Do You Exist?

This may seem somewhat ludicrous to even consider that you may not exist. I mean, you do things, you interact with people, you feel things so obviously you exist, right? Well, not necessarily...

The problem with proving whether or not you exist is that it is very hard to even pin down what existence actually is. It is defined in the dictionary as "objective reality or being", but that poses yet more questions; if we do not know what existence is how can we begin to comprehend reality?

Within mathematics in order to prove something you must only use something that is an already proved axiom. Multiplication works because it is essentially multiple additions (2 × 3 is the same as 2 + 2 + 2). And in fact addition can be proved to be true using something simpler, something more fundamental than itself; logic. You can prove that one plus one equals two (or if you prefer a well explained video), and from this simple fact all other vastly complex mathematics can be used and proved in the knowledge that it is correct.

But existence can not be tackled in the same manner, it is the most fundamental property. Existence precedes even the most basic mathematical principles. We must exist for anything to hold true, including maths. It has to be taken as a given for our principles of logic, maths, science, everything to be true. But this does not prove that we exist. Unfortunately everything else functioning so well because of one assumption does not prove the assumption, it could just be that everything else is radically wrong.

And the main problem of trying to understand our existence - you cannot attack it with maths. You cannot break existence down into a mathematical problem so ones perception of what does, or does not constitute existence is merely subjective. When opinions are the dominant factors of an argument you can never reach a correct answer because essentially, there is none.

I can feel the philosophers amongst you grinding at your teeth, beginning to pound at your keyboard reciting Descartes quote as if it were the ultimate truth. For those of you who do not know of Descartes, he also pondered whether or not he existed but then saw that if he was able to actually ponder his existence then there must on some level be something that exists to do the pondering; hence his quote "I think, therefore I am". It is a well constructed argument, it concisely answers the question using a very logical approach. But that in itself is it's downfall, it uses logic. Logic is something that can only function as a tool if in fact there is existence on some level, so actually he uses the fact that we exist to prove that we exist.

Also, why does the property of thinking create existence? This plays on the difficulty on defining what existence actually is. It almost defines existence as thinking, then citing the fact that we do think to be a proof of existence, which is fundamentally wrong.

So, do you exist? Probably. There is no real means of actually answering the question when we cannot truly define existence. But if you do not exist and your whole life and world is not truly there, you are none the wiser and will never truly know, so why does it matter? Ignorance is bliss as they say. If you wish to believe that you exist, then you exist.

Introduction to Modular Arithmetic and Congruence

Modular arithmetic is an arithmetic system for the integers where the numbers wrap around, like on a clock for example. The numbers start at one, they go round to twelve and then start again at one, this would be an example of modulo 12. However generally in modular arithmetic we start at 0 and go to 11, before starting at 0 again. This means that 7 o'clock would be 6 mod 12 (mod is often used to shorten modulo). In fact because the number line "wraps around" it means that 19 o'clock = 7 o'clock (as you will know if you have used a 24 hour clock), this means that 18 mod 12 = 6 mod 12; because of this a mod m is wrote where a < m.

However it does not have to be just modulo 12, it can be modulo anything (as long at is a positive integer), so let us jump straight into the definition. So that you are aware of the notation I will use a|b means b divides a (this means that b/a is an integer). Let m be a positive integer and let a, b both be integers if m|a-b then a is congruent to b modulo m, wrote mathematically this is: a ≡ b mod m. Make sure you do not think "≡" means equivalent in this case, because it does not!

You can add in modular arithmetic (provided they have the same modulo). Let m be a positive integer and let a, b,c and d be integers if a ≡ b mod m and c ≡ d mod m then a + c ≡ (b+d) mod m. So basically you add the parts preceding the modulo together and then find modulo m of that.
You may also notice that a/m has a remainder of b, this is how we can quickly work out that 19 o'clock is the same as 7 o'clock, or the same as 103 o'clock. You simple divide the number by the modulo and work out the remainder.

An example of why we would use this using our clock as an example is if it is 4 o'clock now, what will the time be in 157 hours? So what we have is 3 mod 12 and 157 mod 12, adding these together we get 160 mod 12, 160/12 = 13 remainder 4. This means that 160 mod 12 = 4 mod 12, so it will be 5 o'clock 157 hours from 4 o'clock.

This should give you an idea of how to do basic addition with congruences, if you do not understand fully read what I have wrote again and then if you still do not understand, post a comment. Now we have the basic principles in place we can begin to go further. Now we will prove that a mod m + b mod m = (a+b) mod m and also how multiplication works in modular arithmetic.

Multiplication between two congruences is just as easy as addition with congruences, I will first provide a definition to you, before proving it. Let m be a positive integer and let a, b,c and d be integers if a ≡ b mod m and c ≡ d mod m then ac ≡ (bd) mod m; this is the same sort of principle as addition. Now to prove them.

First to prove addition:

Now to prove multiplication:

These are the very basics of congruences, but the are integral to modular arithmetic (I mean, think how important addition and multiplication is normally), they are very, very powerful and I will begin to explore them more and more in my upcoming posts. I will prove just one more result as it is pretty easy and uses multiplication.

The proposition is that aⁿ ≡ bⁿ mod m is also true provided  a ≡ b mod m and n is a positive integer. We will prove this by induction. Let P(n) be be the statement of the proposition, then P(1) is obviously true as we already know that a ≡ b mod m. So let us assume that P(n) is true, then we have that aⁿ ≡ bⁿ mod m and that a ≡ b mod m, multiplying the two of them together (using our proved method) we get a × aⁿ ≡ b × bⁿ mod m, this then simplifies to: aⁿ⁺¹ ≡ bⁿ⁺¹ mod m, which is P(n+1). Hence P(n) is true for all values of n!

I hope that you like this post and that is helps you. If you did like it then please also like us on Facebook.

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.