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.

Do numbers exist?

This seems like a bit of an odd topic for a maths fanatic to discuss, and really it isn't very mathematical to even think about it, as you will you soon find out it is more philosophical.


You'd think it is obvious that the answer is "yes, they do exist", but it really isn't that simple. The fundamental idea of numbers started with counting a number of items, animals, children, etc. and from there it spiralled off. In fact the definition of a number is: "an arithmetical value, expressed as a word or symbol representing a particular quantity".


So numbers 'exist' to serve a purpose for counting, arithmetic and calculations. So if we have 'two' dogs, do we count the number of hairs they have as being them? Do we count the number of bugs on them? Do we need to count the number of atoms in the dog towards the 'two'? So is it really 'two' or is it some uncountable number of atoms? Or quarks? But just because we can not clearly state what two is in real life, it does not mean they do not exist.


Numbers do make interpreting what is happening an awful lot easier, but that doesn't mean they 'exist'. They easily could have been created by man to help with problems, the problem arises when you need to define what 'existing' is. To 'exist' you need to have reality or being, do numbers really have this? They are not a physical, touchable thing and could anyone really argue the case of a number having a reality to it?


You could then argue that nothing really, truly exists, as what is reality? But that is a completely different tangent for a future post. I mean, do thoughts really exist? Do words that are spoke? But anyway, I digress...


Numbers can be thought of as tools developed by us a civilization to help understand the world we live in. But if there was a race of intelligent aliens there is almost no doubt that they would have a form of maths, they may work in a different base (in fact the Aztec's worked in base 6, opposed to our base 10), or some of their basic principles may be different (a negative times a negative could still be a negative, for example). But there is almost no doubt that basic principles of maths will be there. And once they are in place more advanced concepts start to develop like how are these numbers distributed? What about a number between 1 and 2? How do we add numbers? How do we multiply numbers? And so on.


So whether they are invented or they exist, they are a necessity. It takes away the "what if?" factor from so many elements and provides concrete and quantifies otherwise incomprehensible things and data. Numbers are so useful for everything, so really it doesn't matter if they exist or not because what they accomplish is real, the data, the facts and the information they provide about the world we live in is real, they have a real impact.


So rather than getting bogged down in the semantics of whether numbers 'exist' or not should not detract from the joy and beauty of mathematics. Maths is fun, do not let your philosophical stand point alter any of that.

Imaginary Exponents: x^i

To learn how we do this I first need to explain a very, very useful mathematical formula. It is called "Euler's formula", and this formula gives us a way to find the value of the imaginary exponential function (e^ix) using methods that we already have well defined and are easy to deal with. Euler's formula is: e^ix = Cos(x)+iSin(x), where x is an angle in radians. I may do a post on the actual maths behind why this is the case, but for the purpose of this post it has no relevance.


Now, we can find e^ix, but what use is this if we want to find 2^i, i^i or just anything raised to the power of i, let's call this a^i. So, we want to find a^x where x = i, so we need to try and remember a^x as something involving e raised to the power of something. So this means we have, a^x = e^y.

This then means that a^x = e^[xln(a)]. So we now have a^x in a form involving e raised to a power. So now we can input when x = i. Now by simply placing this into the equation we get, a^i = e^[i*ln(a)]. We can then turn this into something we can solve using Euler's formula, e^[i*ln(a)] = Cos[ln(a)]+iSin[ln(a)].

So now to actually input some numbers to this. Let's say I want to find 2^i, so from our previously defined formula we now have that: e^[i*ln(2)] = Cos[ln(2)]+iSin[ln(2)]. Using our calculators we will find that this is roughly Cos(0.693147)+iSin(0.693147), which then equates to roughly 0.76924+0.63896i. So 2^i ≈ 0.76924+0.63896i

As you can see, this is a complex number and it will be a lot of the time when we deal with imaginary exponents, but (as you may have thought) there are times when the solution to a^i will be a real answer. This is when iSin(x) = 0, and this will happen at Sin(x) = 0 and if you know your Sine curves you will know that this is at Sin(kπ), where k is any integer. Using our formula derived from Euler's, e^[i*ln(a)] = Cos[ln(a)]+iSin[ln(a)], we can see that if a^i is a real number, ln(a) = kπ. If we make both sides to the power of e, we can clear our logarithm to get: a = e^kπ. This should also then mean that the solution that is real (where a = e^kπ) should be equivalent to Cos(kπ).

