=> √10 = √(3 2 1) Construction 1 Take a line segment AO = 3 unit on the xaxis (consider 1 unit = 2cm) 2 Draw a perpendicular on O and draw a line OC = 1 unit 3 Now join AC with √10 4 Take A as center and AC as radius, draw an arc which cuts the xaxis at point E 5 The line segment AC represents √10 unitsThe point C shows the number √3 2 Represent √5 on the number line Solution Let us draw a number line, mark the center as point O and mark a point Q at number 2 such that it is 2cm from the center ie, l(OQ) = 2 units Now, draw a line QR perpendicular to the number line through the point Q such that l(QR) = 1 unit Draw seg ORThen, extend a line from 0 to the point you just plotted That line is the visual representation of the number 32i Some other properties are represented by the line on the Argand diagram The length of the line represents the modulus of the number √(3 2 2 2) = √13 The line also forms an angle with the positive side of the real axis
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Represent 3 2/5 on the number line- √5 on the number line 5 can be written as the sum of the square of two natural numbers ie, 5 =1 4 =1 2 2 2 On the number line, Take OA = 2 units Perpendicular to OA, draw BA = 1 unit Join OB Using Pythagoras theorem, We have, OB= √5 Draw an arc with centre O and radius OB using a compass such that it intersects the number line Finding √2 On The Number Line 1 Finding √2,√3 and√4 on the number line An animated geometric construction 2 a 2 b 2 =c 2 1 2 1 2 =11=2 So Hypotenuse = √2 This line is 1 unit long 3 a 2 b 2 =c 2 ( √2) 2 1 2 =21=3 So Hypotenuse = √3 4 a 2 b 2 =c 2 ( √3) 2 1 2 =31=4 So Hypotenuse = √4 5
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Now, to represent √32 on the number line Let us take the line BC as number line and point 'B' as zero, point 'C' as '1' and so on Draw an arc with centre B and radius BD, which intersects the number line at point 'E' Then, the point 'E' represents √32 Second method Construction method (fig2) Let A represent number 2, B represent 3 , on the number line Draw a segment BC perpendicular to segment AB & length same as AB (=1) Now ABC is a right angle triangle with sides AB & AC have unit lengths (Constructed) By Pythagoras theorem, diagonal AC length is sqrt (1^2 1^2)= sqrt 2 AC= sqrt 2Exercise 15 Question 1 Classify the following numbers as rational or irrational Solution (i) Since, it is a difference of a rational and an irrational number ∴ 2 – √5 is an irrational number (ii) 3 – = 3 – = 3 which is a rational number (iii) Since, = = , which is a rational number
=> √10 = √(3 2 1) Construction 1 Take a line segment AO = 3 unit on the xaxis (consider 1 unit = 2cm) 2 Draw a perpendicular on O and draw a line OC = 1 unit 3 Now join AC with √10 4 Take A as center and AC as radius, draw an arc which cuts the xaxis at point E 5 The line segment AC represents √10 units => √12 = √{32 (√3)2 } So, for representing √12 on number line, first we have to represent √3 on number line Now, √3 = √(2 1) => √3 = √{(√2)2 12 }The value √ 3 2 ≈ 0866 is greater than the value 1 2 = 05 For t = π 6, the xcoordinate is greater than the ycoordinate Therefore, x must be √ 3 2For t = π 3, the xcoordinate is less than the ycoordinate Therefore, x must be 1 2For special points in Quadrants II, III, and IV, use the values 1 2, √ 3 2, ∧ √ 2 2 with
This is easy Transfer Fig 16 onto the number line making sure that the vertex O coincides with zero (see Fig 17) We have just seen that OB = √2 Using a compass with centre O and radius OB, draw an arc intersecting the number line at the point P Then P corresponds to √2 on the number line (3 √3) 2 is a rational number (b) (5 – √5) 2 = (5) 2 (√5) 2 – 2×5×√5 On further calculation, we get, = 25 5 – 10√5 = 30 – 10√5 which is a irrational number Therefore, (5 – √5) 2 is an irrational number Represent the number √7 on the number line Answer d = 12/√(3^24^2) = 12/5 = 24 The other way is to realize that the shortest distance from a point to a line is along the line through the point and perpendicular to the given line Since our line has slope 4/3, the perpendicular line has slope 3/4 SO, the normal line is y = 3/4 x The two lines intersect where3/4 x = (4x12)/3
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Representation Of Irrational Numbers On The Number Line 2 On Number Line
1 Irrational Numbers on the Number Line Where are the points on the number line that correspond to the irrational numbers?SOLUTION Solution A, B, C, and D The product of two irrational numbers will be either a rational or an irrational number Consider the following example (√3√2)×(√3−√2)= 1 Both (√3√2) and (√3−√2) are irrational numbers and their product is 1 1 is an integer and since, all integers are rational numbers as well, weWrite 5 rational numbers between 5 & 6 7 Write 4 irrational numbers between √ & √ 8 Show that 1 ⃐ can be in the form (rational number) 9 √Is an irrational number Give reason 10 Represent on a number line √(a) 2 √(b) √5 (c) 10 11 Represent 4735 using successive magnification 12
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For each pair of numbers, write two expressions to represent the distance between the numbers on a number line, then determine this distance a) b) 2 Arrange in order from least to greatest a) b) 5 8, √ 72 50, 2 √ 1 16, √ 9 5 3 √ 6, √ 24, 2 √ 6, √ 96 22 1 75 and 375 3 8 and 3 1 4 21 Each radical has index 2Write eachAnswer From the definition of the Argand diagram, we know that the complex number 𝑧 = 𝑎 𝑏 𝑖 will be represented by a point with Cartesian coordinates ( 𝑎, 𝑏) Hence, 𝑍 will be represented by the point 𝐴 ( 8, 1) In our next example, we will identify complex numbers and Ans i) Irrational numbers are √2, √3, √5, √3 etc, which can be shown in real number line therefore every irrational number is a real number ii) All positive numbers as an ex √1, √2, √3, √4 can be represented on the number line but the square root of a negative number, √ (–m ) does not existso it is false
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Q5 Show that 3√5 is an irrational number Ans Let us assume 3√5 is a rational number, 3√5 = p/q, where p and q are coprimes, √5 = p/3q Clearly √5 is irrational, while number on right q ≠ 0 are rational ∴ Irrational = Rational But above deduced can't be right Therefore our supposition is wrong making 3√5 an irrational number Which number line represents the solution set for the inequality x 24?Using Pythagoras theorem We can write the √10 as below √10 = (√91) This can be written as √10 = (√3 2 1 2) The construction steps are shown below Take a line segment AO = 3 units on x axis Here consider 1 unit = 2 cm Draw a perpendicular on O, name it as OC such that OC = 1 unit
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2 √3 = 2 √3 Rational Number Irrational Number = Irrational Number Eg 7 √3 = 7 √3 Non 0 Rational Number / Irrational Number = Irrational Number 3 / √3 = √3 Class 9 Exercise 11 Number System (Multiple Choice Questions) Write the correct answer in each of the following Q1 Every rational number is A) a natural number B) an integer C) a real number D) a whole number Q8 A rational number between √ 2 and √ 3 is;√32(B) Describe the mathematical relationships found in the base10 place value system through the hundred thousands place √34(I) Determine if a number is even or odd using divisibility rules Processing Standards √31(A) Apply mathematics to problems arising in
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