Numbers, Sets and Groups
To count, all you need are the Natural Numbers,
1, 2, 3, 4, 5, 6, ….
And whenever you add together a pair of natural numbers, you get another natural number,
e.g. 7+9=16
The natural numbers are also called the positive integers (an integer is a whole number)
If you subtract natural numbers you get another natural number, e.g. 12-5=7
But the number may not always be positive,
e.g. 5-12=-7
and so we need the negative integers, together with the very important integer, 0 to be able to do all the arithmetic we ever need..... well up to a point!
In all of the above integers have been added to give another integer. Subtracting integers is the same as adding a negative integer. And we have 0, which added to any integer has no effect, e.g, 4+0=4
So the set of integers form a GROUP when added.
Which is all O.K. until you want to multiply,
multiplying two integers makes another integer,
but dividing might need an answer to 5÷7.
The answer to this is not an
integer, so the number system has to be extended to include
fractions such as 
So now we can multiply and divide anything, that is we have the GROUP of Rational Numbers with multiplication.
Rational numbers are all the fractions (whether written as fractions or as decimals) and all the integers.
But then you want to multiply a number by itself.
No problem there, e.g. 92=81,
until you want to reverse the process
2.64575131106459059059050161575363926 to 35 decimal
places.
The decimal continues infinitely and as you can see there is no recurring pattern in the figures. It is an example of an IRRATIONAL number.
The square roots of all integers, that are not square numbers, are irrational.
The number 
is another example of an irrational
number.
All these numbers, Rational and Irrational, are REAL NUMBERS
So does that cover all the different sorts of numbers that there are?
Look again at squaring numbers,
32=9 and –32=9,
so 
So what about 
or, for that matter
the square root of any negative number?
The number system needs to be extended again, but the good news is that only one new symbol is needed,
Yes it looks like the letter i,
but that is how its written. So 
And because, -9 = 9 x –1, 
that is 
This is an example of a COMPLEX NUMBER
And now you can do all the adding, subtracting, multiplying, dividing, squaring and rooting that you could ever want!