CONTENTS NUMBER SYSTEMS. Number Systems

Transcription

1 NUMBER SYSTEMS CONTENTS Introduction Classification of Numbers Natural Numbers Whole Numbers Integers Rational Numbers Decimal expansion of rational numbers Terminating decimal Terminating and recurring decimal Irrational Numbers Insertion of rational numbers between two rational numbers Number line Representation of numbers on number line Irrational number Real numbers by successive magnification Descriptive Problems Training for Competitive Tests Answers Practice Test Project Puzzles

2 This chapter will explore many different types of numbers and how they are classified. There will be a special focus on irrational numbers and how to use them in calculations and representation of these numbers in a number line. The first section will explain how mathematicians classify numbers-natural numbers, whole numbers, integers, rationals, irrationals and real numbers. These classifications are important to set theory and number theory. We will also explain how to arrange numbers in a number line. Classification of Numbers Natural Numbers They are the positive numbers. We use these numbers to count objects.,,,,... are natural numbers. is the smallest natural number. The letter N is used to denote natural numbers. Whole Numbers The whole numbers are the numbers 0,,,, and so on. The letter W is used to denote whole numbers. Note: All natural numbers are whole numbers, but not all whole numbers are natural numbers since zero is a whole number but not a natural number. Integers The integers are... -, -, -, -, 0,,,,... i.e., integers are the collection of positive numbers, negative numbers and zero. It is denoted by the symbol Z. Note: All whole numbers are integers, but not all integers are whole numbers.

3 Rational Numbers The rational numbers include all the integers, plus all fractions. Every rational p number can be written in the form where p and q are integers and q 0 q 5,, etc. are rational numbers. 6 The collection of rational numbers is denoted by Q. Note All natural numbers, whole numbers and integers are rationals, but not all rational numbers are natural numbers, whole numbers, or integers. Note If r and s are any two rational numbers then between r and s. r+s is a rational number Example : Find a rational number between and Rational number between and is + Note There are infinitely many rational numbers between any two given rational numbers. Irrational Numbers Irrational numbers are numbers which are not rationals. i.e., irrational number p cannot be written in the form, p and q are integers and q 0. An irrational q number is a number with a decimal that neither terminates nor repeats.,,, π, are irrational numbers.

4 Real Numbers All the rational numbers and irrational numbers together form the real numbers. It is denoted by the letter R. Our classification look loke. Real Numbers R Rational numbers Irrational numbers Inlegers Z Whole numbers W Natural numbers N Number Line A number line is a horizontal line that has points equally spaced, which correspond to each of the real numbers. Number line extends to right from zero and left from zero. The numbers right to the zero are positive numbers and left to the zero are negative numbers.

5 Note : Every real number is represented by a unique point on the number line. Also every point on the number line that represents on unique real number. Representation of irrational numbers on the number line. Locate on the number line. Step Mark the points -, -, 0,,,... on the number line. Let A denote the point. We can start this procedure if the given number is less than Step Draw a line perpenducular to OA O is the point representing number zero on the number line 5

6 Step Mark the point B which is cm away from A. Step Join OB Note: By pythagores theorem in OAB OB OA + AB + OB 6

7 Step 5 Using a compass with centre O and radius OB, draw an arc intersecting the number line at the point P. Now P corresponds to on the number line Locate on the number line Step Locate on the number line as before. Step Draw a line perpendicular to OB

8 Step Mark the point C cm away from B Step Join OC By Pythagoras theorem in OBC OC OB + BC + OC 8

9 Step 5 Using a compass with centre O and radius OC, draw an arc which intersect the number line at the point Q. Then Q corresponds to Real numbers and their decimal expansions You know every rational numbers (fraction) can be written as a decimal number for example, 0.5, 0.5 etc. In order to represent these number in decimal form, we divide the numerator by denominator. In the case of division, sometimes the remainder becomes zero and in some cases it will not become zero. Consider these cases seperately. Case : The remainder becomes zero 0.5, the remainder becomes zero after some steps. 0.85, the remainder becomes zero 8 after some steps

