9M01_Number Systems_Avanti Module

9M01 – Number Systems 1 Fifth Edition 9M01. Number Systems TABLE OF CONTENTS 9M01. Number Systems 1 9M01.1 Introduction to Rational Numbers......................................................................................................................................2 9M01.2 Decimal Expansion of Real Numbers.................................................................................................................................4 9M01.3 Locating Numbers on Number Line ...................................................................................................................................7 9M01.4 Properties of Real Numbers and Laws of Exponents.................................................................................................8 Advanced Practice Problems.................................................................................................................................................................10 Olympiad Practice Problems.................................................................................................................................................................11

9M01 – Number Systems 2 9M01.1 Introduction to Rational Numbers CONCEPTS COVERED 1. Natural or Counting numbers 2. Whole numbers 3. Integers 4. Rational numbers 5. Equivalent fractions 6. Plotting rational numbers on a number line 7. Comparing and ordering rational numbers 8. Finding rational numbers between rational numbers IN CLASS EXERCISE LEVEL 1 Q1. State True or False: II. Every whole number is a natural number. I. Every integer is a rational number. IV. Every rational number is a whole number. III. Every natural number is an integer. Q2. Represent −4, 3, −2, 0, 4 on the number line. Q3. Guess the value : I. II. III. LEVEL 2 Q4. Represent the following rational numbers on the number line: I. 11 , −5 II. 4 , −3 , 12 22 55 5 Q5. Arrange these numbers in ascending order: I. 7 , −3 , −7 , 9 , 2 II. − 2 , 0.41, −2 1 , 21 III. 7.4, −4 5 , 0.2, −8 2 35 44 5 24 10 4 Q6. Insert three rational numbers between − 2 and − 1. 55 Q7. Insert five rational numbers between − 1 and − 1, and arrange in ascending order. 32 9M01.1

9M01 – Number Systems 3 HOMEWORK LEVEL 1 Q1. Give 3 examples of rational numbers which are: II. Not integers. I. Integers Q2. State True or False: I. Every natural number is a whole number. II. Every integer is a natural number. III. There are infinitely many integers between two integers Q3. Represent 5, 2, −4, 3, 0, −7 and −1 on the number line. Q4. Write three equivalent fractions for the following rational numbers: I. 9 II. 3 III. − 4 5 7 3 Q5. Guess the value: I. II. III. LEVEL 2 Q6. Arrange these numbers in ascending order: I. −6, 5 , 9 , − 21 II. − 36 , 3, −3 6 III. −4.65, 4, 4.76, − 24 , 4 4 8 24 40 40 12 25 25 Q7. Represent the following rational numbers on the number line: I. −4 , 7 , 1 II. 5 , −7 , 9 33 444 Q8. Insert 10 rational numbers between −3 and 8 . 11 11 Q9. Insert two rational numbers between 1 and 1, and arrange in descending order. 34 Q10. Find six rational numbers between 1 and 1 and arrange in ascending order. 56 9M01.1

9M01 – Number Systems 4 9M01.2 Decimal Expansion of Real Numbers CONCEPTS COVERED 1. Irrational numbers 2. Real numbers 3. Decimal expansion of a real number 4. Different types of decimal expansion of rational numbers 5. Difference between decimal expansion of rational and irrational number 6. Conversion of decimal expansion of a rational number to its ������ form ������ IN CLASS EXERCISE LEVEL 1 Q1. Classify the following. Put  mark for all that apply Number Natural Whole Integer Rational Irrational Real ������ ������. ������ −������ ������������������ ������ − ������ √������ ������������ ������������ 0 √������. ������������ ������ √������ Q2. In the following equations, state whether variables ������, ������, ������ etc. represent rational or irrational numbers: I. ������2 = 5 II. ������2 = 9 III. ������2 = 0.04. IV. ������2 = 17 V. ������2 = 3 VI. ������2 = 27 4 VII. ������2 = 0.4 Q3. Give an example of two irrational numbers whose: II. Sum is a rational number IV. Quotient is an irrational number I. Difference is a rational number III. Product is an irrational number 9M01.2

