NUMBER SYSTEMS (A) Main Concepts and Results Rational numbers Irrational numbers Locating irrational numbers on the number line Real numbers and their decimal expansions Representing real numbers on the number line Operations on real numbers Rationalisation of denominator Laws of exponents for real numbers • A number is called a rational number, if it can be written in the form p , where p q and q are integers and q ≠ 0. • A number which cannot be expressed in the form p (where p and q are integers q and q ≠ 0) is called an irrational number. • All rational numbers and all irrational numbers together make the collection of real numbers. • Decimal expansion of a rational number is either terminating or non-terminating recurring, while the decimal expansion of an irrational number is non-terminating non-recurring. • If r is a rational number and s is an irrational number, then r+s and r-s are irrationals. Further, if r is a non-zero rational, then rs and r are irrationals. s • For positive real numbers a and b : a a = (i) ab = ab (ii) b b (iii) ( a + b )( a − b )= a − b (iv) ( a + b )(a − b )= a 2 − b (v) ( a + b ) 2 = a + 2 ab + b • If p and q are rational numbers and a is a positive real number, then (i) ap . aq = ap + q (ii) (ap)q = apq p a p −q (iii) q = a (iv) apbp = (ab)p a (B) Multiple Choice Questions Write the correct answer: 1 1 ⎡⎤ –6 55 ⎛⎞ ⎢⎥ Sample Question 1 : Which of the following is not equal to ⎜⎟ ? ⎢⎥ ⎝ 6 ⎠ ⎢⎥ ⎣⎦ 1 1 11 1 – ⎡⎤ 6 1–1 ⎛ 5 ⎞56 ⎛ 5 ⎞ 5 ⎛ 6 ⎞ 30 ⎛ 5 ⎞ 30 (A) (B) ⎢⎥ (C) (D) ⎜⎟⎜⎟ ⎜⎟ ⎜⎟ ⎝ 6 ⎠ ⎢⎝ 6 ⎠⎥ ⎝ 5 ⎠⎝ 6 ⎠ ⎣⎦ Solution : Answer (A) EXERCISE 1.1 Write the correct answer in each of the following: 1. Every rational number is (A) a natural number (B) an integer (C) a real number (D) a whole number 2. Between two rational numbers (A) there is no rational number (B) there is exactly one rational number (C) there are infinitely many rational numbers (D) there are only rational numbers and no irrational numbers 3. Decimal representation of a rational number cannot be (A) terminating (B) non-terminating (C) non-terminating repeating (D) non-terminating non-repeating 4. The product of any two irrational numbers is (A) always an irrational number (B) always a rational number (C) always an integer (D) sometimes rational, sometimes irrational 5. The decimal expansion of the number 2 is (A) a finite decimal (B) 1.41421 (C) non-terminating recurring (D) non-terminating non-recurring 6. Which of the following is irrational? 4 12 (A) (B) (C) 7 (D) 81 9 3 7. Which of the following is irrational? (A) 0.14 (B) 0.1416 (C) 0.1416 (D) 0.4014001400014... 8. A rational number between and 3 is 2 2 + 3 2 ⋅ 3 (A) (B) (C) 1.5 (D) 1.8 22 p 9. The value of 1.999... in the form q , where p and q are integers and q ≠ 0 , is 19 1999 1 (A) 10 (B) 1000 (C) 2 (D) 9 10. 2 3 + 3 is equal to (A) 2 6 (B) 6 (C) 3 3 (D) 4 6 is equal to (A) 6 5 (B) 5 6 (C) 25 (D) 10 5 1 12. The number obtained on rationalising the denominator of is 7–2 7 + 2 7–2 7 + 2 7 + 2 (A) (B) (C) (D) 3 35 45 1 13. is equal to 9– 8 1 1 (A) ( 3–2 2 ) (B) 2 3 + 2 2 (C) 3–2 2 (D) 3 + 2 2 7 14. After rationalising the denominator of , we get the denominator as 3 3–2 2 (A) 13 (B)19 (C)5 (D)35 32 + 48 15. The value of is equal to 8 + 12 (A) (B)2 (C)4 (D)8 2 –1 16. If 2 = 1.4142, then is equal to 2 +1 (A) 2.4142 (B) 5.8282 (C) 0.4142 (D) 0.1718 17. 