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Chapter 39 4/30/2017 By Chtan FYHS-Kulai 1 4 2, - 4 ? 4/30/2017 By Chtan FYHS-Kulai 2 : real numbers consist of all positive and negative integers, all rational numbers and irrational numbers. 4/30/2017 By Chtan FYHS-Kulai 3 Rational numbers are of the form p/q, where p, q are integers. Irrational numbers are 3 , e, 2 , 4, 4/30/2017 By Chtan FYHS-Kulai 4 4/30/2017 By Chtan FYHS-Kulai 5 Algebra form z a bi Notation Trigonometric form z r cos i sin index form 4/30/2017 By Chtan FYHS-Kulai z re i 6 Complex numbers are defined as numbers of the form : a b 1 or 4/30/2017 a ib By Chtan FYHS-Kulai 7 -1 is represented by i a,b are real numbers. A complex number consists of 2 parts. Real part and imaginary part. 4/30/2017 By Chtan FYHS-Kulai 8 Notes: a ib 1.When b=0, the complex number is Real 2.When a=0, the complex number is imaginary 3.The complex number is zero iff a=b=0 4/30/2017 By Chtan FYHS-Kulai 9 VENN DIAGRAM Representation • All numbers belong to the Complex number field, C. The Real numbers, R, and the imaginary numbers, i, are subsets of C as illustrated below. Complex Numbers a + bi Real Numbers a + 0i Imaginary Numbers 0 + bi 4/30/2017 By Chtan FYHS-Kulai 11 Conjugate complex numbers The complex numbers and a ib a ib are called conjugate numbers. 4/30/2017 By Chtan FYHS-Kulai 12 z a ib z a ib z is conjugate of z . 4/30/2017 By Chtan FYHS-Kulai 13 e.g. 1 Solve the quadratic equation x x 1 0 2 Soln: 4/30/2017 1 1 4 1 3 x 2 2 1 3i 2 By Chtan FYHS-Kulai 14 e.g. 2 Factorise x y 2 z . 2 Soln: x y 2 z x y iz 2 2 2 x y iz x y iz 4/30/2017 By Chtan FYHS-Kulai 15 Representation of complex number in an Argand diagram 4/30/2017 By Chtan FYHS-Kulai 16 (Im) y P(a,b) a ib x (Re) 0 a ib P’(-a,-b) P a,b a ib Argand diagram 4/30/2017 By Chtan FYHS-Kulai 17 e.g. 3 If P, Q represent the complex numbers 2+i, 4-3i in the Argand diagram, what complex number is represented by the mid-point of PQ? 4/30/2017 By Chtan FYHS-Kulai 18 y (Im) Soln: P(2,1) x (Re) 0 Q(4,-3) Mid-point of PQ is (3,-1) 3 i is the complex number. 4/30/2017 By Chtan FYHS-Kulai 19 i 1 i 1 i 1 3 i i i i i i i 1 i 1 i 1 i i i i i i 2 4 5 4/30/2017 6 7 8 9 By Chtan FYHS-Kulai 10 11 12 13 20 Do pg.272 Ex 20a 4/30/2017 By Chtan FYHS-Kulai 21 Equality of complex numbers 4/30/2017 By Chtan FYHS-Kulai 22 The complex numbers a ib and c id are said to be equal if, and only if, a=c and b=d. 4/30/2017 By Chtan FYHS-Kulai 23 e.g. 4 Find the values of x and y if (x+2y)+i(x-y)=1+4i. Soln: x+2y=1; x-y=4 2y+y=1-4; 3y=-3, y=-1 x-(-1)=4, x=4-1=3 4/30/2017 By Chtan FYHS-Kulai 24 Addition of complex numbers 4/30/2017 By Chtan FYHS-Kulai 25 z1 a ib ; z2 c id If then z1 z2 a c ib d 4/30/2017 By Chtan FYHS-Kulai 26 Subtraction of complex numbers 4/30/2017 By Chtan