\sqrt{2} is irrational!

Posted on February 3, 2012 by Nicholas

Firstly, let me remind you that an irrational number is one that can't be written as the ratio of two integers, for example \frac{1}{2} .

We now prove that \sqrt{2} is irrational by contradiction, so let us assume that it can be written as a fraction say \sqrt{2}=\frac{a}{b} , now we can also assume that this fraction is in lowest form.

We can now say that

2=\frac{a^2}{b^2} \Rightarrow 2b^2=a^2.

This means that a is even, so we can write a=2k for some k . Thus we have

2b^2=4k^2 \Rightarrow b^2=2k^2

and so b is even too, this is a contradiction as both a and b are even so the fraction is not in lowest form.

Hence \sqrt{2} is irrational.

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CloudFlare | The web performance & security company

Posted on January 30, 2012 by Nicholas

CloudFlare | The web performance & security company.

We are now using a new DNS provider, I hope this will increase the load times of our websites, any comments are appreciated!

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An infinite number of primes!

Posted on January 29, 2012 by Nicholas

I thought I would begin with a nice proof that there are an infinite number of prime numbers.

What is a prime number? Well it is a number which can only be divided by itself and 1 . For example the first few prime numbers are 2,3,5,7,11,13, \ldots .

Why are there are infinite number of them? Well, lets suppose not, that is there are only a finite number of prime numbers, lets call them p_1,p_2, \ldots ,p_n . Now lets us consider the number N=p_1p_2 \ldots p_n-1 .

So we have created a new number, which is either prime, which would be a contradiction, or it has some prime factors. Now none of p_1,p_2, \ldots ,p_n can be a prime factor of N , so there must be some other prime not on our list, again a contradiction.

Hence there are an infinite number of prime numbers!

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Welcome!

Posted on January 29, 2012 by Nicholas

Hello, I would like to take this time to introduce myself. My name is Nicholas Dale, and I am a student at the University of Sheffield. I am currently studying for a degree in mathematics, where I have an interest in number theory.

I will be using this blog to keep a record of my studies and also share things I find of interest. As you can probably see though, this site is currently under construction, so there may be hiccups along the way, I hope you will bare with me!

Finally, I wish this site to be a community for other like minded people to share there experiences, so I encourage you to become part of the community by signing up to this website, and commenting on my posts, or by participating in the forums!

Nicholas

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