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REAL NUMBERS REAL NUMBERS The real numbers include all the rational numbers (p/q form where q≠0), and all the irrational numbers, such as √3 and π. Real numbers represent points on an infinitely long line called the number line or real number line. Here N represents Natural numbers, Z/I represent Integers, Q represents Rational numbers and R represents Real numbers Real numbers measure continuous variable quantities. For example i. ii. iii. The temperature of Moscow at a given time The speed of a running train. The amount of water in a bottle. REAL NUMBER LINE Real number line represents a straight line with every point on it pointing to a unique real number. Integers are marked on the line at equal distances. Between every two integer lies infinite real numbers (fractions, rational numbers, irrational numbers, decimals) Points to remember i. There are infinite many real numbers between to real numbers or integers ii. iii. iv. v. vi. Integers on number line are equally spaced Real numbers include positive integers, negative integers and zero Infinity is not a real number; it is just a notation to represent uncountable items All fractions and decimals (terminating or non-terminating) are real numbers. There is a unique real number for every number on the number line and for every real number there is a unique point on the number line EUCLID’S DIVISION LEMMA For two positive integers a and b, there exist unique integers q and r satisfying a = bq + r, 0 ≤ r < b. e.g. 495 = 49 x 10 + 5 FUNDAMENTAL THEOREM OF ARITHMETICS Every composite number can be factorized as a product of prime numbers and this factorization is unique e.g. 20580 = 2 x 2 x 3 x 5 x 7 x 7 x 7. We can also say that prime factorization of a natural number is unique. Points to remember i. ii. iii. iv. If p is a prime and p divides x2, then p divides x, where x is a positive integer. Let ‘a’ be a rational number whose decimal expansion terminates, then we can express ‘a’ in the form p/q, where p and q have no common factor except 1(co-prime), and the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers. Let a = p/q be a rational number, such that the prime factorisation of q is of the form 2n5m, where n, m are non-negative integers. Then the decimal expansion ‘a’ terminates. Let a = p/q be a rational number, such that the prime factorisation of q is not of the form 2n 5m, where n, m are nonnegative integers. Then ‘a’ has a decimal expansion which is nonterminating repeating (recurring) PROPERTIES OF REAL NUMBERS 1. Commutative Property of Multiplication a • b = b • a 2. Commutative Property of Addition a + b = b + a 3. Associative Property of Addition a + (b + c) = (a + b) + c 4. 5. 6. 7. 8. 9. Associative Property of Multiplication a • (b • c) = (a • b) • c Additive Identity Property a + 0 = a Multiplicative identity Property a • 1 = a = 1 • a Additive Inverse a + (-a) = 0 Multiplicative Inverse a * 1/a = 1 (a≠0) Distributive Law a • (b + c) = ab + ac