4 3 2 equals 2 11 − 2– 6 (A) 6 (B) (C) (D) 26 2 26 18. The product equals 12 (A) 2 (B)2 (C) 12 2 (D) 32 19. Value of 11 1 (A) (B) (C) 9 (D) 93 81 20. Value of (256)0.16 × (256)0.09 is (A) 4 (B) 16 (C) 64 (D) 256.25 21. Which of the following is equal to x? 21 12 712 5 (A) 7 7(B) x 3 (C) ( x 3 )3 (D) 7 12 x × x x – x 12 ( 4 ) (C) Short Answer Questions with Reasoning Sample Question 1: Are there two irrational numbers whose sum and product both are rationals? Justify. Solution : Yes. 3 + 2 and 3 − 2 are two irrational numbers. ( 3 + 2 )+( 3 − 2 )= 6 , a rational number. ( 3 + 2 ) ×( 3 − 2 )= 7 , a rational number. So, we have two irrational numbers whose sum and product both are rationals. Sample Question 2: State whether the following statement is true: There is a number x such that x 2 is irrational but x 4 is rational. Justify your answer by an example. Solution : True. Let us take x = 42 Now, x2 = , an irrational number. ( 42 ) 2 = 2 x4 = ( 42) 4 , a rational number. = 2 So, we have a number x such that x2 is irrational but x4 is rational. EXERCISE 1.2 1. Let x and y be rational and irrational numbers, respectively. Is x + y necessarily an irrational number? Give an example in support of your answer. 2. Let xbe rational and y be irrational. Is xynecessarily irrational? Justify your answer by an example. 3. State whether the following statements are true or false? Justify your answer. 2 (i) is a rational number. (ii) There are infinitely many integers between any two integers. 3 (iii) Number of rational numbers between 15 and 18 is finite. p (iv) There are numbers which cannot be written in the form , q ≠ 0, p, q both q are integers. (v) The square of an irrational number is always rational. (vi) is not a rational number as 12 and 3 are not integers. 3 15 p (vii) is written in the form , q ≠ 0 and so it is a rational number. 3 q 4. Classify the following numbers as rational or irrational with justification : 9 28 (i) 196 (ii) 3 18 (iii) (iv) 27 343 (v) – 0.4 (vi) (vii) 0.5918 75 (viii) (1 + 5) – (4 + 5 ) (ix) 10.124124... (x) 1.010010001... (D) Short Answer Questions Sample Question 1: Locate on the number line. 13 Solution : We write 13 as the sum of the squares of two natural numbers : 13 = 9 + 4 = 32 + 22 On the number line, take OA = 3 units. Draw BA = 2 units, perpendicular to OA. Join OB (see Fig.1.1). By Pythagoras theorem, OB = 13 Using a compass with centre O and radius OB, draw an arc which intersects the number line at the Fig. 1.1 point C. Then, C corresponds to Remark : We can also take OA = 2 units and AB = 3 units. p Sample Question 2 : Express 0.123 in the form , where p and q are integers and111 37 q q ≠ 0. Solution : Let x = 0.12 3 so, 10x = 1.23 or 10x – x = 1.23 – 0.123 = 1.2333 ... – 0.12333 ... or 9x = 1.11 1.11 111 or x = = 9 900 Therefore, = = 0.123 900 300 Sample Question 3 : Simplify : ( 35 − 52)( 45 + 32) . Solution : (3 5–5 2)( 4 5 + 3 2) = 12 × 5 − 20 2 × 5 + 9 5 × 2 –15 × 2 = 60 − 20 10 + 9 10 –30 = 30 − 11 10 Sample Question 4 : Find the value of a in the following : 6 =3 2 − a 3 3 2 −2 3 6 63 2 + 2 3 × 3 2 −2 3 3 2 − 2 3 3 2 + 2 3 6 ( 3 2 + 2 3) 63 Solution : = ( 2 + 2 3 ) 6 ( 3 2 + 2 3) == = 2 18 − 12 6 ( 3 2 ) 2 −(2 3 ) = 3 2 + 2 3 Therefore, 3 2 +2 3 =3 2 − a 3 or a =– 2 1 3 ⎡ 11 ⎤ Sample Question 5: Simplify : 4 ⎢ ( 3 + 3 ) ⎥ 58 27 ⎣⎦ Solution : 1 ⎡ 1 13 ⎤ 4 ⎡ 1 13 ⎤ 4 1 = 33 ⎢ ( 33 ) ⎥⎢ ( 3 + (3 ) ) ⎥ ⎣58 + 27 ⎦⎣5 (2) 3 ⎦ 1 = ⎡⎤ 4 52 3 ⎣ (+)3 ⎦ 1 = 34 ⎣ () ⎤ ⎡55 ⎦ 1 = [ 4 ]4 = 5 5 EXERCISE 1.3 1. Find which of the variables x, y, z and u represent rational numbers and which irrational numbers: (i) x 2 = 5 (ii) y 2 = 9 (iii) z 2 = .04 (iv) u 2 = 17 4 2. Find three rational numbers between (i) –1 and –2 (ii) 0.1 and 0.11 56 11 (iii) and (iv) and 7 7 45 3. Insert a rational number and an irrational number between the following : 11 (i) 2 and 3 (ii) 0 and 0.1 (iii) and 32 –2 1 (iv) and (v) 0.15 and 0.16 (vi) 2 and 3 52 (vii) 2.357 and 3.121 (viii) .0001 and .001 (ix) 3.623623 and 0.484848 (x) 6.375289 and 6.375738 4. Represent the following numbers on the number line : –3 –12 7, 7.2, , 2 5 5. Locate 5, 10 and 17 on the number line. 6. Represent geometrically the following numbers on the number line : (i) 4.5 (ii) 5.6 (iii) 8.1 (iv) 2.3 p 7. Express the following in the form , where p and q are integers and q ≠ 0: q (i) 0.2 (ii) 0.888... (iii) 5.2 (iv) 0.001 (v) 0.2555... (vi) 0.134 (vii) .00323232... (viii) .404040... 1 8. Show that 0.142857142857... = 9. Simplify the following: 24 54 7 (i) 45– 3 20 + 4 5 (ii) + 89 (iii) 4 12 × 76 (iv) 428 ÷ 37 ÷ 3 7 7 ) 2 33 + 2 27 +(v) (vi) ( 3– 2 3 31 + (vii) 481 –8 3 216 + 15 532 + 225 (viii) 8 2 2 3 3 (ix) – 3 6 10. Rationalise the denominator of the following: 2 40 3 + 2 (i) (ii) (iii) 3 3 3 4 2 16 2 +36 (iv) (v) (vi) 41– 5 2– 3 2 + 3 3 + 2 3 5 + 3 4 3 + 5 2 (vii) (viii) (ix) 3– 2 5– 3 48 + 18 11. Find the values of a and b in each of the following: 5+ 2 3 (i) = a − 6 3 7 + 43 3– 5 19 = a 5–(ii) 3 + 2 5 11 2 + 3 (iii) = 2– b 6 3 2–2 3 7 + 5 7– 5 7 – = a + 5b(iv) 7– 5 7 + 5 11 1 12. If a = 2 + 3 , then find the value of a –. a 13. Rationalise the denominator in each of the following and hence evaluate by taking 2 = 1.414 , 3 =1.732 and 5 = 2.236 , upto three places of decimal. 46 10 – 5 (i) (ii) (iii) 3 6 2 2 1 (iv) (v) 2 + 2 3 + 2 14. Simplify : 4 −12 6 1 38 32 (i) (13 + 23 + 33 )2 (ii) 55 5 2 1 − 1 4 −−2 2 13 (625 ) (iii) (iv) 27 1 1 − 11 2 − 93 × 27 2 64 3 64 3 –64 3 (v) 1 − 2 (vi) 36 × 33 11 83 × 16 3 (vii) − 1 32 3 (E) Long Answer Questions 1 Sample Question 1 : If a = 5 +2 6 and b = , then what will be the value of a a2 + b2? Solution : a = 5 + 2 6 11 15 − 2 6 5 − 2 6 5 − 2 6 × = = 5 − 2 6 = b = = 2 = a 5 +2 6 5 + 2 6 5 − 26 52 −(2 6) 25 − 24 Therefore, a 2 + b2 = (a + b)2 – 2ab Here, a + b = (5 + 2 6) + (5 – 2 6) = 10 ab = (5 + 2 6) (5 – 2 6) = 52 – (2 6)2 = 25 – 24 = 1 Therefore, a2 + b2 = 102 – 2 × 1 = 100 – 2 = 98 EXERCISE 1.4 p 1. Express 0.6 + 0.7 +0.47 in the form , where p and q are integers and q ≠ 0. q 3. If 2 1.414, = 3 1.732 = , then find the value of 3 4 3 – 2 2 3 + 3 3 2+ 2 . 4. If a = 3 2 + 5 , then find the value of 2 2 1 a a + . 5. 6. If 3 2 3 – 2 x + = and y = Simplify : ( ) ( 3 )4 2 256 − − 3 – 3 + 2 2 , then find the value of x 2 + y 2 . 7. Find the value of ( ) 2 3 4 216 − ( ) 3 4 1 256 − + ( ) 1 5 2 243 − +