Therefore if this is correct, then (e^3π)^i should produce a real value. (e^3π)^i = Cos[ln(e^3π)]+iSin[ln(e^3π)]. And when you do work this out, low and behold you get the answer of -1 (which incidentally is the same as Cos(3π)).

Best of the internet

I have strayed off the mathematical path in my last few posts, and this is indeed no different. But I do hope that this will be beneficial, some websites you may know but might not know some of the amazing things these websites can do. The lists will be compromised from the best: games, freeware programs, educational resources and miscellaneous so if you're looking for something specific search for it.

Games:

Addicting Games - Some of the best online games from around the around the internet have been collected, hours have and inevitably will be lost to this collection of games.

Rating: 
It is very, very good collaboration of games that are well designed and as the title of the website says, addicting. The website design is faultless and easy to navigate, and there are some handy features. You can add your own games to the website and could gain 'internet fame' from that, but the reviewing system is also very good and allows members of the website to submit their comments.

PlayR - Ever wanted to relive some of the fond memories you have from the countless hours you spent button mashing on your old handheld consoles? Well with this website you can! Fancy taking a nostalgia trip and playing some of the older renditions of Pokémon, Harry Potter, Mario or even Madden? This website allows you to do exactly that.

Rating: 
The collection is huge, the games (from all the ones I've tried) are in perfect working order, the controls are customisable, the layout is perfect and the memories of the giddy excitement some of the games connote is just immense. The only thing I can really say about it is, I was never a huge fan of too many of these games, except Pokémon (-nerd). 8.5/10


Robot Unicorn Attack - A flash game that centres around a robotic unicorn galloping around in a fantasy world with purple grass, rainbows and sparkling dolphins. And it still makes you feel as if you are the manliness person on the face of the Earth.

Rating:  
This deserves a special mention, it's not a website packed to the brim with games. It's better than that. It doesn't have a plot with twists and turns that keep you on your seat until the end. It's better than that. You make up your own storyline, in your head. You don't need other games, this one is enough; I don't think I have spent more time on any other online game. As an added bonus, the soundtrack is fucking banging (pardon my language but that's the only way to say it). 9.9/10

CrimeBloc - This is a text-based online mafia game, and there is a lot of them about on the internet nowadays because they are great time wasters where you can communicate with people from anywhere in the world one minute and be killing them the next.


Rating:  
What sets this game aside from the rest? Well, the game is coded to perfection, the administrators are very active in the community and with updates, some of the casinos you can gamble on are actually better than ones I've seen on actual gambling websites. With promised updates to come (including a multiplayer poker system, and having beta tested it is nothing short of phenomenal), a growing number of users and something that can literally have you coming back for more for months and months, you simply must sign up to this game. 9/10


Top 150 Flash Games - Pretty self explanatory, the collection of the 150 best online flash games from around the internet.

Rating:  
Yeah you can't comment on the games or gain achievements because well it is just an article with a list of games. But you can not help but lose yourself in the sheer enormity of these games and the fun that could be had. However because it is just a blog post, I can not rate it as highly as some of the games on that list deserve. 8/10

Special mentions - XGen Studios, Miniclip, Mousebreaker, Ijji and Kongregate

Freeware Programs

Avast - This is one of the most well known freeware out there, but it needs a mention here nevertheless. With the rise in the amount of people who frequently use the internet, the amount of viruses and hackers that are around has increased almost exponentially. So there is a huge market out there for anti-virus software.

Rating:  
There is a huge market, but the kind developers of Avast offer it to you for free. Not only is it the best free anti-virus software available, but is actually better than a fair few anti-software products that you have to pay for. Most notably McAfee, it is head and shoulders above it. For finding viruses, for the stability of the product, for not shoving stupid little adverts down your throat even though it's free! 9/10.


OpenOffice - This is the closest piece of software on the market that rivals Microsoft Office. In the modern times a lot of work now has to be done on a computer, be that a presentation, a report or mathematical calculations you need office software. But not all of us want to pay a lot of money in order to just do work. Well your prayers are now answered.