10 In all these cases, the decimal expansion terminates or ends after a finite number of steps. We call the decimal expansion of such numbers terminating. Case : The remainder never becomes zero Consider the following In these examples, we have a repeating block of digits in the quotient. In, the digit is repeating in the quotient whereas in, the digits,,,8,5, are repeating i.e., the remainder will never become zero. We call the decimal expansion of such numbers non terminating and recurring. Notation for non terminating, recurring decimals. We have seen that is repeating endlesly in the decimal representation of. Its not possible to write all the s in the decimal expansion. So to represent these decimals, put a bar on repeating digits. i.e., 0. which means is repeating

11 Also.5.5 here is repeating. Rational numbers and their decimal expansions The decimal expansion of a rational number is either terminating or nonterminating recurring. Example : 0. is rational Irrational numbers and their decimal expansion The decimal expansion of an irrational number is non terminating nonrecurring. Or a number whose decimal expansion is non-terminating nonrecurring is irrational. Example : is irrational Representing Real Numbers on the number line We have seen that every real numbers has a decimal expansion. This helps us to represent it on a number line. Illustration : Locate.66 on the number line. Step Make the points -,-,0,,,, etc on the number line. We looking closely at the number.66 one can see that this lies between and

12 Step We divide the section to into 0 equal parts and make the points.,.,.... and Now the points.66 lies between.6 and. there equal. Step Divide the section.6 to. into 0 equal parts and make the points.6,.6,....6 and - The point.66 lies between.66 and.6 Step Divide the section.66 to.6 into 0 equal parts and make the points.66,.66, and -6 Now we located the points.66 on the number line

13 Illustration : Visualize the representation of. on the number line upto 5 decimal places... upto decimal places Step Make the points -,-,0,,,, on the number line. lies between and 5 Step We divide the section to 5 into 0 equal parts and make the points.,.,...., 5. Now. lies between. and. Step We divide the section. to. into 0 equal parts and make the points.,.,....,.. lies between. and.8

14 Step Divide the section. -.8 into 0 equal parts abd make the points.,.,.8. lies between. and.8 Step 5 Divide the section. -.8 into 0 equal parts abd make the points.,.,....,.8 Now we can locate. Actually. located near to.8 than. Operations on Real Numbers Addition We cannot add any irrational numbers. We can add only if the irrational part is same. Example : + 5 8

15 Substraction As in the case of addition if we want to substract real numbers the irrational parts must be same. Example : 5 - Multiplication We can find the product of any real numbers. a x b ab Example : x x 6 Division a b a b 6 6 Example : Properties The sum of a rational number and an irrational number is irrational. Example : is a rational number, is an irrational the sum + + which is an irrational number The difference of a rational number and an irrational number is irrational Example : 5 is a rational, is irrational the difference 5 - is irrational 5

16 The product of a non-zero rational number with an irrational number is irrational Example : 5 is a rational, is irrational the product 5 x is irrational The quotient of non-zero rational number with an irrational number is irrational Example : 5 is a rational, is irrational 5 the quotient is irrational If we add any two irrationals, the result may be rational or irrational Example : 5, are irrational numbers the sum 5 + is an irrational number, - are irrational numbers the sum which is a rational number If we substract any irrationals, the result may be rational or irrational Example : 5, and are irrational numbers the difference 5 -, an irrational numbers, are irrational numbers the difference - 0, a rational number If we multiply any irrationals, the result may be rational or irrational. 6

17 Example :, are irrational numbers the product x 6, an irrational number, are irrational numbers the product x 8( ) 8 x 6, a rational number If we divide any irrationals, the result may be rational or irrational. Example : 6, are irrational numbers 6 the quotient, an irrational numbers, are irrational numbers, the quotient, a rational number. Identities. (a + b) (a - b) a - b Example : ( + ) ( - ) ( ) - ( ) -. (a + b) (c + d) ac + ad + bc + bd Example : ( + ) ( + 5) x + x 5 + x + x

