Page 1 Real Numbers Page 2 Real Numbers Real Numbers ( 2, -3, , 2, 0 ) 1 3 Learning Numbers is the ?rst step to enter the world of Mathematics. Below are the di?erent form of numbers Terminating ( -3, 7, ) 1 3 (Repeating) (..., -3, -2, -1, 0, 1, 2, 3, ....) (2.5, 1.275, 0.8912) 1.252525..., 0.777777..., 4.32753275... (-1, -2, -3, -4, -5.....) Irrational numbers (0, 1, 2, 3, 4, 5....) Zero (0) (1, 2, 3, 4,5,.....) (Exactly 2 divisors) (Neither prime nor composite) (Minimum 3 divisors) 1 2 -2 4 0.5 , , 2 , 3 , 18 ( ) Decimal form 3.14159..., 0.247912864..., Non-terminating Non-repeating [ ] [ ] ( ) Rational Numbers Integers Fractions / Decimals Whole Numbers Non Terminating Negative Integers Positive Integers Prime Numbers One Composite Numbers REAL NUMBERS 2 Page 3 Real Numbers Real Numbers ( 2, -3, , 2, 0 ) 1 3 Learning Numbers is the ?rst step to enter the world of Mathematics. Below are the di?erent form of numbers Terminating ( -3, 7, ) 1 3 (Repeating) (..., -3, -2, -1, 0, 1, 2, 3, ....) (2.5, 1.275, 0.8912) 1.252525..., 0.777777..., 4.32753275... (-1, -2, -3, -4, -5.....) Irrational numbers (0, 1, 2, 3, 4, 5....) Zero (0) (1, 2, 3, 4,5,.....) (Exactly 2 divisors) (Neither prime nor composite) (Minimum 3 divisors) 1 2 -2 4 0.5 , , 2 , 3 , 18 ( ) Decimal form 3.14159..., 0.247912864..., Non-terminating Non-repeating [ ] [ ] ( ) Rational Numbers Integers Fractions / Decimals Whole Numbers Non Terminating Negative Integers Positive Integers Prime Numbers One Composite Numbers REAL NUMBERS 2 Methods to nd HCF Euclid’s Division Lemma The HCF of (c, d) where c > d, Apply Euclid’s Division Lemma to c and d. c = dq + r, 0 = r < d (where q & r are whole numbers) If r = 0, d is the HCF of (c, d), if r ? 0, then apply the division lemma to d & r, d > r. Continue this process till the remainder is zero. The divisor at this stage will be HCF of (c, d). Every composite number expressed as a product or factors of prime numbers and this factorization, is unique. We nd LCM and HCF of two positive integers using the fundamental theorem of arithmetic, which is also called Prime Factorization method. For any two positive integers c & d. HCF (c, d) x LCM (c, d) = c x d Fundamental Theorem of Arithmetic Step 1 : Step 3 : Step 2 : c x d LCM (c, d) so, HCF (c, d) = Gain an understanding of an important concept in number theory - The Euclid’s Division Lemma and learn its application in nding out the Highest Common Factor (HCF). 2 2 2 5 2 2 2 5 Product of the common & uncommon prime factors involved in the numbers. It is a smallest common dividend for two or more divisors. Example: LCM of (8, 20) = 2 x 2 x 2 x 5 = 40 Example: HCF of (8, 20) = 2 x 2 = 4 LCM ? ? Product of the common prime factors involved in the numbers. It is a greatest common divisor for two or more dividends. HCF ? ? LCM and HCF are the two important concepts in Algebra. This chart will help you understand how to nd LCM and HCF of two or more than two positive integers. REAL NUMBERS 3 Page 4 Real Numbers Real Numbers ( 2, -3, , 2, 0 ) 1 3 Learning Numbers is the ?rst step to enter the world of Mathematics. Below are the di?erent form of numbers Terminating ( -3, 7, ) 1 3 (Repeating) (..., -3, -2, -1, 0, 1, 2, 3, ....) (2.5, 1.275, 0.8912) 1.252525..., 0.777777..., 4.32753275... (-1, -2, -3, -4, -5.....) Irrational numbers (0, 1, 2, 3, 4, 5....) Zero (0) (1, 2, 3, 4,5,.....) (Exactly 2 divisors) (Neither prime nor composite) (Minimum 3 divisors) 1 2 -2 4 0.5 , , 2 , 3 , 18 ( ) Decimal form 3.14159..., 0.247912864..., Non-terminating Non-repeating [ ] [ ] ( ) Rational Numbers Integers Fractions / Decimals Whole Numbers Non Terminating Negative Integers Positive Integers Prime Numbers One Composite Numbers REAL NUMBERS 2 Methods to nd HCF Euclid’s Division Lemma The HCF of (c, d) where c > d, Apply Euclid’s Division Lemma to c and d. c = dq + r, 0 = r < d (where q & r are whole numbers) If r = 0, d is the HCF of (c, d), if r ? 0, then apply the division lemma to d & r, d > r. Continue this process till the remainder is zero. The divisor at this stage will be HCF of (c, d). Every composite number expressed as a product or factors of prime numbers and this factorization, is unique. We nd LCM and HCF of two positive integers using the fundamental theorem of arithmetic, which is also called Prime Factorization method. For any two positive integers c & d. HCF (c, d) x LCM (c, d) = c x d Fundamental Theorem of Arithmetic Step 1 : Step 3 : Step 2 : c x d LCM (c, d) so, HCF (c, d) = Gain an understanding of an important concept in number theory - The Euclid’s Division Lemma and learn its application in nding out the Highest Common Factor (HCF). 2 2 2 5 2 2 2 5 Product of the common & uncommon prime factors involved in the numbers. It is a smallest common dividend for two or more divisors. Example: LCM of (8, 20) = 2 x 2 x 2 x 5 = 40 Example: HCF of (8, 20) = 2 x 2 = 4 LCM ? ? Product of the common prime factors involved in the numbers. It is a greatest common divisor for two or more dividends. HCF ? ? LCM and HCF are the two important concepts in Algebra. This chart will help you understand how to nd LCM and HCF of two or more than two positive integers. REAL NUMBERS 3 Introduction to Real numbers Finding HCF using Euclid's Division Lemma Scan the QR Codes to watch our free videos PLEASE KEEP IN MIND In a layman’s language, LCM of two or more than two numbers means, the smallest number which is divisible by these two numbers or more than two numbers. HCF of two or more than two numbers means, the highest number which divides these two or more than two numbers. If the denominator of a rational number is in the form of 2 n or 5 m or 2 n 5 m , Then the decimal expansion of that rational number is terminating, or else it is non-terminating. Ex: – terminating and – non-terminating. ? ? ? ? ? If a number “n” is not a perfect square, then vn is an irrational number. Arithmetic operations on rational and irrational numbers Rational + Rational = Rational Rational x Rational = Rational Irrational + Irrational = Irrational / Rational Irrational x Irrational = Irrational / Rational Irrational + Rational = Irrational Irrational x Rational = Irrational 2 5 2 7 . . . . . . REAL NUMBERS 4Read More

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