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Our first experience of numbers is at quite an early age when we learn to count. This set of counting numbers is called the natural numbers. In our early arithmetic lessons we learn to add and to multiply and we find that in each case if we take two natural numbers we obtain another natural number. However when we meet the inverse processes, subtraction and division respectively, we do not always have a natural number as a result. This leads us to extend our number system to introduce the negative numbers to obtain the set of integers and fractions to obtain the set of rational numbers. Pythagoras’ theorem then leads to a right-angled triangle whose hypotenuse is of length √2. We see that √2 is not one of the rational numbers and so we extend further to develop the irrational numbers.

 

 

SECONDARY: decimal expansions; terminating, recurring and non-recurring; transcendental numbers.

 

SIXTH FORM: plus continued fractions, Liouville numbers.

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