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 , ,  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).  This number cannot be written like a fraction ( doesn’t count I’m afraid!).   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 .  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 .  However,  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 .  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 , 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, 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!