Month: July 2014

Quadratic Equations and the Quadratic Formula

Okay, first things first.  Since I’m aiming this blog to be accessible to all, this entry is going to be more aimed for those who haven’t studied Maths in a while (or anyone keen enough to be reading this whilst doing GCSEs!).  So if you don’t fall into those categories, feel free to gloss over this one!

 

Last time, I briefly mentioned things called “polynomials”; an equation in terms of powers of x.  A quadratic equation is a polynomial that goes up to the x^2 term.  So, a quadratic equation looks like that:  ax^2 + bx + c, where a, b and c are just numbers.  You may ask yourself:  “What type of number?”, given that I explained the different types of numbers earlier.  Let’s just say that they’re a “real number”, so basically any number.  And, unless I say otherwise, we’ll always assume they are.  Simples.

 

One important thing with maths is being able to solve things.  So, we want to be able to write ‘what x is’ when ax^2 + bx + c = 0.  This is where the ‘quadratic formula’ comes in.  The quadratic formula gives you solutions to any quadratic equation, just so long as you know what the numbers a, b and c are.  It’s actually quite simple to get to this formula, just from the equation.  However, since I want to avoid actually deriving equations and such, I’m not going to go over it here (plus, a WordPress blog isn’t the most suitable place!).   If you’re interested, I’d recommend this link – http://www.mathsisfun.com/algebra/quadratic-equation-derivation.html

 

So, we can now solve the quadratic equation.  All you have to do is plug in some numbers into a formula (and you can even use a calculator to do that, woo!).  However, such formula aren’t always possible.  Indeed, whilst there is a formula for solving cubic equations (polynomials up to a third power, so of the form ax^3 + bx^2 + cx + d= 0), there does not exist any formula for any other polynomials.  Even the ‘cubic formula’ is a very long and complicated (if you’re interesting, just search ‘Cubic formula’ online).  

 

As you can tell, such formula are very rare, making them powerful tools in maths.  So, we can’t rely on there always being a special formula; meaning maths is about learning the different methods and ways of solving things, trying these things and hoping they work.  And if they don’t, then you just try another one.  But let’s not worry about and just take comfort in knowing the quadratic formula is always there to save us…

Magical Modular Maths

Here, I’m going to talk about something called ‘modular arithmetic’.  By ‘arithmetic’, I mean things like adding, subtracting, multiplying and dividing.  It’s quite a weird thing really; when you first think about it, it’s almost like reinventing the wheel, making a mockery of other arithmetic you know.  However, it really is quite useful.

 

Before I go into properly though, let’s look at an example.  Funnily enough, you almost certainly use modular arithmetic in your day to day lives, usually without even thinking.  When you look at a digital clock, it usually displays the time using the 24-hour system (or, at the very least, has the option to do so).  So, you have 01:00 for 1am, 13:00 for 1pm of the same day.  So, a 24-hour clock can show 24 different hours.  However, on your ‘old-fashioned’ clock with hands, you only show 12 different hours.  This is modular arithmetic in action!

 

Let’s look at this in a bit more detail.  When you look at a clock and it reads 14:00, you recognise that it is 2pm.  At 17:00, you recognise it’s 5pm.  As you probably figured out, whenever the hour of the 24-hour clock goes past 12, all you have to do is subtract 12 from the hour to figure out what the hour is in the traditional 12-hour system.  This is us working ‘modulo 12’.

 

Modulo 12 sounds a bit weird and scary, but it’s not.  All we mean when we say ‘modulo 12’ is you want your number to be between 0 and 11.  Similar, when working ‘modulo 10’, we want numbers to be between 0 and 9.  In general, a number ‘modulo n’ is a number between 0 and ‘n-1’.  To work out what a number modulo n is is very simple:

-If the number is bigger or equal to n, you subtract n from that number until you are between 0 and n-1.

-If the small is smaller than 0, you add n to the number until you are between 0 and n-1.

 

So, when looking at a clock, you try to work out the hour modulo 12.  If it says ’04:00′, you know that 4 is between 0 and 11, so you don’t have to do anything and it is 4am.  If it says ’22:00′, you know that’s bigger than or equal to 12, so you take 12 away from it.  Since 22-12=10, and 10 is between 0 and 11, you’re now okay and know the time is 10pm.