18 . (a + b) a + ab + b Example : ( + ) ( ) + x + ( ) Conjugate pairs (Surds) a + b and a - b are conjugate pairs. If we multiply conjugate pairs, the product will be rational number. Example : + and - are conjugate pairs If the product of two irrational numbers is rational, each is called the rationalising factor of the other. Example : Consider If we multiply by, the product is x which is a rational number: is the rationalising factor of Rationalisation The process of making either the denominator or numerater to a rational number is called rationalisation. Illustration Rationalise the denominater of 8

19 Step Divide and multiply given number by x is the rationalising factor of Step Simplyfy the expression in step, if necessary x Now the fraction becomes Illustration Rationalise the denominater of x ( ). i.e., the denominator is a rational number x x + - Given Multiply the numerator and denominater by + 5 ( + ) - ( ) (a + b) (a - b) a - b ( + ) ( - ) - ( ) If the denominater is of the form a + b or a - b multiply and divide by its conjugate pairs

20 5 ( + ) - and ( ) 5 ( + ) Simplify the denominator 5 x + 5 x Open the brackets a x b ab 5 x 0 Now the denominator is a rational number. Laws of exponents for real numbers The following laws will be helpful to simplify the expressions. Let a>0 be a real number and m and n are rational numbers.. a m a n a m + n. (a m ) n a mn a m a n. a m-n, m>n, a 0. a m b m (ab) m Descriptive Problems. Represent as a decimal number a) b) c) 5 0

21 Answer a) b) c)

22 . Insert two rational numbers between a) and b) and Answer a + b a) The rational numbers between a and b is + rational number between and First rational number < < nd rational numbers rational number between and x < < < i.e., and are rational numbers between and + b) Rational number between and Inserting one rational number between + x and

23 i.e., < < Rational number between and 5 i.e., < < < and are rational numbers between and x 5 8 Note : The above insertion of rational number is not unique, as there are infinitive rational numbers between any two given rational numbers.. p Express the following in the form q a) 0. b) 0.5 c) 565 Answer a) Let x () Since one digit is repeating, multiply x by 0. 0x (0...) x 0 0x () () - () 0x x x

24 x x b) Let x () Since digit are repeating, multiply x by 00 00x ( ) x 00 00x () () - () 00x x x 5 x c) Let x () Since digits are repeating, multiply x by x ( ) x () () - () 000x x

25 x 565 x 565. Rationalise the denominator of + - Answer + Consider the given number - Since the denominator is of the form a - b Multiply and divide the given number by the conjugate pair of - So multiply and divide by x + + ( + ) ( + ) - ( ) ( + ) ( + ) - 6 x ( + ) ( + ) - ( + ) ( + ) - 0 () + x

26 { Number Systems Simplify 5 - Answer Consider 5-5 { a Since b -n n b a x

27 Training for Competitive Examinations Very challenging problems are marked with Which of the following numbers lies between and? 6 a) b) c) d) 5. Which of the following numbers is a natural number? a) - b) - c) 0 d). What is the decimal equivalent of? a) 0.85 b) 0.85 c) 0.85 d) is a a) Negative rational number b) Positive rational number c) either positive or negative rational number d) neither positive and negative rational number is 5 + a) rational number b) an irrational number c) an integel d) a natural number 6. Value of. is 5 a) b) c) d)

28 - 5. If a 5 - b when a and b are rational numbers then + 5 a) a b b) a b 0 8 c) a b d) a b 8. The product of and is a) b) c) d) none of these. The value of 6-6 is a) b) c) d) 0. The rationalising factor of 08 is a) b) c) d) 8. Which of the following pairs of numbers has numbers, the difference of which will be an irrational number? a), b), - c), - d) 8, 8. π is a) rational number b) irrational number c) imaginary d) integer. The number ( + 5) is a) rational number b) irrational number c) can t say d) none of these 8

