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Real Numbers

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Real Numbers -Class X CBSE

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    CONTENTS ' Euclid's Division Lemma ' The Fundamental Theorem of Arithmetic ' Revisiting Irrational Numbers ' Revisiting Rational Numbers and Their Decimal Expansions ' Summary
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    Euclid's Division Lemma ' An algorithm is a series of well defined steps which gives a procedure for solving a type of problem. ' A lemma is a proven statement used for proving another statement. THEORM 1 (Euclid's Division Lemma) : Given positive integers a and b, there exist unique integers q and r satisfying WHEREOsr
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    What exactly does the theorem mean? 'Euclid's division algorithm is a technique to compute the Highest Common Factor (HCF) of two given positive integers. Recall that the HCF of two positive integers a and b is the largest positive integer d that divides both a and b.
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    EXAMPLE HCF of the integers 455 and 42. We start with the larger integer, that is, 455. Then we use Euclid's lemma to get 455 = 42 x 10 + 35 Now consider the divisor 42 and the remainder 35, and apply the division lemma to get 42=35x1+7 Now consider the divisor 35 and the remainder 7, and apply the division lemma to get
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    NOTE: 'The flow of the divisor, dividend and remainder! 'The second last remainder, before the remainder becomes zero is the HCF required!
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    To findinq properties of numbers Show that every positive even integer is of the form 24, and that every positive odd integer is of the form 2q + 1, where q is some integer. Solution : Let a be any positive integer and b = 2. Then, by Euclid's algorithm, a = 2q + r, for some integer q >=0, and r = 0 or r = 1, because r < 2. so, a = 2q or 2q + 1. If a is of the form 2q, then a is an even integer. Also, a positive integer can be either even or odd. Therefore, any positive odd integer is of the form 24 + 1.


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