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Number System Chapter 1

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  1. Number SystemChapter 1 9th Grade - Mathematics

  2. Unit 1 –Square Roots • Objectives • Identify square numbers • Find the square root of a perfect square number by factor method • Find the square root of perfect square numbers, decimals and numbers which are not perfect squares by division method • Find the smallest number that has to be added to or subtracted from a given number to make it a perfect square number • Explain the process of finding the square root by division method

  3. Recalling • Squares and Square roots • If a number is multiplied by itself, the product so obtained is called the square of that number, or a perfect square number is the product of two equal factors. Example : 2 x 2 = 4 (—2) x (—2) = 4 • The square root of a given positive real number will have two values : positive and negativewhereas the symbol is used to indicate positive square root only. Example is +2 or—2

  4. Recalling • Squares and Square roots • If a perfect square number has ‘N’ digits then its square root has digits when ‘N’ is even digits when ‘N’ is odd • We can find the square root of a number by using the factor method

  5. Squares & Square roots - Exercises • Find the squares for the following numbers 15 0.6 0.22 0.0082 5 / 14 102/61 • Express the following in n2form 25 10,00,000 • Find the square root using factor method 254842704980138025211600 • Find the value of the following +- x

  6. Square root using Division Method • Recall Factor Method • Procedure • First find prime factors • Write number as product of prime factors • Pair equal primes • Ex: 47,43,68,400 = 2x2 x 2x2 x 3x3 x 3x3 x 5x5 x 11x11 x 11x11 • Write square root • Ex: = 2x2x3x3x5x11x11 = 21780 • Limitations • Lengthy • Not convenient for very large numbers • Unsuitable for numbers that are not perfect squares • Ex: 5 8 10 11

  7. What is Division Method ? • Starting from the unit place and moving to the left, group two digits at a time. • Place a bar over pairs of digits, which forms a group and the left over single digit forms another group. • Begin the division process from the highest place value. In 361, 3 is in the highest place. • The largest square less than or equal to the first group i.e., 3, it is 1. • Take the square root of 1, i.e., 1 as the divisor and quotient, write this quotient 1 above the first group 3

  8. … What is Division Method ? • Find the product of the divisor and quotient, i.e. 1 x 1 = 1 • Subtract this product from the first group, i.e., 3. The remainder is 2 • Bring down the next group 61 to the right of remainder 2. • Double the quotient 1. It is 1+1 = 2 • Enter it in the divisor column with a blank space on the right for the next digit

  9. … What is Division Method ? • Find the next digit in the quotient. Let the digit be ‘a’. • Find ‘a’ such that 2a x a = 261 • Since 261 has a ‘1’ in its units place ‘a’ should be either 1 or 9 • Check: 21 x 1 = 21 29 x 9 = 261 • Hence a = 9 • Therefore, Write 9 in both the divisor and quotient

  10. … What is Division Method ? • Multiply the new digit 9 in the quotient with the second divisor 29 • 9 x 29 = 261 • Now subtract 261 from 261 giving remainder ‘0’ • Therefore, the division process is completed. • The quotient obtained by this division process is 19. • This is the square root of 361 • Therefore, = 19

  11. … What is Division Method ? • Using the following table guess the digit in the units place of the square root of a given square number

  12. Division Method - Examples Example : Note: If the number is large, then continue the steps 3,4,5 and 6 till the last group has been dealt with. The available quotient will be the square root of the given number, In this method, the groups have been made by placing the bars over each pair of digits. The number of bars (or groups) will be equal to the number of digits in the square root.

  13. Division Method - Examples Example : Note: Note : To find the third divisor, double the digit in units place of the second divisor, i.e., 45=40+5 Therefore 40 + 5 x 2 = 50 Or Simply add 5 to 45 i.e., 45 + 5 = 50 then find the next digit i.e.. 8

  14. Division Method - Examples Example :

  15. Division Method - Examples Example : Find Side of a square with area 55,225 square metres Area of Square= side x side = side2 side2 = 55225 side = side = 235 mts

  16. Division Method – Application • Find the least number that should be subtracted from a number to make it a perfect square, For Example – 2361 • Using Division Method the remainder is 57 • So 57 is the least number that should be subtracted from 2361 to make it a perfect square 2304 whose square root is 48

  17. Division Method – Application • Find the least number that should be added to a number to make it a perfect square, For Example – 7390 • Using Division Method the remainder is 165 and quotient 85 • Since remainder is great than 0 the number is between 852 and 862. • So the least number is 862 – 7390 = 6 • So 6 is the least number that should be added to 7390 to make it a perfect square 7396 whose square root is 86

