Yes zero is a rational number as it can be represented in the

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Magdalene Holmes
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1 1 REAL NUMBERS EXERCISE 1.1 Q: 1 Is zero a rational number? Can you write it in the form 0?, where p and q are integers and q Yes zero is a rational number as it can be represented in the form, where p and q are integers and q 0 as etc. Concept Insight: Key idea to answer this question is "every integer is a rational number and zero is a non negative integer". Also 0 can be expressed in form in various ways as 0 divided by any number is 0. simplest is. Q: 2 Find six rational numbers between 3 and 4. There are infinite rational numbers in between 3 and 4. 3 and 4 can be represented as respectively. Now rational numbers between 3 and 4 are Concept Insight: Since there are infinite number of rational numbers between any two numbers so the answer is not unique here. The trick is to convert the number to equivalent form by multiplying and dividing by the number atleast 1 more than the rational numbers to be inserted. Q: 3 APG EDU PORT 1

2 2 REAL NUMBERS Find five rational numbers between. There are infinite rational numbers between Now rational numbers between are Concept Insight: Since there are infinite number of rational numbers between any two numbers so the answer is not unique here. The trick is to convert the number to equivalent form by multiplying and dividing by the number at least 1 more than the rational numbers required. Alternatively for any two rational numbers a and b, between a and b. is also a rational number which lies Q: 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 (i) True, since collection of whole numbers contains all natural numbers. (ii) False, integers include negative of natural numbers as well which are clearly not whole numbers. For example -1 is an integer but not a whole number (iii) False, rational numbers includes fractions and integers as well. For example is a rational APG EDU PORT 2

3 3 REAL NUMBERS number but not whole number. Concept Insight: Key concept involved in this question is the hierarchy of number systems Remember the bigger set consists of the smaller one. Since Mathematics is an exact science every fact has a proof but in order to negate a statement even on e counter example is sufficient. EXERCISE 1.2 Q: 1 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 (iii) Every real number is an irrational number., where m is a natural number. (i) True, since real numbers consists of rational and irrational numbers. APG EDU PORT 3

4 4 REAL NUMBERS (ii) False, Since negative integers cannot be expressed as the square root of any natural number. (iii) False, real number includes both rational and irrational numbers. So every real number can not be an irrational number. Concept Insight: Mentioning the reasons is important in this problem. Real Numbers consists of rational and irrational numbers and not vice versa. Every real number corresponds to a point on number line and vice versa. Recall real number includes negative numbers also. Square root of negative numbers is not defined. Q: 2 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. Square roots of all square numbers is rational. For example Thus the square roots of all positive integers are not irrational Concept Insight: In general only the square root of a prime number is irrational. There are the perfect square numbers. Q: 3 Show how can be represented on the number line. Using Pythagoras Theorem: 5= Taking positive square root we get APG EDU PORT 4

5 5 REAL NUMBERS 1. Mark a point 'A' representing 2 units on number line. 2. Now construct AB of unit length perpendicular to OA. Join OB 3. Now taking O as centre and OB as radius draw an arc, intersecting number line at point C. 4. Point C represents on number line Concept Insight: For a positive integer n, can be located on number line, if is located using Pythagoras Theorem. If is a perfect square then this method is useful. To represent the irrational number key idea is to use Pythagoras theorem and create a length of units by constructing a right triangle of base and perpendicular of length 2 and 1 units. EXERCISE 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) (i) APG EDU PORT 5

6 6 REAL NUMBERS 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? Yes it can be done as follows: APG EDU PORT 6

7 7 REAL NUMBERS 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. (i) Let x = (i) Multiplying by 10 we get 10x = (ii) (ii) - (i) gives 9x = 6 Or x = (ii) Let x = (i) 10x = x = (ii) (ii) - (i) gives 99 x = 43 APG EDU PORT 7

8 8 REAL NUMBERS x = (iii) Let x = (i) 1000x = (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 in the form. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense. Let x = (i) 10x = (ii) (ii) - (i) gives 9x = 9 x = 1 Concept Insight: 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. APG EDU PORT 8

9 9 REAL NUMBERS 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? Terminating decimal expansion will come when denominator q of rational number of 2, 4, 5, 8, 10, and so on......, is either 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. APG EDU PORT 9

10 10 REAL NUMBERS 3 numbers whose decimal expansion is non terminating non recurring are......, Concept Insight: Recall that a non terminating non recurring decimal is an irrational number. to such questions is not unique. Q: 8 Find three different irrational numbers between the rational numbers 3 irrational numbers are 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: APG EDU PORT 10

11 11 REAL NUMBERS (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) 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. EXERCISE 1.4 Q: 1 Visualise on the number line using successive magnification can be represented APG EDU PORT 11

12 12 REAL NUMBERS 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. = We can visualise as in following steps APG EDU PORT 12

13 REAL NUMBERS Concept Insight: Divide the number line between the number to be represented in 10 parts starting the whole number part. EXERCISE 1.

13 13 REAL NUMBERS Concept Insight: Divide the number line between the number to be represented in 10 parts starting the whole number part. EXERCISE 1.5 Q: 1 Classify the following numbers as rational or irrational: (i) As decimal expansion of this expression is non terminating non recurring, so it is an irrational number. APG EDU PORT 13

14 14 REAL NUMBERS (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: APG EDU PORT 14

15 15 REAL NUMBERS 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? 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 APG EDU PORT 15

16 16 REAL NUMBERS integers. In c and d are not integers. Also remember that no measurement is exact. Q: 4 Represent on the number line. (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 ED 2 =EB 2 +DB 2 EDB Using Pythagoras theorem APG EDU PORT 16

17 REAL NUMBERS Concept Insight: This method based on the application of Pythagoras theorem can be used to represent root of any rational number on

17 17 REAL NUMBERS 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: APG EDU PORT 17

18 18 REAL NUMBERS 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 EXERCISE 1.6 Q: 1 APG EDU PORT 18

19 19 REAL NUMBERS Concept Insight: Express the number in exponent notation and use the rule Exponent m must be such that it is divisible by n. Q: 2 APG EDU PORT 19

20 20 REAL NUMBERS Concept Insight: Express the number in exponent notation and use the rule of exponents. Q: 3 APG EDU PORT 20

21 21 REAL NUMBERS Concept Insight: Use the rule of exponents APG EDU PORT 21

Class IX Chapter 1 Number Sustems Maths

Class IX Chapter 1 Number Sustems Maths Exercise 1.1 Question Is zero a rational number? Can you write it in the form 0? and q, where p and q are integers Yes. Zero is a rational number as it can be represented