'Arithmetic of whole numbers or fractions is possible. But how about numbers like pi? How can we cope with irrationality?'
|Module code and title:
|M203, Introduction to pure mathematics
|M203; 01A; 1987
|First transmission date:
|Restrictions on use:
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|David Brannan; Phil Rippon
|BBC Open University
|Decimals; Distributive law.; Irrational; Least upper bound; Lower bound; Rational; Real numbers; Sequences; Series; Surds
|Graph theory; Irrational numbers; Mathematics; Numbers, Real
|Understanding what is meant by an integer is easy enough. Even a fraction is a relatively easy concept. But how can we handle numbers like root 2, or pi or e and how can we do arithmetic on such numbers and work out the answer to sums like root 2 times pi? One way is to represent all real numbers by decimals, and this is the main theme of the programme. The programme starts by reviewing natural numbers, as represented by notches on tally sticks. The natural numbers are then extended to include all the integers, demonstrated by temperatures on a thermometer, and it is shown by an animation that the result "(-1) times (-1) equals +1" is a direct consequence of the distributive law. Extending the number system further, to include rational numbers, is illustrated by considering fractions of a cake; addition and subtraction of fractions is still straightforward enough. Integers and fractions can be represented by a number line which shows the ordering of the rationals, and this property can be proved by representing rationals by decimals, which are either terminating or recurring. Rationals can also be thought of as the sum of the infinite series represented by the decimal. But what about numbers like root 2? It's clearly the length of an isosceles right angled triangle of side 1 . But where does root 2 go on the number line? And how can it be used in arithmetic? The programme shows in an animation that, by reduction ad absurdum, root 2 is not rational, but can be represented by an infinite non-recurring decimal and that by representing all numbers, whether rational or irrational by a decimal expansion, we can readily order the real numbers, and perform all the usual arithmetic operations on them. Two further examples of irrational numbers are now introduced, pi as the area of a unit circle, and a root of the equation x5 + x - 1 = 0, and these numbers are then placed on the number line. So how is root 2 x pi to be calculated using infinite decimals? The solution uses the idea of upper and lower bounds, and the least upper bound property of a monotonic increasing sequence. By truncating the decimal expansions for root 2 and pi, and calculating their products, we can get a sequence of approximations which are bounded above, and so must converge to the answer, root 2 pi. The programme next shows that this idea can be used with the rational numbers 1/3 and 3 to get the recurring decimal 0.999...., which must equal 1! Finally the programme shows that as 0.001 recurring all squared is a rational, it's square must also be rational; but with a recurring block 2,997 digits long! The programme makes extensive use of animations to explain the properties of real numbers, whether rational or irrational, and so would be of interest to any students and teachers who are studying the basis of the real number system. (This programme was re-edited in 1995 as FOUM484K)
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