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Real numbers

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Real numbers

1. 1. Chapter 1 Real numbers Natural Numbers: The counting numbers such as 1, 2, 3, 4, 5 ... are called natural numbers. The set of natural numbers is denoted by N, i.e., N = {1, 2, 3, 4, 5 ...} Whole Numbers: The number 0 and the natural numbers are together called whole numbers. The set of whole numbers is denoted by W, i.e., W = {0, 1, 2, 3, 4, 5 ...} = N {0} Integers: The negative natural numbers and the whole numbers are together called integers. The set of integers is denoted by Z or I, i.e., Z = {... 2, 1, 0, 1, 2, ...} Decimals: A decimal number is simply written by placing a dot between any of the consecutive digits of an integer. E.g., 4580.937 is a decimal number. p Any number which can be expressed in the form , where p and q are integers and q 0, q is called a rational number. 1 4 22 For example, , , , etc., are called rational numbers. These are also called fractions. 2 7 27 p The numbers that cannot be written in the form , where p and q are integers and q 0, q are called irrational numbers. Numbers such as 2 , 15 , 7 , , 3.0214567893215…. are some examples of irrational numbers. All rational and irrational numbers are together called real numbers. Two more important properties satisfied by real numbers are: (1) The set of real numbers is ordered, i.e., if a, b R, then either a < b or a > b or a = b a b a b (2) If a, b R, then R and a b (if a < b) 2 2 A rational number is in the standard form, if the denominator is a positive integer and the only common factor between the numerator and the denominator is 1. 3 12 4 Therefore, , and are in the standard form. 5 7 15 If a rational number is not in the standard form, then it can be reduced to the standard form by dividing the numerator and the denominator by their highest common factor (HCF).
2. 2.  Equivalent rational numbers have the same standard form. 5 25 E.g., and are equivalent rational numbers. 6 30 a c a+c If and are two rational numbers, then is a rational number that lies between b d b+ d the given rational numbers. 1 2 3 1 2 E.g., the rational number lies between and . 9 7 16 9 7 Two positive rational numbers can be compared in fractions. wo negative rational numbers can be compared by ignoring their negative signs and then reversing their order. A decimal expansion is terminating if there is a finite number of digits after the decimal point whereas a decimal point is repetitive and non-terminating if there is no definite number of digits occurring after the decimal. 2 0.666666... is a repetitive and non-terminating decimal 3 12 1.5 is a terminating decimal 8 Every fraction with denominator 10 or 100 or 1000 can be easily converted into decimal form. Example: 56 50 6 6 5 5.6 10 10 10 291 200 91 91 2 2.91 100 100 100 A fraction whose denominator is not 10 or 100 or 1000 can be converted into decimal form by multiplying the numerator and denominator by the same number such that the denominator becomes 10 or 100. Example: 41 41 5 205 200 5 5 2 2.05 20 20 5 100 100 100 9 9 2 18 10 8 8 1 1.8 5 5 2 10 10 10 A decimal in which after a certain point, some digits or a group of digits repeat over and over again is called recurring decimal. We denote the recurring decimal by putting a bar (or dots) on the group of digits that are repeating.
3. 3. Example: Consider the division of 8 by 3. The number 8 can be divided by 3 as follows: 2.666 3 8 6 20 18 20 18 20 18 2 8 2.666... 2.6 3 13 40 Similarly, 0.13 and 13.3 99 3 Conversely, a recurring decimal can be converted into a fraction. This is shown in the following example. Example: Find the fraction represented by the decimal 2.6 Solution: Let x = 2.6 x = 2.666… … (1) On multiplying equation (1) by 10 on both sides, we obtain 10x = 26.666… … (2) On subtracting equation (1) from equation (2), we obtain 9x = 24 24 8 x= 9 3 8 Thus, 2.6 3 If x is a rational number with terminating decimal expansion, then it can be expressed in the p form, where p and q are co-prime (the HCF of p and q is 1) and the prime factorisation of q q is of the form 2n5m, where n and m are non-negative integers.
4. 4. Contrary to this, if the prime factorisation of q is not of the form 2n5m, where n and m are non-negative integers, then the decimal expansion is a non-terminating one. . 7 7 1 1 2 0.58333... 0.0714285714... 12 2 3 14 7 2 p Let x = be any rational number. q (i) If the prime factorisation of q is of the form 2n5m, where n and m are non-negative integers, then x has a terminating decimal expansion. (ii) If the prime factorisation of q is not of the form 2n5m, where n and m are non- negative integers, then x has a non-terminating and repetitive decimal expansion. 17 17 1600 2 52 6 As the denominator can be written in the form 2 n5m, where n = 6 and m = 2 are non-negative integers, the given rational number has a terminating decimal expansion. The four mathematical operations are applied to rational numbers in the same way as in fractions, using BODMAS. E.g., 5 1 10 3 13 (i) 6 4 12 12 1 1 1 1 1 5 (ii) 2 4 5 2 4 1 1 5 2 4 1 4 2 2 5 5 Properties of addition and subtraction of rational numbers: i. If a and b are any two rational numbers and a + b = c, then c will always be a rational number. ii. If a and b are any two rational numbers and a – b = c, then c will always be a rational number. iii. If a and b are any two rational numbers, then a + b = b + a. iv. If a, b and c are any three rational numbers, then a + (b + c) = (a + b) + c. Properties of multiplication of rational numbers: i. a 0=0 ii. a 1 = a ; 1 is the multiplicative identity of the rational number. iii. If x, y and z are any three rational numbers, then x (y + z) = (x y) + (x z).