9M01 – Number Systems 5 LEVEL 2 Q4. If 7������ = 2, then find the decimal expansion of ������. Q5. Find the decimal expansion of 1 1 , 8 and 10. 75 3 Q6. Express 0.159̅ in ������ form, where ������ and ������ are integers and ������ ≠ 0. ������ Q7. Express 1.32̅ + 0. 3̅̅̅5̅ in the form ������, where ������ and ������ are integers and ������ ≠ 0. ������ PUZZLE In the figure above, if one box is inside another, it means that all its elements belong to both boxes. For example: Box B has contents of Box A along with its own contents. Now, match the columns so that the above arrangement of boxes is correct. Column A Column B A Rational Numbers B Irrational Numbers C Whole Numbers D Natural Numbers E Integers HOMEWORK LEVEL 1 Q1. Explain how irrational numbers differ from rational numbers. Q2. What do you mean by terminating and non-terminating decimal representation? Explain with examples. Q3. State true or False: I. Every rational number is an irrational number. II. Every real number is a rational number. III. Every rational number is a real number Q4. Give an example of each, of two irrational numbers whose: II. Sum is an irrational number IV. Quotient is a rational number I. Difference is an irrational number III. Product is a rational number 9M01.2

9M01 – Number Systems 6 LEVEL 2 Q5. Write the following real numbers in the decimal form. Is the decimal representation terminating or non- terminating? In case it is non-terminating, is it repeating or non-repeating? I. 3 II. 9 2 11 Q6. Express 5. 2̅ in the form of ������, where ������ and ������ are integers and ������ ≠ 0 ������ Q7. Write the decimal expansion of each of the following numbers and say what kind of decimal expansion each has: I. 36 II. 4 1 III. 2 100 8 9 IV. 2 VI. 329 11 400 V. 3 13 Q8. Express 0. ̅2̅3̅̅5̅ in the form of ������ where ������ and ������ are integers and ������ ≠ 0. ������ Q9. Write the decimal expansion of 1. Hence, write the decimal expansions of 2 , 3 , 4 , 5 and 6. 7 7777 7 Q10. What property must ������ satisfy such that rational numbers in the form ������⁄������ (������ ≠ 0), where ������ and ������ are integers with no common factor other than 1 will have terminating decimal representations (expansions)? 9M01.2

9M01 – Number Systems 7 9M01.3 Locating Numbers on Number Line CONCEPTS COVERED 1. Plotting irrational numbers on a number line 2. Geometric representation of √������ for any given positive real number ������ 3. Finding a rational number between any two irrational numbers 4. Finding an irrational number between any two irrational numbers 5. Finding an irrational number between any two rational numbers 6. Plotting real numbers on a number line with the help of their decimal expansion IN CLASS EXERCISE LEVEL 1 Q1. Represent √3 and −√2 on the number line. Q2. Give two rational numbers lying between 0.51 511 5111 51111 … and 0.5353353335 … Q3. Find two rational numbers (in the ������ form) and two irrational numbers between 0.34 344 3444 34444 3 … and ������ 0.36 366 3666 36666 3.... Q4. Find two irrational numbers between 0.1 and 0.12. LEVEL 2 Q5. If ������ = 4.5, find √������ geometrically. Q6. Visualize 2.65 on the number line, using successive magnification. Q7. Visualize 9. 37 on the number line, upto 4 decimal places. Q8. Find a point corresponding to 3 + √2 on the number line. HOMEWORK LEVEL 1 Q1. Represent √10 on the number line. Q2. Represent √7 and −√5 on the number line. Q3. Find two rational and two irrational numbers lying between 0.51511511151111 … and 0.52522522252222 … . Q4. Find two rational and two irrational numbers lying between 0. 2̅ and 0.2526232120 … . Q5. Find a rational number and also an irrational number lying between the numbers 0.3030030003 … and 0.3010010001 … . LEVEL 2 Q6. Represent √9.3 on the number line. Q7. Represent √4.7 on the number line. Q8. Visualize 0. ̅5̅̅3̅ on number line upto 3 decimal places. Q9. Visualize 2.67 on the number line, using successive magnification. Q10. Solve 0. 6̅ + 0.15̅ and visualize on number line upto 3 decimal places. 9M01.3