29 The fraction ( + 6) ( + ) has a value equal to a) b) c) d) 5. Which of the following pairs of numbers has numbers, the product of which will be a rational number? a) -, + b) -, - c) -, d) -, 6. Which of the following statement is false? a) all natural numbers are integers b) all whole numbers are natural numbers c) all whole numbers are integers d) all irrational numbers are real numbers.. Which of the following statement is false a) Smallest natural numbers is b) Largest integer cannot be determined c) Smallest integer is 0 d) There are infinite numbers lying between numbers. 8. Which of the following irrational numbers is represented by the following diagram? a) b) c) d)

30 . Which of the following statement is true? a) Every irrational number is a real number b) Square roots of all integers are real c) every real number is an irrational number d) every point on the number line is of the form n. where n is a natural number 0. If - x x, then x is a) rational number b) irrational number c) natural numbers d) none of these The value of is 8 + a) 0 b) c) d) What is the percentage of least number in the greatest number if,, 6 6, are arranged in ascending order? 6 6 a) % b) 0% c) 0% d) 5%. If x + 8, then x + x a) 6 b) 8 c) d) 6. The average of the middle two rational numbers if 5,, 5, are aranged in asending order is 0

31 a) 0 b) c) d) Which of the following is not irrational? a) π - (-π) b) 5 + c) ( + )( - ) d) (5 + )( 5) 6. The product of and is a) 6 b) c) d) none of these If x and y then x + y + xy a) 6 b)6 - c) d) Find the values of a and b if a + b 5 -. Find the least rationalising factor of 8-50 a) b) c) 8 d) 50 0 Simplify : a) 8-5 b) c) d)

32 Answers. c It is clear that < <. d. a b - - So it is a positive rational number 5. b x 5-5 -

33 ( 5 - ) ( 5 - ) ( 5) ( 5 - ) - ( 5) - 5() + - (a - b) a - ab + b Since there is an irrational number 5 in the nd term after simplification, its an irrational number 6. b Let x () Since the one digit is repeating, multiply x by 0 0x (...) (0) 0x () () - () 0x - x x 8.5 x x

34 . a x ( - 5)( - 5) - ( 5) ( - 5)( - 5) - x 5 ( - 5)( - 5) - 0 ( - 5)( - 5) - () - ( 5) ( 5) Given a 5 - b - 5 i.e., + a 5 - b Equating the coefficient of 5 a

35 Equating constant term, - -b or b i.e., a and b 8. a product. ( x ) (88) x x x d 6 ( ) 6 x x x x 5

36 6-6 - ( - ) ( - ) (-) a 08 (08) ( ) 08 Now we have to find out the rationalising factor. i.e., if we multiply that number with rational number. Consider the first option, the result will be a. In this case the result is a rational number is a rationalising factor of

37 . c Lets find out the difference of given numbers Consider (a) - 0, a rational number Consider (b) - (-) +, a rational number Consider (c) - (- ) + 6, an irrational number So option (c) is the answer.. b π is considered as an irrational number. b ( + 5) ( ) + ( )( 5) + ( 5) Since the irrational number 5 is present in the expanded form, the given expression is an irrational number.. d ( + 6) ( + ) ( +. ) ( + )

38 . ( + ) ( + ) Taking outside ( + ) + ( - ) - multiply and divide by - ( + ) ( - ) ( + ) x( - ) Since ab a. b ( + ) ( - ) - ( ) (a + b)(a - b) a - b ( + ) ( - ) - ( + ) ( - ) ( + ) ( - ) + ( + ) ( - ) Since a a. a ( + ) ( - ) Since a a 8

39 ( +) ( - ) ( ) - (a + b)(a - b) a - b - x 5. a Let s find out the product of the given numbers Consider (a) ( - ) ( + ) - ( ) -, a rational number i.e., option (a) is the answer 6. b Consider the number 0, which is a whole number, but not a natural number. i.e., All whole numbers are not natural numbers.

