Number Systems. Exercise 1.1. Question 1. Is zero a rational number? Can you write it in the form p q,
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1 s Exercise. Question. Is zero a rational number? Can you write it in the form p q, where p and q are integers and q 0? Solution Yes, write 0 (where 0 and are integers and q which is not equal to zero). Question. Find six rational numbers between and 4. Solution There can be infinitely many rationals between and 4, one way is 8 to take them and 4. (Q6+ ) First rational number between and 4 q (rational number between 8 and ) 8 < < Second rational number between and 4 q (rational number between and ) 9 < < < Third rational number between and 4 8 q (rational number between + 8 and ) < < < < 8 8 Similarly, < < < < < < < Hence, the six rational numbers 9 0 0,,,,, are all lying between and
2 Question. Find five rational numbers between and 4. Solution Let a and b 4 a + b A rational number between a and b 4 A rational number between and < < 0 Now, a rational number between and ( 6+ ) 0 4 < < < 0 0 Similarly,,, are rational numbers between and 4. Hence, required rational numbers are,,,, Question 4. State whether the following statements are true or false. Give reasons for your answers. (i) Every natural number is a whole number. (ii) Every integer is a whole number. (iii) Every rational number is a whole number. Solution (i) True, because natural numbers are,,, 4,..., and whole numbers are 0,,,, 4,,...,. or The collection of whole numbers contain all the natural numbers. (ii) False (Qnegative integers are not included in the list of whole numbers.) (iii) False Q 6 0,, are not whole numbers. 9 0
3 s Exercise. Question. State whether the following statements are true or false. Justify your answers. (i) Every irrational number is a real number. (ii) Every point on the number line is of the form m, where m is a natural number. (iii) Every real number is an irrational number. Solution (i) True (QReal numbers Rational numbers + Irrational numbers.) (ii) False (Q no negative number can be the square root of any natural number.) (iii) False (Qrational numbers are also present in the set of real numbers.) Question. Are the square roots of all positive integers irrational? If not, give an example of the square root of a number that is a rational number. Solution No, the square roots of all positive integers are not irrational. eg.., 6 4 Here, 4 is a rational number. Question. Show how can be represented on the number line. Solution We know that, P Draw of right angled triangle OQP, such that OQ units PQ unit and OQP 90 Now, by using Pythagoras theorem, we have OP OQ + PQ + OP 4 + O 0 Q R
4 Now, take O as centre OP as radius, draw an arc, which intersects the line at point R. Hence, the point R represents. Question 4. Classroom activity (constructing the square root spiral ). Solution Take a large sheet of paper and construct the square root spiral in the following fashion. Start with a point O and draw a line segment OP of unit length. Draw a line segment PP perpendicular to OP of unit length (see figure). P P P O Constructing square root spiral Now, draw a line segment PP perpendicular to OP. Then draw a line segment PP 4 perpendicular to OP. Continuing in this manner, you can get the line segment Pn Pn by drawing a line segment of unit length perpendicular to OP n. In this manner, you will have created the points P, P, K, P n, K, and joined them to create a beautiful spiral depicting,, 4,K.
5 s Exercise. Question. Write the following in decimal form and say what kind of decimal expansion each has (i) 6 (ii) (iii) 4 (iv) (v) (vi) Solution (i) Clearly, can be written as (Terminating decimal) (ii) Dividing by, we get ) (Non-terminating repeating) (iii) We have, Dividing by 8, we get 4. 8 ) (Terminating)
6 (iv) We have, / Dividing by, we get ) (Non-terminating repeating) (v) We have, / Dividing by, we get 0.88 ) (Non-terminating repeating) (vi) We have, 0 / 400 Dividing 9 by 400, we get ) / (Terminating)
7 Question. You know that Can you predict what the decimal expansions of 4 6,,,, are, without actually doing the long division? If so, how? [Hint Study the remainders while finding the value of carefully.] Solution We have, Question Express the following in the form p, where p and q are q integers and q 0. (i) 0. 6 (ii) 0.4 (iii) 0.00
8 Solution (i) Let x (i) Multiplying Eq. (i) by 0, we get On subtracting Eq. (ii) from Eq. (i), we get 0x (ii) ( 0x x) ( ) ( ) 9x 6 x 6 / 9 x / (ii) Let x (iii) Multiplying Eq. (iii) by 0, we get Multiplying Eq. (iv) by 0, we get On subtracting Eq. (v) from Eq. (iv), we get 0x (iv) 00x 4. (v) ( 00 x 0x) ( 4....) ( 4....) 90x 4 x 4 90 (iii) Let x (vi) Multiplying Eq. (vi) by (000), we get 000x (vii) On subtracting Eq. (vii) by Eq. (vi), we get ( 000x x) ( ) ( ) 999x x 999 Question 4. Express in the form p. Are you surprised by q your answer? With your teacher and classmates discuss why the answer makes sense. Solution Let x (i) Multiplying Eq. (i) by 0, we get 0x (ii) On subtracting Eq. (ii) by Eq. (i), we get ( 0 x x) ( ) ( ) 9x 9 x 9 9 x
9 Question. What can the maximum number of digits be in the repeating block of digits in the decimal expansion of? Perform the division to check your answer. Solution The maximum number of digits in the repeating block of digits in the decimal expansion of is 6 we have, ) Thus, , a block of 6 digits is repeated.