9M01 – Number Systems 8 9M01.4 Properties of Real Numbers and Laws of Exponents CONCEPTS COVERED 1. Different properties of real numbers 2. Different laws of exponents IN CLASS EXERCISE LEVEL 1 Q1. Classify the following as rational or irrational: I. 5√2 II. 4 III. 2√5 IV. 4������ V. √147 2√3 3√5 √75 Q2. Evaluate each of the following removing radical signs and negative indices wherever they occur: 1 II. (125)− 1 III. (27)− 2 IV. (64)− 3 3 3 2 I. (64)3 25 LEVEL 2 Q3. Simplify the following expressions: II. (3 + √3)(5 − √2) I. (11 + √7)(3 + √2) Q4. Simplify the following: II. (2√5 − 3√2)2 III. (√11 − √5)2 I. (√5 − √3)2 Q5. Simplify each of the following: II. 4+√5 + 4−√5 I. 3 + 2 4−√5 4+√5 5−√3 5+√3 Q6. If both ������ and ������ are rational numbers, find the values of ������ and ������ in each of the following equalities: I. √3−1 = ������ + ������√3 II. 3+√7 = ������ + ������√7 III. 5+√3 = 47������ + √3������ √3+1 3−√7 7−4√3 Q7. If ������ = 3 − 2√2, find ������2 + ������12. HOMEWORK LEVEL 1 Q1. Simplify each of the following: I. (625)− 1 5 4 II. (256)4 III. (243)− 4 5 81 IV. 5√(32)−3 32 Q2. Simplify each of the following: I. (√5)−3(√2)−3 II. (25)− 1 × 3√16 3 III. (3√8)− 1 IV. (√4)−7(√2)−5 2 9M01.4

9M01 – Number Systems 9 LEVEL 2 Q3. Simplify the following expressions: II. (3 + √3)(5 − √2) III. (√5 − 2)(√3 − √5) I. (4 + √7)(3 + √2) Q4. Simplify the following expressions: II. (√5 − √3)2 I. (√3 + √7)2 Q5. Simplify: √5 − 2 √5 + 2 − √5 + 2 √5 − 2 Q6. If both ������ and ������ are rational numbers, find the values of ������ and ������ in each of the following equalities: I. 5+2√3 = ������ + ������√3 II. √5+√3 = ������ + ������√15 III. √2+√3 = ������ − ������√6 7+4√3 √5−√3 3√2−2√3 Q7. If ������ = √3+√2 and = √3−√2 , find ������2 + ������2. √3−√2 √3+√2 Q8. If ������ = 3 + 2√2, then find the value of √������ − 1 . √������ ADVANCED QUESTIONS Q1. Solve: ( 3√3 + 3√2) 2 + 2 − 1 (33 23 63) Q2. Simplify: 5√4√(24)3 − 55√8 + 25√4√(23)4. 243������ ×9×(3− ������ 3 = 1. 3) −(81)������ 81 Q3. Find the value of ������ − ������ in the equation 92������×23 9M01.4

9M01 – Number Systems 10 Advanced Practice Problems Q1. Five – eighth of three – tenth of four – ninth of a number is 45. What is the number? Q2. 55% of a number is more than one–third of that number by 52. What is two–fifth of that number? Q3. A number gets reduced to its one–third when 48 is subtracted from it. What is two–third of that number? Q4. Simplify √−√3 + √3 + 8√7 + 4√3. Q5. Arrange the following in ascending order: 31/6, 21/5, 51/5. Q6. Which of the following is smallest? A) 4√5 B) 5√4 C) √4 D) √3 Q7. The value of 351 + 2171 ÷ 5 7 − 4 8 5 1 is _____________. 4170 − 12172 11+ + − 3 5 8 2 Q8. If ������ = 4 and ������ = 5, then find ������2−������2 . ������ 5 ������ ������2+������2 6 Q9. A student was asked to multiply a given number by 4. By mistake he divided it by 4. His answer was 36 more 55 than the correct answer. Find the number. Q10. If three consecutive numbers are given such that twice the first, 3 times the second and 4 times the third together make 182, find the numbers. A.P.P.