  18. Division Method - Exercises • Find the number of digits in the square root of these perfect squares 225 2304 6924 17424 44100 75625 • Square root using division method 441 3844 99856 219024 • Least number to subtract from these to make it perfect square 27092815423550847401 • Least number to be added to these to make it perfect square 11513479636082809215

  19. Exercises - Application • Length of square whose are is 41,209 SqMts • Commander wants army of 15,376 soldiers to assemble in square form. How many soldiers in each row ? • 13456 students are sitting in a stadium where the number of students per row is same as number of rows. How many rows ? • Gardener arranges plants in rows to form a square. Doing this 5 plants are left out. If total plants are 64014 then find number of plants in each row ?

  20. Square root of Decimal Numbers • Some decimal numbers are their squares are given. Observe the number of digits after the decimal point in the square. What do you observe ? • We can conclude that number of digits in the decimal part of a decimal square is twice the number of digits in the decimal part of the decimal number.

  21. Arriving at square root of decimal numbers • Take an example • Ignoring decimal point we get is 144 • Find which is 12 • Place the decimal point using the property we saw in the above rules i.e. = 1.2 • Another example • Ignoring decimal point we get is 36 • Find which is 6 • The square root should have 4 digits in the decimal parti.e. = 0.0006

  22. Square root of decimals by division method • First step is to group the digits into given number of pairsEx: 252.70729 • For whole numbers we move from units digits to the lefti.e. 252  2 52 • In case of decimal we move from decimal to the righti.e. .70729  7 07 29 • After grouping the process of finding the square root is the same as for whole numbers

  23. Square root of decimals by division method • Grouping the digits in pairs • The square root of 81 i.e., the first group is 9. Write 9 in both divisor and quotient columns. Now multiply these 9s and enter the product 81 below the first group.

  24. Square root of decimals by division method • Subtract 81 from 81, giving a remainder 0.Now bring down the next group 54.Since the integral part is exhausted and 54 is the decimal part of the number, place a decimal point in the quotient.i.e. 9 .

  25. Square root of decimals by division method • Double the quotient 9 and enter it in the divisor column. Since we have to write a new digit in the divisor column and by doing so, the divisor will be more than the dividend 54, we bring down the next group 09 also. So place one zero in the quotient column and one zero in the divisor column. Bring down the next group 09.

  26. Square root of decimals by division method • Let the new digit in the quotient be ‘a’find a such that 180ax a = 5409x is either 3 or 9check: 1803 x 3 = 5409Write 3 in both divisor and quotient columns.Subtracting 5409 we get ‘0’.Quotient: 9.03 which is

  27. - Examples Example : Note: There are two zeros after decimal point. Hence write 0 after decimal point of the quotient and continue the steps of finding square root by division method.

  28. - Exercises • Determine the number of digits in the decimal part of the square root: 84.8241 0.008281 1227.8016 144.0976312516 • Find the square root of the following by division method:73.96235.31560.000017640.00002116 Application of concept • Find length of one side of square play ground with area 150.0625 square metres • A number is multiplied by itself giving a product 47.0596 • Find the perimeter of a square garden whose area is 1227.8016 square metres

  29. Finding square root of non-perfect squares Find the square root of 2 • Think of the largest square less than or equal to 2. It is 1. • Take its square root. = 1 • Write 1 as divisor and also as quotient. • The product of the divisor and quotient is 1 • Write 1 below 2 and subtract • The remainder is 1

  30. Finding square root of non-perfect squares • Double the quotient 1. It will be 2. Write 2 in the divisor column (Note: The divisor 2 is bigger than the remainder 1. In this situation we cannot proceed further as the remainder is smaller than the divisor. In order to proceed further to find the square root of 2, we have to write a pair of zeroes after placing a decimal point) • Place a decimal point after 1 in the quotient. Write two zeroes after placing a decimal point next to the dividend 2.Bring down these two zeros (a group)

  31. Finding square root of non-perfect squares • Let the new digit in the quotient and in the divisor column be ‘a’Find ‘a’ such that 2a x a ≤ 100By Inspection, we will find that 24 x 4 = 96 < 100 • Subtract 96 from 100. The remainder is 4. Double the quotient without considering the decimal point.

  32. Finding square root of non-perfect squares Note: Again we have arrived at a situation, where we cannot proceed further as the remainder is smaller than the divisor. To proceed further, write two more zeroes after 2. • Bring down these two zeroes (i.e., a group). Let the new digit in the quotient and in the divisor column be ‘a’. Now find ‘a’ such that28a x a ≤ 400 • The new digit in the quotient column and in the divisor column will be 1. Write the product of 281 and 1 below 400.