40 . c... -, -, -, are integers less than 0 i.e., the smallest integer cannot be determined. 8. a The point P the length OB But OAB is a right triangle. By Pythagoras theorem, OB OA + AB OB 8 i.e., P is the point. a 0. d - x x -() x (x) -6 x x -6 Square root of a negative number is not a real number. So the answer is option (d). c

41 Consider the st term ( ) (- ) ( ) - ( ) ( - ) + - Similarly

42 a The numbers are,, and Since the denominators are same, we can arrange these numbers in ascending order of numerators.,,, Least number 6 Greatest number 6 percentage of least number in the greatest number

43 6 percentage of in 6 6 x 00 % 6 x 6 x 00 x 6 x 00 % 00 % % %. b x Since 8 x + + x ( ) - - () x -

44 x + ( + ) + ( - ) x + ( ) + () ( ) + () ( ) + () - ( ) - () ( ) + () (- ) + ()() + + ()() d 5 The given numbers are,, and 5 We have to arrange these numbers in ascending order. So first find out the decimal expansions approximated to digit of these numbers <0.<0.56<0.5 5 i.e., < < < 5

45 In ascending order, the numbers are,, 5, 5 Middle two rational numbers are and 5 Average of middle two rational numbers () + 5(5) 5() c We have to simplify each expressions given in the options to identify the rationals. π - (-π) π + π.π, an irrational number 5 + which cannot be simplified further. It is an irrational number. ( + )( - ) ( ) - ( ) -, a rational number i.e., Option (c) is not irrational 5

46 6. a.... ( ) 6. (6. ) (6) 6. d x ( 5 - ) ( 5) - ( 5 - ) ( 5 - ) 6

47 x ( 5) - ( 5)() y ( 5 + ) 5 - ( 5 + ) y ( 5) + ( 5)() x ( - 5) - ()( 5) + ( 5) (5) x () y ( + 5) + ()( 5) + ( 5) (5) y ()

48 xy ( - 5)( + 5) - ( 5) 8-6(5) 8-80 xy () () + () + () x + y + xy b Given (5 + ) 5 - ( ) (5 + ) 5 - () (5 + ) 5-8 (5 + ) 5 +(5)( )+( ) 8

49 i.e., a + b Equating the coefficient of, 0 b Equating the constant terms, a. b Given Consider () The conjugate pair is a rationalising factor. To find the least rationalising factor, we have to simply the given expressions, x x 50 x 5 x 5 5 a + b

50 In the simplified expression, the rational number is present. To rationalise this, multiply it by. i.e., the rationalising factor is which is also least. 0. d Consider the first term, ( - ) ( ) - ( - ) () - ( - ) - ( - ) ( - )

51 Consider the nd term, ( + ) ( ) - ( + ) () - ( + ) ( + )

52 PRACTICE TEST Practice makes perfect. Test the level of your accuracy and speed by answering the given problems. Please note the time you spend for each problem. Write all the necessary steps. But if you are thorough with the ideas you can skip some steps. Please send the answers to our office in the business envelope which is free of cost. We will provide you proper guidance. If you want to get a proper evaluation and assessment, you have to do it sincerely.. Which of the following is not a rational number? a) b) 6 c) - d). What is the decimal equivalent of +? a) 0. b) 0. c) 0. d).. Which of the following statement is true? a) 5 < < < b) 5 < < < c) 5 < < < d) 5 < < <. If x + and xy, then + x y a) 6 b) c) d) 5. The rationalising factor of - is a) + b) + c) + d) none of these 5