11 Olympiad Practice Problems 9M01 – Number System INSTRUCTIONS Answer each of the following questions. 1. 5ଶ௫ିଵ െ (25)௫ିଵ = 2500, then the value of ‫ ݔ‬is _________ . A) 2 B) 5 C) 3 D) 1 D) ξ2 2. The value of ቀ ଵ + ଵ + ‫ ڮ‬+ ଵ ቁ is D) None of these ξଶାξଷ ξଷାξସ ξ଻ାξ଼ D) 45 A) 0 B) 1 C) 2ξ2 3. The last digit of the value of (205)ଵସ + (2001)ଶ଴଴ହ + (2002)଼ଽ is A) 2 B) 8 C) 9 4. If 3ᇣହᇧ+ᇧᇧ3ᇧହᇧ+ᇧᇧ3ᇧହᇤ+ᇧ3ᇧହᇧ+ᇧᇧ‫ڮ‬ᇧᇧ+ᇧ3ᇥହ = 3௫/ଶ, then ‫ ݔ‬is _________ ଽ times A) 9 B) 7 C) 14 5. If ‫ ݔ‬and ‫ ݕ‬are positive real numbers, then which of the following is incorrect? A) ‫ ֜ ݕ > ݔ‬െ‫ < ݔ‬െ‫ݕ‬ B) ‫ ֜ ݕ > ݔ‬െ‫ > ݔ‬െ‫ݕ‬ C) ‫ ֜ ݕ < ݔ‬ଵ > ିଵ D) None . ௫௬ 6. If ‫ݔ‬ = ξ5 െ 2, then value of ‫ݔ‬ଶ + ଵ and ‫ݔ‬ଶ െ ଵ is _________ . ௫మ ௫మ A) 18, െ8ξ5 B) 22, െ16ξ5 C) 18,16ξ5 D) None of these 7. Which of the following pair of numbers has the terminal decimal representation? A) ଵ , ଶ B) ଵ , ଶ C) ଵ , ହ D) ଵ , ଷ ହ ଵ଴య ହ଻ ଷ଻ ଼଺ 8. Find the value of 2ܽ െ ܾ if ξଷାଵ െ ξଷିଵ = ܽ + ܾξ3 ξଷିଵ ξଷାଵ A) െ2 B) 0 C) 1 D) 2 9. Set of all imaginary numbers is a subset of _________ . B) Set of complex numbers. D) All of these A) Set of real numbers. C) Set of natural numbers. 10. If ܰ = ඥξଵ଺ିଷାඥξଶହିଵ ,then ܰcan be simplified as: ටඥସ଴ା(଺×ସ) A) ටଷξଶ B) ଷξଶ ଶ ଶ C) ଷξଶ D) None of these ସ 11. Which statement is not correct? A) If ‘ܽ’ is a rational number and ‘ܾ’ is irrational, then ܽ + ܾ is irrational. B) The product of a non-zero rational number with an irrational number is always irrational. C) Addition of any two irrational numbers can be rational. D) Division of any two integers is an integer. Avanti – 9M01– Number System – Olympiad Practice Problems ss

