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CHAPTER 01, REAL NUMBERS, , , , e Natural numbers: Counting numbers are, called Natural numbers. These numbers, are denoted by N = {1, 2, 3, ......... }, , e Whole numbers: The collection of natural, numbers along with 0 is the collection of, Whole number and is denoted by W., , e Integers: The collection of natural numbers,, their negatives along with the number zero, are called Integers. This collection is, denoted by Z., , e Rational number: The numbers, which are, obtained by dividing two integers, are, called Rational numbers. Division by zero, is not defined., , e Coprime: If HCF of two numbers is 1, then, the two numbers area called relatively, prime or coprime.
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>, The Fundamental Theorem of Arithmetic :, , Every composite number can be expressed, (factorized) as a product of primes and this, , factorization is unique, apart from the order in, which the prime factors occur., , Ez: 24=2x2x2x3=3x2x2x9, , let r= 7? 470 to be a rational number,, such that the prime factorization of βqβ is of the, form 2m+5n, where m, n are non-negative, , integers. Then x has a decimal expansion which, is terminating., , A vet L= =" q #0 be a rational number,, , such that the prime factorizationof q is not of, the form 2m+5n, where m, n are non-negative, integers. Then x has a decimal expansion which, is non-terminating repeating., , 5 VP is irrational, which p is a prime. A number, is called irrational if it cannot be written in the, , form Γ© where p and q are integers and, , q #0., , 6 If a and b are two positive integers, then, HCF(a, b) x LCM(a, b) =a xb
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i.e., (HCF x LCM) of two intergers = Product of, intergers., , 7. A rational number which when expressed in, the lowest term has factors 2 or 5 in the, denominator can be written as terminating, decimal otherwise a non-terminating recurring, , decimal. In other words, if the rational number , is, such that the prime factorization of b is of, form 2β’β.5β, where m and n are natural, , a, , numbers, then 7 has a terminating decimal, , expansion., , 8. We conclude that every rational number can, be represented in the form of terminating or nonterminating recurring decimal.
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5. Check whether 6" can end with the digit 0 for any, natural number n., , Sol., , If any number ends with the digit 0, it should be, divisible by 10 or in other words, it will also be divisible, by 2and5as10=2x5, , Prime factorisation of 6" = (2 x3)", , It can be observed that 5 is not in the prime, factorisation of 6"., , Hence, for any value of n, 6" will not be divisible by 5., , Therefore, 6" cannot end with the digit 0 for any natural, number n.
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6. Explain why 7x 11x13+13and7x6x5x4x3x, 2 x 1 +5 are composite numbers., , Sol., , Numbers are of two types - prime and composite., Prime numbers can be divided by 1 and only itself,, whereas composite numbers have factors other than 1, and itself., , It can be observed that, , 7*11* 13413 =13x(7x11+1), , = 13 x (77 + 1)= 13 x 78= 13 x13 x 6, , The given expression has 6 and 13 as its factors., Therefore, it is a composite number., 7X6xX5x4x3x2x14+5, =5x(7x6x4x3x2x1+1), , = 5x (1008 + 1)= 5 x1009, , 1009 cannot be factorized further, , Therefore, the given expression has 5 and 1009 as its, factors., , Hence, it is a composite number.