Rating:  
This software is quite simply, amazing. If you want to create: text documents, spreadsheets, presentations, databases or even small drawings you can do all of that and more. You can save all the documents in a number of different file types, and the best bit? Because it is open you can get numerous extensions for the different programs which can enhance all of your work. 10/10

Linux Mint - The commonly overlooked OS that is Linux. It used to be the case that you had to be a computer genius to be able to run Linux effectively. With the rise in its popularity and with Linux rapidly having programs released specifically for it as well being released alongside Windows and Mac OS X.

Rating:  
There are hundreds and hundreds of free distros that can provide you with Linux in a brilliant form. But I have chosen this version because it comes ready to use. Many distros require you to install a lot of essentials, but with this you are ready to experience Linux first hand within minutes of installation. Once you are more adept with Linux you can venture into some of the more elegant and more difficult distros (such as: Fedora, Ubuntu Studio and openSUSE). But this as a starter package has to get 8/10.

Miro - Simple ideas are rarely executed very well at all. Miro however is a very simple idea that has been executed with such brilliance. A video and audio player that allows you to subscribe to and download podcasts and other online media. It also allows you to view your own videos that are on your computer.


Rating:  
A beautifully designed piece of software is geared more towards watching videos, and with the sidebar you can quickly switch between videos on your computer, and the central pane allows you to search for and view videos. The reason for placing this in above other media players is the ability to download video from torrents and watch them all within Miro and that id you have to exit a video suddenly it allows you to go back to that video at the same place. 8.9/10

Special mentions - Freeware list, Notepad ++, GIMP and Browser list

Education

Wolfram Alpha - Need some help with your homework? Not certain how to answer a certain problem? Want to know how the population of Papa New Guinea ranks in the world? You can find all of that and much, much more here. The amount of data that Wolfram Alpha can show you about almost anything is simply staggering.


Rating:  
I do not think there is a single computational engine that rivals Wolfram Alpha, the speed it can compute what you enter rivals Google for impressiveness. It can: intergrate; differentiate; give pi to over 10,000 digits; give e to over 10,000 digits; multiply matrices; calculate volumes and areas; plot graphs; factorise; show roots and much more. This is one of the greatest technological advances of the past year. 10/10

Ask a Mathematician - A blog where a mathematician and a physicist openly answer any mathematical questions you have, the posts are very informative and interesting. Incredibly advanced topics are explained in a very manageable and clear way.


Rating:  
I've not seen a single mathematical blog that regularly and openly answers any questions you have as good as this one. An easy to look at and simple design doesn't take anything away from the content, which is of an incredibly high calibre. Questions that are emailed are answered incredibly quickly which ensures you always come back to the blog for more. 8.5/10

Special mentions - Wiki Answers, Yahoo Answers and Maths Revision

Miscellaneous


Grooveshark - A website that allows you to search and listen to music all for free on the internet. You can search for artists, albums and song title. This all adds to an amazing website.

Rating:
The design is pretty much perfect, everything is clearly laid out and is very easy to get to grips with. It allows you to discover music where you already know the artist or album name, but you can also go into Groovesharks different stations that automatically directs you to a song of that genre. The ability to upload your own music means the database is constantly growing. 8/10

last.fm - Very similar to Grooveshark in that it's a music streaming website that allows you to discover new music that the intelligent system thinks you'd like.

Rating:
Yet again a faultless design, that is easy on the eye and very easy to navigate. Where these two sites differ is last.fm starts creating a profile of the types of music you like and randomly chooses music it thinks you'll enjoy, if you don't you can skip the song and it then develops more of an understanding for what you like. I'm never without this open because you've set a profile up you really don't have to do much work at all. 9.5/10

StumbleUpon - You're very unlikely to have never seen this website before. If you haven't prepare to lose hours and hours of your life. Seriously, this is like crack. You simply tick boxes of subjects you think would interest you (and there is a lot of them) press stumble and you're on your way to having no life outside your computer.

Rating:
If you install the recommended StumbleUpon add-on from FireFox then it is simply a button press away from being thrown onto a site it's sure you'll love. If you like the site, press the thumbs up button, if not thumbs down. This helps to develop a personality that StumbleUpon can then use to find more websites it thinks you'll like. A resounding 10/10, that really doesn't justify the awesomeness of stumble upon.

Special mentions - io9, Keyboard Shortcuts and YouTube