FYHS-Kulai 27 z1 a ib ; z2 c id If then z1 z2 a c ib d 4/30/2017 By Chtan FYHS-Kulai 28 Do pg.274 Ex 20b 4/30/2017 By Chtan FYHS-Kulai 29 Multiplication of complex numbers 4/30/2017 By Chtan FYHS-Kulai 30 e.g. 5 If z 3 i, find the values of 2 (i) z (ii) z z Soln: (i) z 3 i 3 i 9 1 6i 8 6i 2 (ii) z z 3 i 3 i 9 1 10 4/30/2017 By Chtan FYHS-Kulai 31 z1 a ib ; z2 c id If then z1 z2 a ib c id ac bd iad bc 4/30/2017 By Chtan FYHS-Kulai 32 Division of complex numbers 4/30/2017 By Chtan FYHS-Kulai 33 If then z1 a ib ; z2 c id z1 a ib c id z2 a ib c id c id c id ac bd i bc ad c 4/30/2017 2 By Chtan FYHS-Kulai d 2 34 e.g. 6 Express 2i 3i in the form a ib . Soln: 2 i 2 i 3 i 6 1 5i 3i 3i 3i 9 1 5 5i 1 1 i 10 2 4/30/2017 By Chtan FYHS-Kulai 35 e.g. 7 If x iy 2 i 3 i, find the values of x and y. 4/30/2017 By Chtan FYHS-Kulai 36 e.g. 8 If z=1+2i is a solution of the equation 2 z az b 0 where a, b are real, find the values of a and b and verify that z=1-2i is also a solution of the equation. 4/30/2017 By Chtan FYHS-Kulai 37 The cube roots of unity 4/30/2017 By Chtan FYHS-Kulai 38 If z is a cube root of 1, z 1 or 3 z 1 0 3 4/30/2017 By Chtan FYHS-Kulai 39 z 1 0 3 z 1z 2 z 1 0 z 1 ; z z 1 0 2 1 3i z 2 The cube roots of unity are 1 1 1, 1 i 3 , 1 i 3 2 2 4/30/2017 By Chtan FYHS-Kulai 40 Notice that the complex roots have the property that one is the square of the other, 2 2 1 1 1 2 1 i 3 4 1 i 2 3 3 2 1 i 3 1 1 1 2 1 i 3 4 1 i 2 3 3 2 1 i 3 4/30/2017 By Chtan FYHS-Kulai 41 let 1 3i 2 So the cube roots of unity can be expressed as 1, , 4/30/2017 2 By Chtan FYHS-Kulai 42 If we take then 1 3 i 2 2 1 3 i 2 2 or vice versa. 4/30/2017 By Chtan FYHS-Kulai 43 (1) As z is a solution of 1 z 1 0 3 3 (2) As z is a solution of z z 1 0 2 1 0 2 4/30/2017 By Chtan FYHS-Kulai 44 1 2 (3) (4) (5) 4/30/2017 3n 1 3n … etc 2 3n 1 2 3n By Chtan FYHS-Kulai 1 2 45 e.g. 9 Solve the equation z 1 1 . 3 4/30/2017 By Chtan FYHS-Kulai 46 e.g. 10 If is a cube root of unity, show that 1 4 4/30/2017 2 By Chtan FYHS-Kulai 47 Soln: We have , 1 0 2 1 0 4 3 2 0 2 2 4 2 1 4 4/30/2017 2 By Chtan FYHS-Kulai 3 48 Do pg.277 Ex 20c 4/30/2017 By Chtan FYHS-Kulai 49 4/30/2017 By Chtan FYHS-Kulai 50 z x iy y 0 P(x,y) x y Is called the principal value x Argand diagram r is called the modulus of z, θ(in radians) is called the argument of z. 4/30/2017 By Chtan FYHS-Kulai 51 From the Argand diagram, x r cos , y r sin z x iy r cos ir sin r cos i sin This is called the (r,θ) or modulus-argument form of the complex number 4/30/2017 By Chtan FYHS-Kulai 52 z x iy r cos ir sin r cos i sin This is called the (r,θ) or modulus-argument form of the complex number or modulus-amplitude form of the complex number 4/30/2017 By Chtan FYHS-Kulai 53 x r cos , y r sin r z x y 2 2 y arg z tan x 1 modulus 4/30/2017 argument (or amplitude) By Chtan FYHS-Kulai 54 y arg z tan x 1 OR y am z tan x 1 4/30/2017 By Chtan FYHS-Kulai 55 One important formulae : z z zz 2 2 Refer to Example 15 below 4/30/2017 By Chtan FYHS-Kulai 56 Representation of cube roots of unity in Argand diagram B y 1 3 , 2 2 0 1 3 C , 2 2 4/30/2017 A1,0 2 3 By Chtan FYHS-Kulai x ABC is an equilateral triangle. 