 

But what about adding?  Say the time is now 10am, and you want to know what time it is in 5 hours time.  Simple.  You add the numbers first, then find the result modulo 12.  So, 10+5=15.  You now subtract 12, and get 3.  So, it’ll be 3pm.  Simple!

 

The clock is a very simple example, and everyone is used to it.  However, it’s no different to anything else; you can work with modulo any whole number bigger than zero.  You can write down, for example, 66 modulo 10 if you want.  Keep subtracting 10 from 66 until you get a number between 0 and 9.  You can even do it with really large numbers, like figuring out what 9351 modulo 123 is (though I’d recommend spending your time doing better things…).

 

So really, modular arithmetic isn’t anything difficult; it’s just doing usual arithmetic, then doing some more adding or subtracting.  Already, you can sorta see it has uses (telling the time), but does it have more?  Well, yes!  It crops a lot, usually because it makes things easier to look at (so, rather than deal with really large numbers, you can look at them modulo a small number; big numbers are scary).  So this is not all something with direct applications, but also something that helps with much more difficult maths.

 

My ‘main problem’ for my research project involves looking at equations modulo a number, but I’ll do more into that another time…

Why the Square Root of 2 is Irrational

Okay, this is a bit of a sidetrack to the overall goal of this blog, but it was a suggestion of something for me to do.  Plus, it’s quite a cool little thing to talk about.  In the previous entry about different ‘types’ of numbers, I said how \sqrt2 was irrational, which means we cannot write it like a fraction \frac{a}{b}, where a and b are whole numbers.  But how do we know this?  Can we be sure that this is the case?  Well yes, and I’m going to explain how.

 

In maths, you always like to be sure about things.  You can’t guess and say ‘Well this is probably true’; you need to be 100% certain.  There’s different methods of checking statements in maths, and I’m going to talk about one of these now, which we can use to show \sqrt2 is irrational.  What we do is take a statement which we think is false.  Then, we assume that it is true.  We then look at what this means, and hope that it contradicts something we already know to be true.  Unsurprisingly, we call this a ‘contradiction method’.  It seems a bit weird, but in the world of maths, it’s actually quite a useful trick.

 

Let’s look at a non-mathsy examples.  Someone, let’s call him Elmer, claims that ‘All rabbits have three eyes’.  Everyone else thinks that this is false, but Elmer is having none of this, so we need to show he’s wrong.  So, let’s assume that Elmer is correct, and all rabbits do have three eyes.  But what does this mean in the big picture?  Well, if we find a random rabbit, anywhere in the world, it will have three eyes.  So someone finds a random pet rabbit called Bugs, which only has two eyes.  This contradicts us assuming ‘all rabbits have three eyes’, which means it cannot possibly be true.  So Elmer is wrong.  He went a bit loony over it and hunted rabbits for the rest of his days.

 

So, let’s go to the big problem – showing \sqrt2 is irrational.  Just as a bit of a warning, there’ll be a bit of potentially scary maths here!  Since we think this is true, we take the opposite of this and assume that that is true.  So, we will assume that ‘\sqrt2 is rational’.  This means that \sqrt2 = \frac{a}{b}, with a and b being whole numbers..  We now square both sides, giving us:  \frac{a^2}{b^2} = 2.  Next, we multiply both sides by b^2, so we get the equation 2b^2 = a^2.  We haven’t done anything particularly ground breaking here, just a bit of simple rearranging.

 

Now we use a little subtle point.  When we write a fraction, we like to write it in ‘lowest terms’.  This means a and b can be divided by the same number, we should do this, since it gives us the same number anyway.  So, for example, we don’t write \frac{a}{b} = \frac{2}{4}, we write \frac{a}{b} = \frac{1}{2}.  Instead of \frac{a}{b} = \frac{4}{6}, we write \frac{a}{b} = \frac{2}{3}.  And so on.  One thing this means is that a and b both can’t be even (since if they were, they’d both be divisible by 2).  Bare this in mind.

 

We know that a^2 is even, since it is a number multiplied by 2.  This means that a is also even (if you don’t believe me on that one, check for yourself!).  Since a is even, this means we can write it as 2 multiplied by a different number, say a=2c.  Now we stick this into 2b^2 = a^2, getting 2b^2 = (2c)^2  So, 2b^2 = 2c \times 2c = 4c^2.  Divide by 2 gives us b^2 = 2c^2.  This means that b^2 is even and so is b.

 

This means that a and b are both even, which we know cannot be the case!  So we have contradicted the statement that ‘\sqrt2 is rational’, which mean that ‘\sqrt2 is irrational’.  This is what we wanted; yay!

