the output too, if it does not follow this then it is said to be discontinuous. Also, just to note, if the inverse function of a continuous function is also continuous it is said to be bicontinuous.


A good way to picture the rules of topology is to picture plasticine, if you have a shape that can be twisted, stretched or deformed in anyway without being torn and put together into a new form then your original shape is topologically equivalent to the new shape you would make.


The reason that topology is useful is that it examines the properties of a shape or region and investigates the properties that are independent of geometric properties the shape or region has. This is the great advantage that topology utilises is that it does not rely on angles or distances, the only thing you need to know is the type of "continuous information" of the object.


Two objects with the same topological properties are said to be homeomorphic. The most well known example of this is that a torus and coffee cup are homeomorphic, they are (topologically speaking) the same shape. As a lot of objects are homeomorphic, it means that if we have a theorem that applies to one object in topology then it also applies to every other object that is homeomorphic to it.

An animation depicting how you
create a coffee cup from a torus.

The point of topology is not to be confusing or just strange, like all Maths it serves some purpose. Be that to merely solve more problems in Maths with a greater ease or to answer problems in the "real world", there is always a purpose to a Mathematical field. And of course, topology is no different. Topology was created to investigate the relationships between objects, and that is exactly what topology does! In fact the origins of topology stemmed from the famed Leonhard Euler when he tried to solve the problem Seven Bridges of Konigsberg, however it truly took off when George Cantor and Henri Poincaré. Poincaré (the famous Mathematician) also famously said "mathematics is not the study of the objects themselves, but the relations between them".


Previously unanswerable questions are now possible, because of the aid of topology. Some that you may wonder why anyone ever wanted to know and others that are pretty useful and interesting. This list should show you what I mean by that.

1. If a coffee is stirred gently then is there always a drop of coffee that finishes in the same place as it started.
2. It is not possible for the velocity of the wind to be non-zero at every point on Earth. This means that at some point at Earth, this means that at any time at all there is no wind whatsoever.
3. There are always two points on the equator that are 180° apart in longitude that have the same temperature.
4. A tennis ball can never be combed in such a way that a cowlick does not appear.
5. It is always possible to cut two irregular pancakes on a plate in half using just one slice.

And that is it for a basic introduction to topology, I will try to get more posts done on topology in the future, so please stay tuned for that! Be sure to comment any questions, tell your friends about us and like us on Facebook!

Proof by Contradiction: Infinite Prime Numbers

There are many complex and obscure methods of proving that there are infinite prime numbers, but this one is certainly the easiest and it is a demonstration of how unpredictable numbers can be dealt with in an efficient manner, it is also an easy introduction to proofs by contradiction.

First, just to clarify what proof by contradiction actually means:
1.) We assume that what we are trying to prove to be false is in fact true.
2.) If we find a contradiction in our assumed hypothesis that we have assumed then it can not be true. And that means that our assumed hypothesis is false, proving what we originally wanted.

So using this as our basis we are going to assume that there is in actual fact a finite amount of prime numbers, how many there is does not matter just that at some point there are no more prime numbers. If we multiply all of these prime numbers together we will get:

2 × 3 × ... × p_n-1 × p_n

Where p_n is the last prime number and this number is clearly not prime as is has every single prime number as a factor. However if we add 1 to this number then any of the prime numbers that we try to divide it by will give a remainder of 1, this means that it too is prime!
p = 2 × 3 × ... × p_n-1 × p_n + 1

If this number is also prime it means that our original statement about there being finite prime numbers is incorrect, therefore there are an infinite number of prime numbers!

This took literally two lines of working to prove that there are an infinite number of prime numbers, this massive, beautiful and unfathomable concept was proved so efficiently and beautifully. This is what Maths is about, how concisely you can express complicated statements. Maths is solely about explaining the world around us and where this leads us is some amazing, incomprehensible places.

Proof by Induction: Sum of Odd Numbers

I will introduce an integral part of mathematics in this blog post, proof by induction. It is fundamental to many proofs and is a pretty simple concept to grasp, I will demonstrate how this proof works with a simple example.

The first basis of proving by induction requires some intelligent intuition to our problem. This means that we look at the pattern of what is happening and try to prove this for all values. For the sum of square numbers this goes:

1 = 1
1+3 = 4
1+3+5 = 9
1+3+5+7 = 16

You may have noticed from this that the sum the answers is always a square number. The sum of the first n odd numbers appears to always equal n², but it just appears that way for the examples that we have given, we do not know that is true for all values, yet.

So our assumption is the sum of the first n odd numbers equals to n²:
1+3+5+...+(2n-1) = n<sup>2

This is our assumption, but if we can prove that n+1 odd numbers equals to (n+1)2 then that means it will hold true for the sums of any number of odd numbers. To begin to check this we add n+1th odd number onto both sides of our assumption.</sup>

1+3+5+...+(2n-1)+(2n+1) = n²+2n+1

As you can see the right side does factorise to give:

1+3+5+...+(2n-1)+(2n+1) = (n+1)²

Which means that our original assumption was correct! Ergo the sum of the first n odd numbers does equal to n² formally in mathematical notation this is:

And that is it for an example of how you use the proof by induction, now it just needs to be put in terms that are applicable for all problems.

Principle of Mathematical Induction
If for each positive integer n we have a statement P(n). If we prove the following two things then the statement is true:
a.) P(1) is true.
b.) For all n, if P(n) is true then P(n+1) is also true. This then means that P(n) is true for all positive integers n.

If that is pretty hard to compute look back to our example, our assumed statement is that P(n) = n² our assumption is true for P(1), 1 = 1² and we then proved that assuming P(n) is true that P(n+1) is also true. Therefore if P(1) is true, then P(2) is true too, etc.

Stay tuned for my future blog posts where I will utilise mathematical induction to find general formulas or proofs to many things.