Ex 1.5

                                                Maths NCERT Book Solutions

                                                                  Chapter 1

                                                                   Ex.1.5

Q: 1

Classify the following numbers as rational or irrational:

Answer

(i)    As decimal expansion of this expression is non terminating non recurring, so it is an irrational number.    (ii)    It can be represented in  form so it is a rational number.   (iii)     As it can be represented in  form, so it is a rational number.   (iv)       As decimal expansion of this expression is non terminating non recurring, so it is an irrational number. (v)     As decimal expansion is non terminating non recurring, so it is an irrational number.   Concept Insight: Do the simplifications as indicated and see whether the number is terminating, non terminating recurring or neither terminating nor repeating.  Remember Sum/difference/Product of a rational and irrational number may or may not be  irrational. Q: 2 Simplify each of the following expressions:     Answer   Concept Insight: Apply the algebraic identities (a+b)2, (a-b)2,(a+b)(a-b) etc to simplify the given expressions. Equivalent Identities used here are     Q: 3 Recall,  is defined as the ratio of the circumference (say c) of a circle to its diameter (say d). That is,  . This seems to contradict the fact that  is irrational. How will you resolve this contradiction?   Answer There is no contradiction. Since    here circumference or diameter are not given to be integers . When we measure a length with scale or any other instrument, we only get an approximate rational value. We never get an exact value. c or d  may be irrational. So, the fraction   is irrational. Therefore,  is irrational. Concept Insight: A rational number is the number of the form   where p and q are   integers. In    c and d are not integers. Also remember that no measurement is exact. Q: 4 Represent   on the number line.   Answer (i)    Mark a line segment OB = 9.3 on number line.   (ii)    Take BC of 1 unit.   (iii)    Find mid point D of OC and draw a semicircle on OC while taking D as its centre.   (iv)     Draw a perpendicular to line OC passing through point B. Let it intersect semicircle at E. Length of perpendicular BE =  . (v)    Taking B as centre and BE as radius draw an arc intersecting number line at F. BF is  i.e point F represents   on number line     Verification: In EDB ED2=EB2+DB2      Using Pythagoras theorem     Concept Insight: This method based on the application of Pythagoras theorem can be used to represent root of any rational number on the number line.   The key idea to represent  is to create a length of   units. In ODB DB =    Q: 5 Rationalise the denominators of the following:     Answer           Concept Insight: Rationalisation of denominator means converting the irrational  denominator  to rational  i.e .  removing the radical sign from  denominator.A number of the form   can be converted to rational form by multiplying with its conjugate. Remember the algebraic identities

Ex 1.4

                                                Maths NCERT Book Solutions

                                                                  Chapter 1

                                                                   Ex.1.4

Q: 1

Visualise 3.765 on the number line using successive magnification.

Answer

3.765 can be represented       Concept Insight: Divide the number line between the number to be represented in 10 parts starting the whole number part. Q: 2 Visualise   on the number line, up to 4 decimal places.   Answer  = 4.2626 We can visualise 4.2626 as in following steps    Concept Insight: Divide the number line between the number to be represented in 10 parts starting the whole number part.

Ex 1.3

                                               Maths NCERT Book Solutions

                                                                  Chapter 1

                                                                   Ex.1.3

Q: 1

Write the following in decimal form and say what kind of decimal expansion each has: (i)             (ii)             (iii)  (iv)              (v)            (vi)

Answer

(i)         terminating  (ii)          non terminating repeating     (iii)          Terminating  (iv)           non terminating repeating  (v)    non terminating repeating decimal  (vi)          Terminating decimal Concept Insight: The decimal expansion of a rational number is either terminating or non terminating recurring. Decimal expansion  terminates in case the prime factors of denominator includes 2 or 5 only. Q: 2 You know that  . Can you predict what the decimal expansion of      are, without actually doing the long division? If so, how?   Answer Yes it can be done as follows:   Concept Insight: Multiples of the given decimal expansion can be obtained by simple multiplication with the given constant. Cross check the answer by performing long division. Q: 3 Express the following in the form  , where p and q are integers and .   Answer (i)                Let x = 0.666 ...    (i)       Multiplying by 10 we get          10x = 6.666 ...    (ii)       (ii)  - (i) gives            9x = 6       Or  x =  (ii)              Let x = 0.4777 ...   (i)           10x = 4.777 ...         100x = 47.777 ... (ii)         (ii) - (i) gives         99 x = 43                x =   (iii)                 Let x = 0.001001 ...(i)           1000x = 1.001001 ...(ii)           (ii) - (i) gives               999x = 1                   x =   Concept Insight: The key idea to express a recurring decimal in the p/q form is to multiply the number by the 10n where n = number of digits repeating. This is done to  make the repeating block a whole number part of the decimal. By subtracting the two expressions x can be expressed in the P/q form Q: 4 Express 0.99999 ..... in the form . Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense.   Answer Let x = 0.9999 .. .. ..(i)   10x = 9.9999 ... ...(ii) (ii) - (i) gives   9x = 9     x = 1     Concept Insight: .9999999 ..... is nothing but 1 when expressed in p/q form. Q: 5 What can be the maximum number of digits be in the repeating block of digits in the decimal expansion of ? Perform the division to check your answer.   Answer Expressing   in the  decimal form we                There are 16 digits in repeating block of decimal expansion of . Concept Insight: Maximum number of digits that can repeat will be 1 less than the prime number in denominator. Q: 6 Look at several examples of rational numbers in the form   where p and q are   integers with no common factors other than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy?   Answer Terminating decimal expansion will come when denominator q of rational number  , is either of 2, 4, 5, 8, 10, and so on ... ... Terminating decimal may be obtained in the situation where prime factorisation of the denominator of the given fractions are having power of 2 only or 5 only or both. Concept Insight: A rational number in its simplest form will terminate only when prime factors of its denominator consists of 2 or 5 only. Q: 7 Write three numbers whose decimal expansions are non-terminating non-recurring.   Answer 3 numbers whose decimal expansion is non terminating non recurring are ... ... , 0.505005000051509 ... ... ... 0.72012009200011500007200000 ... ... ... 7.03124509761202 ... ... ... ... ... ... Concept Insight: Recall that a non terminating non recurring decimal is an irrational number.  Answer to such questions is not unique. Q: 8 Find three different irrational numbers between the rational numbers    Answer   3 irrational numbers are - 0.73073007300073000073 ... ... ... 0.75075007500075000075 ... ... ... 0.79079007900079000079 ... ... ... Concept Insight: There is infinite number of rational and irrational numbers between any two rational numbers. Convert the number into its decimal form to find irrationals between them. Alternatively following result can be used to answer Irrational number between two numbers x and y   Q: 9 Classify the following numbers as rational or irrational:     Answer (i)  As decimal expansion of this number is non-terminating non recurring. So it is an irrational number. (ii)     Rational number as it can be represented in   form. (iii)   0.3796 As decimal expansion of this number is terminating, so it is a rational number.   (iv)    As decimal expansion of this number is non terminating recurring so it is a rational number.   (v)     As decimal expansion of this number is non terminating non repeating so it is an irrational number. Concept Insight:  A number is rational if its decimal expansion is either terminating or non terminating but recurring. A number which cannot be expressed in p/q  form is irrational. Square root of prime numbers is always irrational.