భభభ 12 12. The value of ቀ௫௫೜ೝቁ೜ೝ × ቀ௫௫೛ೝቁೝ೛ × ቀ௫௫೛೜ቁ೛೜ is equal to _________ . B) 0 D) 1 A) ‫ݔ‬ C) ‫ݔ‬௣௤ା௤௥ା௥௣ 13. The number ‫ = ݔ‬11.2141414 … … can be expressed in the form ‫ = ݔ‬௣, where ‫ ݌‬and ‫ ݍ‬are positive integers having no ௤ common factors then find the value of (‫ ݌‬െ 6‫)ݍ‬. A) 4756 B) 2581 C) 2973 D) None of these 14. Find out the last digit of the following number. (102)ଷଷ × (88)ହଵଶ × (91)ଶ଴଴ଽ × (2016)ଶ଴ଵ଼ A) 2 B) 4 C) 8 D) 6 15. The ascending order of the surds యξ2, లξ7, వξ5 is _________ . A) యξ2, వξ5, లξ7 B) లξ7, యξ2, వξ5 C) వξ5, యξ2, లξ7 D) వξ5, లξ7, యξ2 D) 0 16. Value of ଶ೙శరିଶ(ଶ೙) = _________ . ଶ(ଶ೙శయ) A) 2௡ାଵ െ ଵ B) 2௡ାଷ െ ଵ ଼ ଼ C) 2௡ାସ(1 െ 2௡) D) ଻ ଼ 17. Consider the following statements : Assertion (A) : ܽ଴ = 1, ܽ ് 0 Reason (R) : ܽ௠ ÷ ܽ௡ = ܽ௠ି௡; ݉, ݊ being integers. For the given two statements. A) Both (A) and (R) are true and (R) is correct explanation of (A). B) Both (A) and (R) are true and (R) is not the correct explanation of (A). C) (A) is true, but (R) is false. D) (A) is false, But (R) is true. 18. If ܾܽ + ܾܿ + ܿܽ = 0, then the value of ଵ + ଵ + ଵ will be _________ . ௔మି௕௖ ௕మି௖௔ ௖మି௔௕ A) െ1 B) ܽ + ܾ + ܿ C) ܽଶ + ܾଶ + ܿଶ 19. Find the rationalising factor of లඥ‫ݔ‬ଶ‫ݕ‬ସ‫ ݖ‬is _________ . A) ඥ‫ݔ‬ସ‫ݕ‬ଶ‫ݖ‬ହ B) ඥ‫ݔ‬ଷ‫ݕ‬ଶ‫ݖ‬ହ C) ඥ‫ݔ‬ଵ଴‫ݖ଺ݕ‬ଽ D) None of these 20. The denominator of ൬௔ାඥ௔మି௕మ + ௔ିඥ௔మି௕మ൰ is _________ . ௔ିඥ௔మି௕మ ௔ାඥ௔మି௕మ A) ܽଶ B) ܾଶ C) ܽଶ + ܾଶ D) ܽଶ െ ܾଶ D) ି଺ 21. Solve for ‫ ݌‬is 81(9௣) = 6561ଷ௣ାଶ ଵଵ A) ିହ B) ିଽ C) ି଻ D) ଵଽଶξହ ଵଵ ଵଵ ଵଵ ଶ଴଴ 22. If ‫ݔ‬ = ξହ, then find ‫ݔ‬ଷ + ଵ = _________ . ௫య D) ସ ଶ ହ A) ଵ଼ଽ B) ଵ଼ଽξହ C) ଵ଼ଽξହ ଶ଴଴ξହ ସ଴ ଶ଴଴ 23. Which of the following numbers has the recurring decimal representation? A) ହ B) ଷ C) ଵ ଺ ଶ଴ ଶ଴଴ 24. If ‫ݔ‬ = ଷିξହ, find the value of ‫଺ݔ‬ + ௫ଵల. ଶ A) 322 B) 326 D) None of these C) 364 Avanti – 9M01– Number System – Olympiad Practice Problems ss

25. Which of the following statements are true? 13 1. The additive inverse of ௔ is ௕. ௕௔ 2. The multiplicative inverse of ௔ is ௕ ௕௔ 3. Multiplication can’t be distributed over subtractions of rational numbers. 4. Multiplication can be distributed over addition of rational numbers. A) Statements (1) and (3) are correct. B) Statements (2) and (4) are correct. C) Statements (2) and (3) are correct. D) Statements (1) and (4) are correct. 26. 3.456 can be expressed in rational form as _________ . A) ଽ଴଴ B) ଷଵଵଵ C) ଵଷଵଵ D) None of these D) None of these ଵଷଵଵ ଽ଴଴ ଽଽ଴ D) 1 27. The value of : C) Less than ଽଽ ଵ + ଵ + ଵ + ‫ ڮ‬+ ଵ is ଵ଴଴ ଵ×ଶ ଶ×ଷ ଷ×ସ ଽଽ×ଵ଴଴ C) 2 A) Equal to ଽଽ B) Greater than ଽଽ ଵ଴଴ ଵ଴଴ 28. On simplifying (଻଻଻଻଻ହ×଻଻଻଻଻଻)ାଵ + (ଶଶଶସ×ଶଶଶଶ)ାଵ we get _________ . (଻଻଻଻଻଺)మ (ଶଶଶଷ)మ A) 4 B) 3 29. If ଽ೙ ×ଷమ×ቆଷష೙మ ቇషమ = ଵ. ଷయ೘×ଷయ ଶ଻ then find (݊ െ ݉) : A) െ2 B) െ3 C) െ ଶ D) െ ଷ ଷ ଶ 30. Set of composite numbers is a subset of _________ . B) Set of natural numbers and odd numbers. D) All of these. A) Set of natural numbers and even numbers. C) Set of whole numbers and real numbers. Avanti – 9M01– Number System – Olympiad Practice Problems ss