 

If you didn’t follow all of that, don’t worry too much about it.  The rest of this blog won’t really focus on this, nor will it contain as many equations.  Also, it’s difficult to follow things like this if you don’t write it down yourself.  But if you did follow it, give yourself a pat on the back.

 

 

 

Different Types of Numbers

In the last entry, I mentioned things called ‘algebraic numbers’.  But what’s the difference between an ‘algebraic number’ and a ‘number’?  Well, that’s the thing – there’s different ‘types’ of numbers!  Let me explain…

 

When you’re first taught Maths in primary school (or nursery or whenever), the first thing you’re taught is how to count and do simple adding and subtracting.  You learn numbers like 1, 2, 3, 4 and so on.  You also learn about ‘nothing’; ie:  0.  All whole numbers which are not less than zero are called ‘natural numbers’.  In other words, the numbers 0, 1, 2, 3, 4, 5… forever on.  So even 284844929103 is a natural number.  

 

But then, as you carry on through school, you come across a problem.  You can do things like ‘3-1=2’, ‘2-1=1’, ‘5-3=2’, etc.  But what happens if you want to do ‘1-2’ or ‘2-4’?  You can try anything and everything, but you’re not going to get a natural number when you calculate them.  So, there’s a need for “negative natural numbers”; -1, -2, -3… – an exact mirror copy of the natural numbers, but with a minus number in front.  We call these ‘whole numbers’ or ‘integers’.

 

But again, you quickly find in school that even whole numbers aren’t good enough for you.  There’s times you need a number that’s “bigger than 0, but smaller than 1”.  Think of a pie for example.  You have 1 pie and 3 people, how much pie do they get?  You certainly can’t give them ‘1 pie’ or ‘3 pies’ or ‘-1 pies’; you need numbers which ‘aren’t whole’.  These numbers are called ‘rational numbers’, and are ones that can be written in a fraction, so \frac{1}{3}, \frac{2}{17}, \frac{-5}{3} and so forth.  Note that all whole numbers are rational!

 

That’s all well and good you may think.  Or is it?  Can all numbers be written as fractions?  Well, no.  One important example would be \pi (pi).  This number cannot be written like a fraction (\frac{\pi}{1} doesn’t count I’m afraid!).  \pi is of course an important number which links the circumference of a circle to its diameter, so we need this to exist.  So, numbers which cannot be written in a fraction are called ‘irrational numbers’, surprisingly enough!  

Another way to look at this is to look at the equation x^2 - 2 = 0.  We want to know what x is.  Mathematicians always want to be able to solve equations whenever possible.  We can ‘rearrange’ the equation and see that x = \sqrt2.  However, \sqrt2 is not rational.  So, to be able to know what x is, there needs to numbers which aren’t rational, which is why ‘irrational numbers’ need to exist.

 

Most people will be familiar with all of this (though perhaps the thought process is a bit different).  We call the collection of all rational and irrational numbers the ‘real numbers’.  At first, this seems like a strange name.  However, let’s look at the equation x^2 = -1.  You would then try to ‘square root both sides’, but you were probably taught in secondary school “you can’t square root a minus number”.  Well, that’s a bit of a lie.  

 

As I said earlier, mathematicians want to be able to solve equations wherever possible.  So, if some equations can’t be solved using ‘real numbers’, we may as well create ‘imaginary numbers’ to solve them.  It’s a bit of a cheat in a way, but no more of a cheat than what was done earlier with ‘making’ irrational numbers.  So, we say \sqrt-1 = i, where ‘i’ stands for imaginary.  This gives us solutions to pretty much all equations you can think of!  

 

But what about “algebraic numbers”?  Where do they fit into all of this?  Well, “algebraic numbers” are numbers which solve any equation which can be written in powers of x (an equation like this is called a polynomial).  So, an algebraic number could be any type of number; it could be a whole number, rational, irrational, or complex.  Indeed, we’ve seen that with all the above examples.  However, not ALL numbers are algebraic numbers.  For example, \pi is not algebraic.  It’s quite difficult to actually prove this, so just take my word for it, yeah?  

 

So that’s what algebraic numbers are.  Bringing this back to my project for a minute, algebraic number theory is the in-depth study of these numbers.  As time goes on, I’ll explain exactly what algebraic numbers have to do with my project, but that’s still a fair while away yet.  Instead, we have to look to things away from algebraic numbers before we can understand what is going on!

