Page 1 Real Numbers Real Numbers are combination of five different types of numbers as described in detail below. Natural Numbers, commonly referred to as counting numbers i.e. 1, 2, 3, 4, 5, 6, 7, ....etc. Whole Numbers are combination of natural numbers and zero (0) i.e. 0, 1, 2, 3, 4, 5, 6, 7, ....etc. Integers are combination of Natural and Whole Numbers and their negatives. For e.g. …-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7... Continue infinitely in both directions. Rational Numbers are made of ratios of integers and referred to as fractions. Rational numbers take on the general form of a/b where a and b can be any integer except b ? 0. Thus, the Rational numbers include all Integers, Whole numbers and Natural numbers. Irrational Numbers are made of special, unique numbers that cannot be represented as a ratio of Integers. Examples of Irrational numbers include (3.14159265358979...) and the square root of 2 (1.4142135623730950...). Algorithm -An algorithm is a series of well defined steps which gives a procedure for solving a type of problem. Lemma - A lemma is a proven statement used for proving another statement. Euclid’s Division lemma: For any two given positive integers a and b there exist unique integers q and r such that a=bq + r, where 0= r<b. Here a, b, q and r are respectively called as dividend, divisor, quotient and remainder. Euclid’s Division Lemma can be used to find H.C.F of two positive integers. Euclid’s division Algorithm: It is a technique to compute Highest Common Factor(H.C.F) of two given positive integers, consider c and d are two positive integers, with c > d. We use following the steps to find H.C.F of c and d: Step I: Apply Euclid’s division lemma, to c and d, so we find whole numbers, q and r such that c =dq +r, 0=r<d Step II: If r=0, d is the H.C.F of c and d. If r?0 apply division lemma to d and r Page 2 Real Numbers Real Numbers are combination of five different types of numbers as described in detail below. Natural Numbers, commonly referred to as counting numbers i.e. 1, 2, 3, 4, 5, 6, 7, ....etc. Whole Numbers are combination of natural numbers and zero (0) i.e. 0, 1, 2, 3, 4, 5, 6, 7, ....etc. Integers are combination of Natural and Whole Numbers and their negatives. For e.g. …-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7... Continue infinitely in both directions. Rational Numbers are made of ratios of integers and referred to as fractions. Rational numbers take on the general form of a/b where a and b can be any integer except b ? 0. Thus, the Rational numbers include all Integers, Whole numbers and Natural numbers. Irrational Numbers are made of special, unique numbers that cannot be represented as a ratio of Integers. Examples of Irrational numbers include (3.14159265358979...) and the square root of 2 (1.4142135623730950...). Algorithm -An algorithm is a series of well defined steps which gives a procedure for solving a type of problem. Lemma - A lemma is a proven statement used for proving another statement. Euclid’s Division lemma: For any two given positive integers a and b there exist unique integers q and r such that a=bq + r, where 0= r<b. Here a, b, q and r are respectively called as dividend, divisor, quotient and remainder. Euclid’s Division Lemma can be used to find H.C.F of two positive integers. Euclid’s division Algorithm: It is a technique to compute Highest Common Factor(H.C.F) of two given positive integers, consider c and d are two positive integers, with c > d. We use following the steps to find H.C.F of c and d: Step I: Apply Euclid’s division lemma, to c and d, so we find whole numbers, q and r such that c =dq +r, 0=r<d Step II: If r=0, d is the H.C.F of c and d. If r?0 apply division lemma to d and r Real Numbers Step III: Continue the process till the remainder is zero. The divisor at this stage will be the required H.C.F The Fundamental theorem of Arithmetic: Every composite number can be expressed (factorised) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur. Ex. 28 = 2 x 2 x 7 ; 27 = 3 x 3 x 3 Theorem : Sum or difference of a rational and irrational number is irrational. Theorem : The product and quotient of a non-zero rational and irrational number is irrational. Theorem : If p is a