Describe the structure of the Real Number System, defining each type of number which it comprises and making clear the relationship between them.

The first numbers that we are introduced to from an early age are 1, 2, 3, 4 etc.

These are called the 'Natural numbers', and can be placed on a number line from 1

to infinity. The natural numbers, in order, are always 1 whole number larger or

smaller than the next.

When we add or subtract the natural numbers the answer is always a natural

number. If we use subtraction or division however, we would without any other

system, not always be able to obtain an answer within the natural number system.

The sum 8 minus 10 for example would be impossible therefore a new number

system is needed. Suggate (1998, p.40) uses temperatures below freezing as an

example. In this instance we record how many degrees below O°c it is by counting

backwards from 0, to the left, using the 'negative' numbers. The integers are all

positive and negative whole numbers including 0 but the positive integers are also

natural numbers.

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When taking into account these numbers it is not always possible to calculate while

keeping the sum within the integer system. Therefore another system is needed. If

we take for example 1 cake which is whole and divide it into 4 parts. Each piece is

considered ¼ of the original whole cake. Therefore there are 4 ¼ pieces. These

fractions or 'rational numbers' tell us this by the bottom numbers - the denominator.

The number on top of the fraction is the numerator and tells us about how many

parts we are dealing with. With fractions there are different ways to write the same

amount, e.g. 2/4 is equal to ½ . However we can also write fractions as decimals.

Therefore ½ , half a whole number, can be expressed as 0.50 ( a half of 1)...