The Big Idea

So, in the introduction, I avoided giving any reference to what my project is actually called, or in general what it’s about.  I’ll change that here.  Kind of.

Broadly speaking, I’ll be researching a topic in ‘algebraic number theory’.  I expect that will mean very little to most people reading this at the moment, so let me explain.  Algebraic number theory (ANT) is a branch of number theory (shocking!).  Number theory itself is the in-depth study of integers (whole numbers) and what interesting properties they have.  A particular collection of integers that is of interest, for example, are prime numbers (2, 3, 5, 7, 11…); numbers that are divisible by themselves and 1.

Some of number theory is very easy to understand. For a famous example, Pythagoras’ Theorem can be seen to boil down to which whole numbers work for the equation x^2 + y^2 = z^2. However, this doesn’t work when we look at x^3 + y^3 = z^3, x^4 + y^4 = z^4 or indeed any power greater than 2; something known as ‘Fermat’s Last Theorem’. A less-well known number theory statement is Goldbach’s conjecture: Every even number greater than 2 is the sum of two primes (4=2+2, 6=3+3, 8=5+3…). It’s a really simple statement to understand. However, how do you prove these things? Do we know that, say, 79494819480 is the sum of two prime numbers?  It’s not exactly easy to show…

This is what I love about number theory. Even though some of it is very simple, what’s going on ‘behind the scenes’ if you will is very complicated, and even the easiest and most obvious of things requires lots of difficult maths to prove, so that we can be sure it’s correct!

So, that’s number theory in a nutshell. But what about ANT? Well, that’s a bit more complicated to explain. Earlier, I said that number theory is about whole numbers? ANT decides that’s a bit boring, so decides to study things called algebraic numbers. I’ll explain what they are properly later. However, for the time-being, it’s worth saying that all whole numbers are algebraic numbers, but not all algebraic numbers are whole numbers. So, for example, 2 is a whole number and an algebraic one, but \sqrt2 is an algebraic number, but not a whole one (again, I’ll explain why later). 

Basically, ANT broadens the numbers we are interested in. Why? By doing this, we can get answers to number theory problems which we may not have gotten otherwise. It’s sorta like you’re driving along a road and then all of a sudden large pile of bricks is put in your way. You could either try and move all the bricks so you can take the obvious route, or you can just leave them alone, and take a detour that adds 10 minutes extra driving time. It may seem weird at first, but it makes sense when you look at the small details.

And that’s algebraic number theory. As I’ve said, my project is an algebraic number theory one. However, my project isn’t about trying to get an answer in number theory, it’s simply ANT for the sake of ANT. It’s not one about trying to get past the ton of bricks, it’s one about enjoying the detour you have to take. What exactly I’m doing, I’ll be explaining much later.

Hopefully that’s given a ‘big idea’ to what it is I’m doing. I’ll soon be going into the more ‘nitty gritty’ stuff…yay!

Introduction – What’s this all about?

Over the next 6-7 weeks, I’ll be doing a small research project. Of course, if I try and explain what exactly I’m researching, it will go over most people’s heads, which I personally think is a shame, as all maths at this level is really interesting. This is why I’ve decided to set up this blog.

 

Essentially, I’ll try and explain my project in the simplest (and, hopefully, most interesting) way possible. Fingers crossed it will be accessible to most people, even if you hated GCSE Maths! Like most maths, this blog will end up being cumulative in a way; posts will depend on information on from the previous ones. I guess it could be like a story. As such, I’ll try and keep everything relatively short and brief, whilst still capturing the key essence of what I’m talking about.

 

This will include several entries, as there is a lot of stuff to talk about. However, if you stick with it, you’ll hopefully see what degree-level maths is (it’s NOT just loads of long equations and number crunching), why it’s interesting, why my project is interesting and maybe even learn something.

 

I don’t think there’s many blogs with an aim like this, so it should be quite a cool thing for me to do at least. And if it’s successful, then who knows, I might well continue it after my project is finished.

 

Just to make one thing clear, I won’t do any scary maths, showing off loads of complicated examples. In fact, I’ll do as few examples as possible. Instead, the aim here is to simply explain what things are, why they’re important, and give very basic examples (which should be quite easy / obvious to understand).

 

If, at any point, you have questions, comments, see a mistake, whatever, let me know! I don’t bite!