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Transcript
Number Systems 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 q are integers and q ≠ 0. • A number which cannot be expressed in the form q ≠ 0) is called an irrational number. , where p and , (where p and q are integers and • 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 nonrecurring. • 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/ s are irrationals. • For positive real numbers a and b : • If p and q are rational numbers and a is a positive real number, then Exercise: 1) 2) 3) 4) 5) 6) What are natural numbers? What are whole numbers? What are integers? What do you mean by the word “Zahlen” ? What are rational numbers? Are the following statements true or false? Give reasons for your answers. (i) Every whole number is a natural number. (ii) Every natural number is a whole number. (iii) Every integer is a whole number (iv) Every integer is a rational number. (v) Every rational number is an integer. (vi) Every rational number is a whole number. (vii) There are infinitely many rational numbers between any two given rational numbers. (viii) Every irrational number is a real number (ix) Every point on the number line is of the form √ , where m is a natural number. (x) Every real number is an irrational number. 7) Find five rational numbers between 1 and 2. 8) Find six rational numbers between 3 and 4. 9) Is zero a rational number? Can you write it in the form p/q , where p and q are integers and q ≠ 0? 10) What are irrational numbers? 11) Every real number is represented by a unique point on the number line. Also, every point on the number line represents a unique real number. True or false? 12) Locate √2 on the number line. 13) Locate √3 on the number line. 14) 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. 15) Show how √5 can be represented on the number line. 16) Find the decimal expansions of , , . 17) Show that 3.142678 is a rational number. In other words, express 3.142678 in the form , where p and q are integers and q 0. 18) Show that 0.3333... = 0. 3 can be expressed in the form integers and q 0. integers and q 0. integers and q 0. , where p and q are 19) Show that 1.272727... = 1.27 can be expressed in the form , where p and q are 20) Show that 0.2353535... = 0 .235 be expressed in the form , where p and q are 21) Find an irrational number between and . 22) Write the following in decimal form and say what kind of decimal expansion each has : 23) You know that = 0.142857 . Can you predict what the decimal expansions of , , , , are, without actually doing the long division? If so, how? 24) Express the following in the form , where p and q are integers and q 0. 25) Express 0.99999 .... in the form . Are you surprised by your answer? With your teacher and classmates discuss why the answer makes sense. 26) 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. 27) 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? 28) Write three numbers whose decimal expansions are non-terminating non-recurring. 29) Find three different irrational numbers between the rational numbers and 30) Classify the following numbers as rational or irrational : . 31) Visualize the representation of 5.37 on the number line upto 5 decimal places, that is, up to 5.37777. 32) Visualise 3.765 on the number line, using successive magnification. 33) Visualise 4 .26 on the number line, up to 4 decimal places. 34) Check whether below numbers are irrational or not. 35) Add 2√2 + 5 √3 and √2 - 3√3 . 36) Multiply 6√5 by 2√5 . 37) Simplify the following expressions: 38) Rationalise the denominator of 1/√2 . 39) Rationalise the denominator of 40) Rationalise the denominator of 41) Rationalise the denominator of . √ √ √ √ . . 42) Classify the following numbers as rational or irrational: 43) Simplify each of the following expressions: 44) Recall, π is defined as the ratio of the circumference (say c ) of a circle to its diameter (say d). That is, π = c/d⋅ This seems to contradict the fact that π is irrational. How will you resolve this contradiction? 45) Represent √9.3 on the number line. 46) Rationalise the denominators of the following: 47) Simplify: 48) Find: 49) Find: 50) Simplify : Multiple Choice Questions: 51) Which of the following is not equal to 52) Every rational number is (A) a natural number (B) an integer (C) a real number (D) a whole number 53) 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 54) Decimal representation of a rational number cannot be (A) terminating (B) non-terminating (C) non-terminating repeating (D) non-terminating non-repeating 55) 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 56) .The decimal expansion of the number √2 is (A) a finite decimal (B) 1.41421 (C) non-terminating recurring (D) non-terminating non-recurring 57) Which of the following is irrational? 58) Which of the following is irrational 59) A rational number between √2 and √3 is 60) The value of 1.999... in the form , where p and q are integers and q ≠ 0 , is 61) 2√ 3 + √3 is equal to 62) √10 x √15 is equal to 63) The number obtained on rationalising the denominator of 64) √ √ is is equal to 65) After rationalising the denominator of (A) 13 66) The value of √ √ (B) 19 √ √ 67) If√ 2 = 1.4142, then 68) √ (A) 2.4142 √2 equals to is equal to √ √ (C) 5 is equal to (B) 5.8282 69) The product √2 √2 √32 equals √ √ , we get the denominator as (C) 0.4142 (D) 35 (D) 0.1718 70) Value of 81 is 71) Value of (256)0.16 × (256)0.09 is (A) 4 (B) 16 (C) 64 72) Which of the following is equal to x? (D) 256.25