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# For practical mensuration the whole range of h aving

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Unformatted text preview: our assumption that S2 is rational is wrong. We say S2 is irrational. The term IRRATIONAL literally means “ having no ratio ”. For practical mensuration the whole range of h aving the rationals is more than sufficient. But from the theoretical point of view this is not enough. For example, we need to define the length of the diagonal of a square of side of unit length. In f act, between any two r a tional n umber s there ar e infinitely m a n y irr ir rational numbers. So we cannot say that the set of rationals Q is COMPLETE. We cannot say that each and every point on the number line is some rational number. The set of rational numbers Q i s DENSE but not COMPLETE. 10 number Ir r a tional n umber s: Look at the number line again and consider a portion of it, say from 0 to 1. ...... − 3 0 1 2 3 Try to visualize it as a collection of infinitely many points, so many that we cannot even begin to ENUMERATE them. Infinitely many points represent r ational number s and infinitely many points represent ir r a tional n umbers. ir 1. − 2 − 1 Between any two irrational numbers there are INFINITELY many irrational numbers. The numbers: S2, S5, 3SS3 + S2, S3S5 + S7 and many other expressions involving rational numbers under the radical sign S a re irrational. These irrational numbers are said to be expressed in terms of radicals. The decimals help us classify the rational and irrational numbers. Rational numbers are represented by TERMINATING DECIMALS, 1 e.g. − = 0.25 4 1 or INFINITE REPEATING DECIMALS, e.g. − = 0.333 . . . 3 I r r a t i o n a l n u m b e r s a r e r epr esented by NON- TERMINATING NONREPEATING DECIMALS. Examples: S 2 = 1 .41421356237 . . . e = 2 .718281828459045235360 . . . π = 3 .14159265358979323846264338327950 . . . 11 2. Are the irrationals COUNTABLE ? Without giving a formal proof let us try to get an intuitive feel for how many ir r ational irr ir numbers there could be as campared to the COUNTABLE set of rationals Q . How many r ational numbers can you create with TERMINATING (finitely many digits) decimal expansion? Inf...
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