  33. Finding square root of non-perfect squares • Subtract 281 from 400. Double the quotient without considering the decimal point and write it in the divisor column. • As the remainder is smaller than the divisor, write two more zeroes, after 2.0000. Bring down these two zeroes. (i.e., a group)

  34. Finding square root of non-perfect squares • Find a such that, 282a x a <= 11900 By inspection we find that a = 4write this 4 both in the quotientand in the divisor columns. • Subtract the product of 2824and 4 from 11900 Note: Since is an irrational number, the division process will be non-terminating and non-recurring. Hence we continue the division process for required number of decimal places. i.e., If the square root value is required for ‘n’ decimal places we continue the division process up to (n+ 1) decimal places and then correct it to ‘n’ places. In the above example the quotient found is 1.414 which has 3 decimal places. We can correct it to 2 decimal places as 1.41. = 1.41 (correct to 2 decimal places)

  35. - Examples Example : corrected to 2 places Example : corrected to 2 places Note : We have to continue division up to 3 decimal places. Already there is one pair of digits after decimal point. Therefore we take two more pairs of zeroes. • = 2.236 up to 3 decimal places • = 2.24 corrected to 2 places

  36. - Exercises • Find the square root of the following corrected to 2 decimal places: 3 6 7 10 11 20 24 25.36 0.8 0.0041 Application of concept • Find approximate length (correct to 1 decimal place) of square plot whose area is 325 sq mts • Find the approximate length of the side of a square whose area is 12.0068 sq mts (correct to 2 decimal places)

  37. Unit 2 – Real Numbers • Objectives • Recall different sets of numbers • Differentiate the numbers as rational and irrational • Realisethe need of a new set of numbers called set of real numbers • Explain the meaning of real numbers • Explain the properties of real numbers

  38. Recalling Different Sets of Numbers • W is the set of Whole Numbers written asW = {0, 1, 2, 3, 4, 5 . . . . . } • The set of whole numbers is not closed under operation subtraction.For example 2-10 = -8 • This problem can be understood by defining the set of Integers, which is representedby Z.Z = { . . . . -4, -3, -2, -1, 0, I, 2, 3, 4, . . . .)Now, we have 2 – 10 = - 8 Zwe also know that W  Z • A further extension of set of integers is needed, because many division problems withintegers do not have answer in Z.

  39. Rational Numbers • = 2 is not an integer. To include such numbers consider the set of rational numbers defined as follows: • A Number is called rational number if it can be written in the form where a,b  Z and b  0 (also a and b have no common factor other than 1) • Examples: = 0.1 = 0.5 = 3.33… etc • Therefore a rational number is either a terminating decimal or a non-terminating recurring decimal • Now: W Z Q

  40. Irrational Numbers • There are some numbers such as ,,, , etc. which cannot be expressed by the form a/b (where a and b are integers and b  0) • Such numbers are called irrational numbers • A Number is called irrational number if it has non-terminating and non-recurring decimals • Examples: = 1.414213…… = 2.64575…… = 3.1416……

  41. Real Numbers • The two sets of numbers discussed earlier, Rational and Irrational numbers are grouped into one set called ‘Real Numbers’ • Therefore the set of real numbers ‘R’ contains all terminating decimals, non-terminating recurring decimals and non-terminating non-recurring decimals • This entire set of numbers represented by decimals is called the set of real numbers • To every point on the number line there is a corresponding real number and vice-versa • Thus the collection of all points on the number line can be thought of as the system of real numbers There is nonumber, that is bothrational and irrational.

  42. Relationship between numbers – Venn Diagram The relationship between the set of real numbers and its subsets can be represented using Venn diagrams as shown below.

  43. Relationship between numbers – Flow Diagram The relationship between the set of real numbers and its subsets can be represented using Venn diagrams as shown below.

  44. Properties of real numbers • Properties of Equality • If a = a, Reflexive property • If a = b and b = a, Symmetric property • If a = b and b = c then a = c, Transitive property • If a = b then a+c = b+c and ac = bc • If ac = bc and c  0, then a = b

  45. … Properties of real numbers

  46. … Properties of real numbers

  47. … Properties of real numbers

  48. … Properties of real numbers

  49. Real Numbers - Exercises • State the basic property of R used in each of the following statements:

  50. Real Numbers - Exercises • a. Give the additive inverse of each of the following:b. Give the multiplicative inverse of each of the following: • Does the commutative property hold good on the set R for the following operations. a) Subtraction b) Division. Justify your answer. • Is addition distributive over multiplication? Justify your answer.