10 Question 6. Look at several examples of rational numbers in the form p q ( q 0 ). Where, p and q are integers with no common factors other that and having terminating decimal representations (expansions). Can you guess what property q must satisfy? Solution Consider many rational numbers in the form p q q( 0 ), where p and q are integers with no common factors other that and having terminating decimal representations. Let the various such rational numbers be 6 9 9,,,,,, etc In all cases, we think of the natural number which when multiplied by their respective denominators gives 0 or a power of From the above, we find that the decimal expansion of above numbers are terminating. Along with we see that the denominator of above numbers are in the m n form, where m and n are natural numbers. So, the decimal representation of rational numbers can be represented as a terminating decimal. Question. Write three numbers whose decimal expansions are non-terminating non-recurring. Solution Question 8. rational numbers and 9. Find three different irrational numbers between the
11 Solution To find irrational numbers, firstly we shall divide by and 9 by, So, Thus, Thus, The required numbers are Question K Classify the following numbers as rational or irrational (i) (ii) (iii) 0.96 (iv) (v) Solution (i) (irrationalqit is not a perfect square.) (ii) (rational) (whole number.) (iii) 0.96 rational (terminating.) (iv) rational (non-terminating repeating.) (v) irrational (non-terminating non-repeating.)
12 s Exercise.4 Question. Visualise.6 on the number line, using successive magnification. Solution We know that,.6 lies between and 4. So, let us divide the part of the number line between and 4 into 0 equal parts and look at the portion between. and.8 through a magnifying glass. Now.6 lies between. and.8 [Fig. (i)]. Now, we imagine to divide this again into ten equal parts. The first mark will represent., the next. and soon. To see this clearly,we magnify this as shown in [Fig. (ii)]. Again.6 lies between.6 and. [Fig. (ii)]. So, let us focus on this portion of the number line [Fig. (iii)] and imagine to divide it again into ten equal parts [Fig. (iii)]. Here, we can visualise that.6 is the first mark and.6 is the th mark in these subdivisions. We call this process of visualisation of representation of numbers on the number line through a magnifying glass as the process of successive magnification. So, we get seen that it is possible by sufficient successive magnifications of visualise the position (or representation) of a real number with a terminating decimal expansion on the number line (i). M M N N (ii) N P (iii) N Question. Visualise 4. 6 on the number line, upto 4 decimal places. Solution We adopt process by successive magnification and successively decrease the lengths of the portion of the number line in which 4.6 is located. Since 4.6 is located between 4 and and is divided into 0 equal parts [Fig. (i)]. In further, we locate 4. 6 between 4. and 4. [Fig. (ii)].
13 To get more accurate visualisation of the representation, we divide this portion into 0 equal parts and use a magnifying glass to visualise that 4.6 lies between 4.6 and 4.. To visualise 4.6 more clearly we divide again between 4.6 and 4. into 0 equal parts and visualise the repsentation of 4.6 between 4.6 and 4.6 [Fig. (iii)]. Now, for a much better visualisation between 4.6 and 4.6 is agin divided into 0 equal parts [Fig. (iv)]. Notice that 4.6 is located closer to 4.6 then to 4.6 at M M (i) M P P P M (ii) P N N P (iii) N P (iv) N
14 s Exercise. Question. Classify the following numbers as rational or irrational. (i) (ii) ( + ) (iii) (iv) (v) π Solution (i) IrrationalQ is a rational number and is an irrational number. is an irrational number. (QThe difference of a rational number and an irrational number is irrational) (ii) + (rational) (iii) (rational) (iv) (irrational) Q 0 is a rational number and 0 is an irrational number. is an irrational number. (QThe quotient of a non-zero rational number with an irrational number is irrational). (v) π (irrational)q is a rational number and π is an irrational number. x is an irrational number. (QThe product of a non-zero rational number with an irrational number is an irrational) Question. Simplify each of the following expressions (i) ( + )( + ) (ii) ( + )( ) (iii) ( + ) (iv) ( )( + ) Solution (i) ( + ) ( + ) ( + ) + ( + ) (ii) ( + )( ) ( ) [ Q( a + b)( a b) a b ] 9 6 (iii) ( + ) ( ) + + ( ) [ Q( a + b) a + ab + b ] (iv) ( )( + ) ( ) ( ) [Q( a b)( a + b) a b ] π
15 Question. Recall, π is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is π c. This seems to contradict d the fact that π is irrational. How will you resolve this contradiction? Solution Actually c d which is an approximate value of π. Question 4. Represent 9. on the number line. Solution Firstly we draw AB 9. units. Now, from B, mark a distance of unit. Let this point be C. Let O be the mid-point of AC. Now, draw a semi-circle with centre O and radius OA. Let us draw a line perpendicular to AC passing through point B and intersecting the semi-circle at point D. D The distance BD 9.. Draw an arc with centre B and radius BD, which intersects the number line at point E, then the point E represents 9.. Question. (i) Solution (ii) (iii) (iv) (i) Rationalise the denominator of the following (ii) (iii) (iv) A ( ) ( 6) (Multiplying and dividing by ) (Multiplying and dividing by + 6) (Multiplying and dividing by + ( ) ( ) ) ( ) (Multiplying and dividing by + ) O 9. units B C unit E
16 s Question. Find Exercise.6 (i) 64 (ii) (iii) Solution (i) 64 ( 8 8) 8 m n mn 8 [ Q( a ) a ] (ii) ( ) (iii) ( ) Question. Find (i) 9 (ii) (iii)64 (iv) Solution (i) 9 ( ) (ii) ( ) 4 (iii) 6 4 ( 4 ) 4 8 (iv) Question. (i) ( ) Solution (i) (ii) (iii) 4 Simplify (ii) (iii) / 4 / (iv) 8 (Qx x x 4 a b a + b ) a b [Q( x ) 4 Q x x a b x ab x ] a b (iv) 8 ( 8) ( 6 ) a a a [Qx y ( xy) ]
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