57 Geometrically, if P1, P2, P3 represent the number z1, z2 and z1+z2. Then, you see the following diagram : y z1 z1+z2 0 4/30/2017 z2 By Chtan FYHS-Kulai x 58 Multiplication and division of two complex numbers (in modulusargument form) 4/30/2017 By Chtan FYHS-Kulai 59 If z1 r1 cos1 i sin 1 z2 r2 cos2 i sin 2 then z1 z2 r1r2 cos1 2 i sin 1 2 4/30/2017 By Chtan FYHS-Kulai 60 z1 r1 cos1 2 i sin 1 2 z 2 r2 4/30/2017 By Chtan FYHS-Kulai 61 e.g. 11 If find 4/30/2017 z cos i sin 1 ? z By Chtan FYHS-Kulai 62 e.g. 12 If 4i log 3 m 5 , find the value of m . 4/30/2017 By Chtan FYHS-Kulai 63 e.g. 13 If 4/30/2017 1 z i, 1 z then 1 z ? By Chtan FYHS-Kulai 64 e.g. 14 If 4/30/2017 z i 1 i 1 i , then z ? By Chtan FYHS-Kulai 65 Miscellaneous examples 4/30/2017 By Chtan FYHS-Kulai 66 e.g. 15 Evaluate 4 i 6 2i 5 7 ans : 22 14i 4/30/2017 By Chtan FYHS-Kulai 67 e.g. 16 z 8 6i Given find 4/30/2017 2 100 z 16 z z 3 By Chtan FYHS-Kulai , . 68 e.g. 17 z 5 , 3 4i z is an imaginary number, z ? 4/30/2017 By Chtan FYHS-Kulai 69 e.g. 18 If 1 3 i 2 2 , then 1 2 4/30/2017 By Chtan FYHS-Kulai 13 ? 70 e.g. 19 Prove that 3n 4/30/2017 2 3n By Chtan FYHS-Kulai 2 71 e.g. 20 Find the value 1 1 2 2 ans : 4 4/30/2017 By Chtan FYHS-Kulai 72 e.g. 21 Prove that 4/30/2017 3n 1 2 3n 1 By Chtan FYHS-Kulai 1 73 e.g. 22 Simplify cos i sin 3 cos 2 i sin 2 4/30/2017 By Chtan FYHS-Kulai 74 e.g. 23 If z1 r1 cos1 i sin 1 z2 r2 cos2 i sin 2 Show that z1 z2 r1r2 4/30/2017 By Chtan FYHS-Kulai 75 e.g. 24 If z cos i sin Prove that 1 z i tan 2 1 z 2 4/30/2017 By Chtan FYHS-Kulai 76 e.g. 25 If z cos i sin Prove that 4/30/2017 1 1 By Chtan FYHS-Kulai 1 2 z i tan 1 2 z 77 Addendum 4/30/2017 By Chtan FYHS-Kulai 78 (1) In general, z is a complex number then, z a represent a circle with centre at (0,0) and radius “a”. 4/30/2017 By Chtan FYHS-Kulai 79 (2) amz represent a straight line with gradient=tanθ. 4/30/2017 By Chtan FYHS-Kulai 80 (3) If 4 points P1, P2, P3, P4 are concyclic, then z3 z1 z 4 z1 am am z3 z 2 z4 z2 4/30/2017 By Chtan FYHS-Kulai 81 y P2 P1 P3 0 P4 x P1, P2, P3, P4 are concyclic 4/30/2017 By Chtan FYHS-Kulai 82 (4) If 3 points P1, P2, P3 formed an equilateral triangle, z2 z3 z3 z1 z1 z2 2 4/30/2017 2 By Chtan FYHS-Kulai 2 0 83 Do pg.280 Ex 20d & Misc 20 4/30/2017 By Chtan FYHS-Kulai 84 Do pg.127 Ex 6a Pg. 130 Ex 6b Pg. 138 Ex 6d q1-q10, q12, q14, q16 No need to do Pg. 135 Ex 6c, pg. 139 Misc 4/30/2017 By Chtan FYHS-Kulai 85 The end 4/30/2017 By Chtan FYHS-Kulai 86