Number System - Worksheet Question 1: Express the following in the form p/q, where p and q are integers and q 0. Question 2: Express 0.99999... in the form p/q. Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense. Question 3: What can the maximum number of digits be in the repeating block of digits in the decimal expansion of 1/17? Perform the division to check your answer. Question 4: 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 than 1 and having terminating decimal representations (expansions). Can you guess what property q must satisfy. Question 5: Write three numbers whose decimal expansions are non-terminating non-recurring. Question 6: Find three different irrational numbers between the rational numbers 5/7 and 9/11. Question 7: Visuallise 3.765 on the number line, using successive magnification. Question 8: Visualise 4. 26 on the number line, up to 4 decimal places. Question 9: Classify the following numbers as rational or irrational: (i) 2-5 (ii) (3 + 23) - 23 (v) 2π Question 10: Simplify each of the following expressions: (i) (3 + 3)(2 + 2) (ii) (3 + 3)(3-3) (iii) ( 5 + 2) 2 (iv) ( 5-2)( 5 + 2)
Solution (i) Answer: [Since only one digit is repeating, so multiply x with 10.]
(ii) Answer: [Since only one digit is repeating, so multiply x with 10.]
(iii) Answer: Since, there are three repeating decimal digit, so multiply x with 1000
2.
3. Thus, maximum number of digits in the repeating block is 17.
4. Answer: For having terminating decimal expansions, the denominator should have either 2 or 5 or both as factor. So, q must have either 2 or 5 or both. Examples: Non-terminating repeating Non-terminating repeating Non-terminating repeating Non-terminating repeating
5. Answer: Non-terminating non-recurring numbers are known as irrational numbers. Irrational numbers cannot be expressed in the form of p/q where q 0. Following are the possible numbers: 0.72012001200012000001 0.73013001300013000001 0.7501500150001500001.. 6. Answer: 5/7 = 0.714285714285.. and 9/11 = 0.8181818 < Possible irrational numbers between them can be as follows: 0.72012001200012000001 0.73013001300013000001 0.7501500150001500001.. Note: Non-terminating non-recurring numbers are known as irrational numbers. Irrational numbers cannot be expressed in the form of p/q where q 0. Numbers given above cannot be expressed in the form of p/q and hence are irrational. 7. Answer: Step 1: Consider a number line. Step 2 Visualise line between numbers 3 and 4. Step 3 Magnify space between numbers 3 and 4 into 10 equal parts. Step 4 Visualise between number 3.7 and 3.8 minutely using a magnifying glass. Step 5 Magnify and divide space between numbers 3.7 and 3.8 into 10 equal parts. Step 6 Visualize the number 3.765 using a magnifying glass. First Magnification: Second Successive magnification: Third Successive magnification: Fourth Successive magnification:
8. Answer: Step 1- Consider a number line. Step 2 Take space between numbers 4 and 5. Step 3 Divide the space between 4 and 5 into 10 equal parts. Step 4 Consider the space between numbers 4.2 and 4.3 using a magnifying glass. Step 5 Divide space between numbers 4.2 and 4.3 into 10 equal parts. Step 6 Consider space between numbers 4.26 and 4.27. Step 7 Divide space between numbers 4.26 and 4.27 into 10 equal parts. Step 8 Consider space between numbers 4.262 and 4.263 Step 9 Divide space between numbers 4.262 and 4.263 into 10 equal parts. Step 10 Visualise 4.2626 between the numbers 4.262 and 4.263. First magnification: Second Successive magnification: Third Successive magnification: Fourth Successive magnification: Fifth Successive magnification to visualize 4.2626:
9. (i) Answer: Since this equation contains an irrational number 5 thus, given number is an irrational number. (ii) Answer: Given, (3 + 23) - 23 = 3 + 23-23 = 3 Thus, it is rational number. (iii) Answer: Thus, it is a rational number. (iv) Answer: Since given number contains an irrational number as numerator, thus, it is an irrational number. (v) Answer: Since, given number an irrational number, π as factor, thus, given number is an irrational number.
10. (i) Answer: Given, (3 + 3)(2 + 2) = 3 x 2 + 2 3 + 3 2 + 3 x 2 = 6 + 2 3 + 3 2 + 6 (ii) Answer: Given, (3 + 3)(3-3) = (Since, (a + b)(a b) = a 2 b 2 ) Hence, we get 3 2 ( 3) 2 = 9 3 = 6 (iii) Answer: Given ( 5 + 2) 2 (Since, (a + b) 2 = a 2 + b 2 + 2ab) Hence, we have; ( 5) 2 + ( 2) 2 + 2 x 5 x 2 = 5 + 2 + 2 10 = 7 + 2 10 (iv) Answer: Given, ( 5-2)( 5 + 2) (Since, (a + b)(a b) = a 2 b 2 ) Hence, we have; ( 5) 2 ( 2) 2 = 5 2 = 3