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CBSC ClassX Mathematics Real Numbers

An algorithm is a series of well defined steps which gives a procedure for solving a type of problem.<br>

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CBSC ClassX Mathematics Real Numbers

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  1. MATHS REAL NUMBERS Real Numbers 1. An algorithm is a series of well defined steps which gives a procedure for solving a type of problem. 2. A lemma is a proven statement used for proving another statement. 3. Euclid’s Division Lemma Given positive integers a and b, there exists unique integers q and r satisfying www.topperlearning.com 2

  2. MATHS REAL NUMBERS 4. Euclid’s Division Algorithm This algorithm is a technique to compute the H.C.F. of two given positive integers. According to this algorithm, the HCF of any two positive integers ‘a’ and ‘b’, with a > b, is obtained by following the steps given below: Step 1: Apply Euclid’s division lemma, to ‘a’ and ‘b’, to find q and r, such that a = bq + r, 0 r < b. Step 2: If r = 0, the HCF is b. If r 0, apply Euclid’s division lemma to b and r. Step 3: Continue the process till the remainder is zero. The divisor at this stage will be HCF (a,b). Also, note that HCF (a, b) = HCF (b, r). www.topperlearning.com 3

  3. MATHS REAL NUMBERS Euclid’s Division Algorithm can be summarized as follows: 5. Euclid’s Division Algorithm is stated for only positive integers but it can be extended for all integers except zero, i.e., b ≠ 0. www.topperlearning.com 4

  4. MATHS REAL NUMBERS 6. The numbers which can be represented in the form of ,where and p and q are integers are called rational numbers. 7. Any real number that cannot be expressed in the form of is called an irrational number. 8. Rational and irrational numbers together constitute real numbers. 9. There are more irrational numbers than rational numbers between two consecutive numbers. 10. Terminating fractions are the fractions which leaves remainder 0 on normal division. www.topperlearning.com 5

  5. MATHS REAL NUMBERS 11. Recurringfractions are the fractions which never leave a remainder 0 on normal division. 12. Properties of irrational numbers: i. The sum, difference, product and quotient of two irrational numbers need not always be an irrational number. ii. Negative of an irrational number is an irrational number. iii. Sum of a rational and an irrational number is irrational. iv. Product and quotient of a non-zero rational and irrational number is always irrational. 13. If p is a prime and divides a2, then p divides a, where ‘a’ is a positive integer. www.topperlearning.com 6

  6. MATHS REAL NUMBERS 14. If p is a prime, then is an irrational number. 15. The decimal expansion of rational number is either terminating or non-terminating recurring (repeating). 16. The decimal expansion of an irrational number is non-terminating, non-recurring. 17. If the decimal expansion of rational number terminates, then we can express the number in the form of , where p and q are co prime, and the prime factorization of q is of the form 2n5m, where n and m are non negative integers. www.topperlearning.com 7

  7. MATHS REAL NUMBERS 18. If x =is a rational number, such that the prime factorization of q is of the form 2n5m, where n, m are non- negative integers. Then, x has a decimal expansion which terminates. a. If the denominator of a rational number is of the form 2n5m, then it will terminate after n places if n >m or after m places if m > n. 19. Let x =be a rational number, such that the prime factorization of q is not of the form 2n5m, where n, m are non-negative integers. Then, x has a decimal expansion which is non-terminating repeating. 20. A number ends with the digit zero if and only if it has 2 and 5 as two of its prime factors. www.topperlearning.com 8

  8. MATHS REAL NUMBERS 21. 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. 22. The highest common factor (HCF) of two numbers is the smallest power of each common prime factor in the number. www.topperlearning.com 9

  9. MATHS REAL NUMBERS 23. The lowest common multiple (LCM) of two numbers is the product of the greatest power of each prime factor, involved in the numbers. 24. Relationship between HCF and LCM of two numbers: If a and b are two numbers, then: HCF(a, b)  LCM (a, b) = a  b 25. Relationship between HCF and LCM of three numbers: www.topperlearning.com 10

  10. MATHS REAL NUMBERS Important Links View Past session Syllabus Download www.topperlearning.com 11

  11. MATHS REAL NUMBERS Watch Video Lesson Buy class 10 th pack Visit us on: http://www.topperlearning.com www.topperlearning.com 12

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