Numbers Behaving Badly?
When a person is irrational they do things out of the ordinary without warning. An example would be a teacher who normally leaves school in a clapped out old Fiesta one day out of the blue leaves the car park by unicycle wearing only a pile of dog-eared maths books to hide their modesty.
Irrational numbers are not quite so unpredictable, however they are not totally predictable either!
The most famous example
of an irrational
number is pi (the symbol for which is the Greek letter 
).
You'll hear more about pi when looking at circles, for the time being it's enough
to know that pi is approximately equal to 3.14159265358979……..going on and on
and on. The numbers after the decimal place are never repeated, hence it's impossible
to know exactly the value of pi. This hasn't stopped people trying to find out
- pi has been calculated to more that 2,000 million decimal places using powerful
computers and still no repeating pattern has been seen!
Another example of an irrational number is the square root of 2 which is approximately equal to 1.414213…..
This doesn't mean that irrational numbers are no use - to make use of them we simply approximate - it is normally sufficient to approximate pi to 3.14159 (at least in this world!). Quite often pi is approximated by the rational number 22/7 see below.
Again we see opposites coming into the scene…..the opposite of an irrational number is a rational number.
Rational numbers are numbers which can be expressed as a whole number (can be called an integer also) divided by another whole number.
Hence 0.625 is a rational number as it is equivalent to 5/8.
TYPES OF NUMBERS
Numbers are given particular names as we have seen above. As these names are popular with examiners it's a good move to know what they mean:
INTEGERS are …..-3, -2, -1, 0, 1, 2, 3, 4
RATIONAL NUMBERS are e.g. 3/4, 22/6 etc. (can be expressed as a fraction)
IRRATIONAL NUMBERS are e.g. pi, square root of 2 etc. (can't be expressed as a fraction)
PRIME NUMBERS are strange numbers as they can only be divided by themselves or 1. There are lots of low valued prime numbers 2, 3, 5, 7, 11, 13, 17, 19, 23…etc.
A formula has never been discovered which would enable you to predict the next prime number in a sequence of primes.
Another curious thing about prime numbers is that ALL integers are either prime numbers or can be made by multiplying prime numbers together:
Identifying the prime numbers which were multiplied together to give an integer we use the grand title "decomposition into prime factors". Have a look at the following example to see what this means:
Take the integer 264
First divide by 2 (the easiest way to start breaking down a number into other numbers which can be multiplied together to make the number we are breaking down). The numbers we multiply together are called factors - if you factorise all you do is break a number down into other numbers which can be multiplied together to give that number. This gives us 132.
Now divide by 2 again - this gives us 66.
And divide by 2 again - to give 33
Now divide by 3 to give 11.
Have you noticed that the numbers in purple are all prime numbers - have you also noticed that 2 x 2 x 2 x 3 x 11 = 264?
And you can do the same sort of thing for ANY integer (that isn't a prime number already) - spooky isn't it?