5. 5. iv. If x, y and z are any three rational numbers, then x (y – z) = (x y) – (x z). v. If the multiplication of two numbers gives the result as 1, then the two numbers are called reciprocals or multiplicative inverses of each other. Two irrational numbers m a and n b can be compared directly when their powers are equal or their bases are equal. If the powers are not equal, then we make the powers equal. Example: Compare the numbers 2 and 3 5. Solution: 1 1 2 2 and 5 5 2 3 3 We first make the exponents of both the numbers equal. We do this by taking the LCM of the denominators of the exponents. LCM of 2 and 3 is 6. 1 1 3 3 2 2 3 6 1 1 2 2 3 3 2 6 1 3 1 1 2 22 26 23 6 86 6 8 1 2 1 1 3 3 6 2 6 6 6 5 5 5 5 25 25 Now, the exponents of both the numbers are equal. So, they are comparable. Therefore, 6 8 6 25 because 8 25. Another method for comparing irrational numbers is by making their bases equal. For example, consider the irrational numbers 3 4 and 4 8 . 1 1 2 Here, 3 4 43 22 3 23 1 1 3 4 8 84 23 4 24 2 3 Now, we know that 3 4 2 3 23 24 3 4 4 8 To find irrational numbers between rational numbers, we follow two steps. Step 1 Find the decimal representation (up to 2 or 3 places of decimal) of any two rational numbers. Let those decimal representations be a and b and let a < b. Step 2
6. 6. Choose the required non-repetitive and non-terminating decimal numbers between a and b. Let us take an example. Example: Find an irrational number between 0.12 and 0.15. Solution: The rational numbers are already in the decimal form. Let a = 0.12 and b = 0.15 We can see that a < b. There are infinite number of irrational numbers between a and b. Clearly, the number 0.12101001000 … lies between 0.12 and 0.15. Hence, the required irrational number between 0.12 and 0.15 is 0.12101001000 … Rounding off to decimal places: In order to round off a given number to a certain number of decimal places, we follow the steps given below. (i) Work out the number to one more place than what is required. (ii) If the extra digit is 5 or more, then add 1 to the number before it. The number obtained will be the required number. (iii) If the extra digit is less than 5, then do not change the number before it. That will be the required number. For example, 82.4912 can be rounded off correct to one decimal place as 82.5. Some rules in determining the significant figures in a number are as follows: (a) All digits in a whole number are significant. For example, 428 has three significant figures. (b) If a number is greater than 1, then all the non-zero digits and all the zeros between them are significant. For example, 5.204012 has seven significant figures. (c) If a number lies between 0 and 1, then the initial zeroes are not significant. For example, 0.0042 has two significant figures. (d) The final zero(s) of an approximated number when expressed as a decimal is/are significant. For example, the number 48.4956 when approximated to two decimal places is 48.50. Here, the number 48.50 has four significant figures. Rounding off to significant figures To round off a given decimal number to a certain number of significant figures, we proceed in the same way as we do while rounding off to certain number of decimal places. For example, 6.0809 can be rounded off correct to four significant figures as 6.081. By using Pythagoras Theorem, an irrational number can be represented on the number line. Example: Locate 6 on the number line.
7. 7. Solution: It is seen that 2 6 5 12 To locate 6 on the number line, we first need to construct a length of 5 . 5 22 1 By Pythagoras Theorem, OB2 OA 2 AB2 22 12 4 1 5 OB 5 Steps: a. Mark O at 0 and A at 2 on the number line, and then draw AB of unit length perpendicular to OA. Then, by Pythagoras Theorem, OB 5 b. Construct BD of unit length perpendicular to OB. Thus, by Pythagoras Theorem, 2 OD 5 12 6 c. Using a compass, with centre O and radius OD, draw an arc intersecting the number line at point P. d. Thus, P corresponds to the number 6 . Representation of real numbers on the number line Example: Visualise 3.32 on the number line, up to 4 decimal places. Solution:
8. 8. 3.32 3.3232...... 3.3232 approximate up to 4 decimal places Now, it is seen that 3 < 3.3232 < 4. Divide the gap between 3 and 4 on the number line into 10 equal parts and locate 3.3232 between 3.3 and 3.4 [as 3.3 < 3.3232 < 3.4]. To locate the given number between 3.3 and 3.4 more accurately, we divide this gap into 10 equal parts. It is seen that 3.32 < 3.3232 < 3.33. We continue the same procedure by dividing the gap between 3.32 and 3.33 into 10 equal parts. It is seen that 3.323 < 3.3232 < 3.324. Now, by dividing the gap between 3.323 and 3.324 into 10 equal parts, we can locate 3.3232.Contribute to this Revision Note:If you find anything of importance missing from this note, email it to us at revision-notes@meritnation.com,and we’ll add it to this note under your name!