53 6. The greatest amoung,, 6 is a) b) c) 6 d) Cannot be determined. If x and y, then xy a) b) + 8 c) d) none of these 8. If + a - b then a) a, b - b) a -, b c) a, b d) a, b -. The least rationalising factor of 5 is a) 5 b) c) 5 d) 5 0. If x +, xy then x + y a) 00 b) 68 c) 5 d) 00. If a - 0 and b 8 - then a) a b b) a + b 0 c) a > b d) a < b. Which of the following numbers does not lie between - and? - -5 a) b) c) - d). Rationalising factor of a + a a) a - a b) a + a c) a - a d) a + a - - 5

54 . The product of which of the following pairs of numbers will be a rational number? a) -, + b) -, - c) -, + d) -, 5. Which of the following statement is not true? a) Every real number is represented by a unique point on the number line b) Every point on the number line represents a unique real number. c) Irrational numbers cannot be represented on the number line. d) There are infinity many numbers on the number line. 6. What is the fractional equivqlent of 0.6? 5 a) b) c) d) To what set of numbers does belong? 6 5 a) irrationals b) integers c) whole numbers d) rationals 8. Which is the smallest integer? a) b) 0 c) - d) cannot be determined. Which of the following is an irrational number? a) x b) ( - 8) ( + 8) c) d)

55 p 0. In the expression, p and q are integers q 0, and q n x 5 m, where n and q m are natural numbers, then p a) the decimal expansion of terminates q p b) is a non recurring decimal. q p c) is a non terminating non recurring decimal q d) Can t say Project Write a note about golden number. Is it an irrational number? Most people are familiar with the number, π, since it is one of the most common irrational numbers known to man. But there is another irrational number that has the same propensity for popping up which is not as well known as π. This wonderful number is golden number, a Greek letter φ (read as phi) is used to denote this number. Consider the solutions of the equation x - x - 0 This is a quadratic equation Solutions x -b ± b - ac a -(-) ± (-) - ()(-) () -(-) ± + + ± 5 a coefficient of x b coefficient of x - c constant - 55

56 + 5 i.e., roots are, x or x - 5 We consider the first root to be the golden number (φ) + 5 Golden number φ We know 5 is an irrational number, + 5 is an irrational number. Golden number is an irrational number. Let s discuss these in detail Consider φ We know 5 ~ φ φ.68 If objects are in the ratio.68, we say that they are in golden proportion, or.68 is known as golden ratio. Golden ratio in geometrical figures Draw a rectangle with length 5.5 and breadth. approximately b l b ~.68 l 56

57 If the ratio of length and breadth is φ, then that rectangle is known as golden rectangle. Golden ratio can be seen in other geometrical figures also. Golden ratio in Nature Golden ratio is most pleasing to eyes. In nature, so many flowers exhibit golden ratio. For example, Consider the head of a daisy, one can discover that the individual florets of the daisy (and of a sunflower as well) grow in two spirals extending out from the centre. The first spiral has arms, while the other has. thier ratio ~.68. i.e., they exhibit golden ratio. Golden ratio can be found in patterns on butterflies wings. We can observe φ in many things. The golden ratio, also known as the divine proportion. Though it is an irrational number, it is most pleasing to human eyes. 5

58 Puzzles Number Systems Puzzles Do you know this magic? If you perform the following operations, you will always arrive at 00. Always 00 Let s think a single digit number other than zero Add to it Strike out the last digit Add 5 to the result Multiply it by Subtract 5 from it 58

59 Puzzles Number Systems A cross word puzzle for you Across. An angle greater than 80 0 but less than 60 0 is called angle.. Additive identity is is of 0. The smallest prime number is. 0. An angle less than 0 0 is called. Numbers is used for counting.. A whole number. Down. angle has its own supplement.. () 0. The letter used to denote integers 5

60 Puzzles Number Systems. Vertically opposite angles are 6. The never intersecting lines are called. The first power of 0 is 8. How many zeroes are there in 0 6?. rd power of a number is also known as Matchstick Puzzle Move two matchsticks to change three into six A prime Number Game : Here are seven prime numbers 5,,,,,,. Can you arrange these prime numbers in the seven circles so that the rows and diagonals add up to the same prime number? 60