prime and p divides a 2 , then p divides “a” where a is a positive integer. Theorem : If p is a prime number then v is an irrational number. Theorem: Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form of where p and q are co-prime and the prime factorisation of q is the form of 2 n .5 m where n, m are non negative integers. Example: 0.7= = × = Theorem: Let x = be a rational number such that the prime factorisation of q is not of the form of 2 n .5 m , where n, m are non negative integers. Then x has a decimal expansion which is non terminating repeating (recurring). Example: = × =1.1666…… Theorem: For any two positive integers p and q, HCF (p, q) x LCM (p, q) = p x q Example: 4 & 6; HCF (4, 6) = 2, LCM (4, 6) = 12; HCF x LCM = 2 x 12 =24 ? p x q = 24 Page 3 Real Numbers Real Numbers are combination of five different types of numbers as described in detail below. Natural Numbers, commonly referred to as counting numbers i.e. 1, 2, 3, 4, 5, 6, 7, ....etc. Whole Numbers are combination of natural numbers and zero (0) i.e. 0, 1, 2, 3, 4, 5, 6, 7, ....etc. Integers are combination of Natural and Whole Numbers and their negatives. For e.g. …-7, -6, -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7... Continue infinitely in both directions. Rational Numbers are made of ratios of integers and referred to as fractions. Rational numbers take on the general form of a/b where a and b can be any integer except b ? 0. Thus, the Rational numbers include all Integers, Whole numbers and Natural numbers. Irrational Numbers are made of special, unique numbers that cannot be represented as a ratio of Integers. Examples of Irrational numbers include (3.14159265358979...) and the square root of 2 (1.4142135623730950...). Algorithm -An algorithm is a series of well defined steps which gives a procedure for solving a type of problem. Lemma - A lemma is a proven statement used for proving another statement. Euclid’s Division lemma: For any two given positive integers a and b there exist unique integers q and r such that a=bq + r, where 0= r<b. Here a, b, q and r are respectively called as dividend, divisor, quotient and remainder. Euclid’s Division Lemma can be used to find H.C.F of two positive integers. Euclid’s division Algorithm: It is a technique to compute Highest Common Factor(H.C.F) of two given positive integers, consider c and d are two positive integers, with c > d. We use following the steps to find H.C.F of c and d: Step I: Apply Euclid’s division lemma, to c and d, so we find whole numbers, q and r such that c =dq +r, 0=r<d Step II: If r=0, d is the H.C.F of c and d. If r?0 apply division lemma to d and r Real Numbers Step III: Continue the process till the remainder is zero. The divisor at this stage will be the required H.C.F The Fundamental theorem of Arithmetic: Every composite number can be expressed (factorised) as a product of primes, and this factorization is unique, apart from the order in which the prime factors occur. Ex. 28 = 2 x 2 x 7 ; 27 = 3 x 3 x 3 Theorem : Sum or difference of a rational and irrational number is irrational. Theorem : The product and quotient of a non-zero rational and irrational number is irrational. Theorem : If p is a prime and p divides a 2 , then p divides “a” where a is a positive integer. Theorem : If p is a prime number then v is an irrational number. Theorem: Let x be a rational number whose decimal expansion terminates. Then x can be expressed in the form of where p and q are co-prime and the prime factorisation of q is the form of 2 n .5 m where n, m are non negative integers. Example: 0.7= = × = Theorem: Let x = be a rational number such that the prime factorisation of q is not of the form of 2 n .5 m , where n, m are non negative integers. Then x has a decimal expansion which is non terminating repeating (recurring). Example: = × =1.1666…… Theorem: For any two positive integers p and q, HCF (p, q) x LCM (p, q) = p x q Example: 4 & 6; HCF (4, 6) = 2, LCM (4, 6) = 12; HCF x LCM = 2 x 12 =24 ? p x q = 24 Real Numbers Types of Decimals: Decimals Terminating Non-terminating Non-terminating repeating Non-terminating non-repeating Rational numbers Irrational numbers Real numbersRead More

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