1/31/20161By Chtan FYHS-Kulai Chapter 39 1/31/20162By Chtan FYHS-Kulai 1/31/20163By Chtan FYHS-Kulai consist of all positive and negative integers, all rational numbers and irrational numbers. 1/31/20164By Chtan FYHS-Kulai Rational numbers are of the form p/q, where p, q are integers. Irrational numbers are 1/31/2016By Chtan FYHS-Kulai5 1/31/2016By Chtan FYHS-Kulai6 Notation Algebra form Trigonometric form index form 1/31/20167By Chtan FYHS-Kulai Complex numbers are defined as numbers of the form : or 1/31/2016By Chtan FYHS-Kulai8 is represented by are real numbers. A complex number consists of 2 parts. Real part and imaginary part. 1/31/2016By Chtan FYHS-Kulai9 Notes: 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 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. Real Numbers a + 0i Imaginary Numbers 0 + bi Complex Numbers a + bi 1/31/2016By Chtan FYHS-Kulai11 1/31/2016By Chtan FYHS-Kulai12 Conjugate complex numbers The complex numbers and are called conjugate numbers. 1/31/2016By Chtan FYHS-Kulai13 is conjugate of. 1/31/2016By Chtan FYHS-Kulai14 e.g. 1 Solve the quadratic equation Soln: 1/31/2016By Chtan FYHS-Kulai15 e.g. 2 Factorise. Soln: 1/31/2016By Chtan FYHS-Kulai16 Representation of complex number in an Argand diagram 1/31/2016By Chtan FYHS-Kulai17 x y 0 P(a,b) P(-a,-b) Argand diagram (Re) (Im) 1/31/2016By Chtan FYHS-Kulai18 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? 1/31/2016By Chtan FYHS-Kulai19 Soln: x y 0 P(2,1) Q(4,-3) Mid-point of PQ is (3,-1) is the complex number. (Re) (Im) 1/31/2016By Chtan FYHS-Kulai20 1/31/2016By Chtan FYHS-Kulai21 Do pg.272 Ex 20a 1/31/2016By Chtan FYHS-Kulai22 Equality of complex numbers 1/31/2016By Chtan FYHS-Kulai23 The complex numbers and are said to be equal if, and only if, a=c and b=d. 1/31/2016By Chtan FYHS-Kulai24 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 1/31/2016By Chtan FYHS-Kulai25 Addition of complex numbers 1/31/2016By Chtan FYHS-Kulai26 If then 1/31/2016By Chtan FYHS-Kulai27 Subtraction of complex numbers 1/31/2016By Chtan FYHS-Kulai28 If then 1/31/2016By Chtan FYHS-Kulai29 Do pg.274 Ex 20b 1/31/2016By Chtan FYHS-Kulai30 Multiplication of complex numbers 1/31/2016By Chtan FYHS-Kulai31 e.g. 5 If, find the values of (i) (ii) Soln: (i) (ii) 1/31/2016By Chtan FYHS-Kulai32 If then 1/31/2016By Chtan FYHS-Kulai33 Division of complex numbers 1/31/2016By Chtan FYHS-Kulai34 If then 1/31/2016By Chtan FYHS-Kulai35 e.g. 6 Express in the form. Soln: 1/31/2016By Chtan FYHS-Kulai36 e.g. 7 1/31/2016By Chtan FYHS-Kulai37 e.g. 8 If z=1+2i is a solution of the equation 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. 1/31/2016By Chtan FYHS-Kulai38 The cube roots of unity 1/31/2016By Chtan FYHS-Kulai39 If is a cube root of 1, 1/31/2016By Chtan FYHS-Kulai40 1/31/2016By Chtan FYHS-Kulai41 Notice that the complex roots have the property that one is the square of the other, 1/31/2016By Chtan FYHS-Kulai42 let So the cube roots of unity can be expressed as 1/31/2016By Chtan FYHS-Kulai43 If we take then or vice versa. 1/31/2016By Chtan FYHS-Kulai44 (1) As is a solution of (2) As is a solution of 1/31/2016By Chtan FYHS-Kulai45 (3) (4) (5) etc 1/31/2016By Chtan FYHS-Kulai46 e.g. 9 Solve the equation. 1/31/2016By Chtan FYHS-Kulai47 e.g. 10 1/31/2016By Chtan FYHS-Kulai48 Soln: 1/31/2016By Chtan FYHS-Kulai49 Do pg.277 Ex 20c 1/31/201650By Chtan FYHS-Kulai 1/31/2016By Chtan FYHS-Kulai51 x y 0 P(x,y) x y r Is called the principal value Argand diagram r is called the modulus of z, (in radians) is called the argument of z. 1/31/2016By Chtan FYHS-Kulai52 From the Argand diagram, This is called the (r,) or modulus-argument form of the complex number 1/31/2016By Chtan FYHS-Kulai53 This is called the (r,) or modulus-argument form of the complex number or modulus-amplitude form of the complex number 1/31/2016By Chtan FYHS-Kulai54 modulus argument (or amplitude) 1/31/2016By Chtan FYHS-Kulai55 OR 1/31/2016By Chtan FYHS-Kulai56 One important formulae : Refer to Example 15 below 1/31/2016By Chtan FYHS-Kulai57 Representation of cube roots of unity in Argand diagram y x 0 is an equilateral triangle. 1/31/2016By Chtan FYHS-Kulai58 Geometrically, if P1, P2, P3 represent the number z1, z2 and z1+z2. Then, you see the following diagram : y x 0 z2 z1 z1+z2 1/31/2016By Chtan FYHS-Kulai59 Multiplication and division of two complex numbers (in modulus- argument form) 1/31/2016By Chtan FYHS-Kulai60 If then 1/31/2016By Chtan FYHS-Kulai61 1/31/2016By Chtan FYHS-Kulai62 e.g. 11 If find 1/31/2016By Chtan FYHS-Kulai63 e.g. 12 If, find the value of m. 1/31/2016By Chtan FYHS-Kulai64 e.g. 13 If, then 1/31/2016By Chtan FYHS-Kulai65 e.g. 14 If, then 1/31/2016By Chtan FYHS-Kulai66 Miscellaneous examples 1/31/2016By Chtan FYHS-Kulai67 e.g. 15 Evaluate 1/31/2016By Chtan FYHS-Kulai68 e.g. 16 Given, find. 1/31/2016By Chtan FYHS-Kulai69 e.g. 17 is an imaginary number, 1/31/2016By Chtan FYHS-Kulai70 e.g. 18 If, then 1/31/2016By Chtan FYHS-Kulai71 e.g. 19 Prove that 1/31/2016By Chtan FYHS-Kulai72 e.g. 20 Find the value 1/31/2016By Chtan FYHS-Kulai73 e.g. 21 Prove that 1/31/2016By Chtan FYHS-Kulai74 e.g. 22 Simplify 1/31/2016By Chtan FYHS-Kulai75 e.g. 23 If Show that 1/31/2016By Chtan FYHS-Kulai76 e.g. 24 If Prove that 1/31/2016By Chtan FYHS-Kulai77 e.g. 25 If Prove that 1/31/2016By Chtan FYHS-Kulai78 Addendum 1/31/2016By Chtan FYHS-Kulai79 In general, z is a complex number then, represent a circle with centre at (0,0) and radius a. (1) 1/31/2016By Chtan FYHS-Kulai80 (2) represent a straight line with gradient=tan. 1/31/2016By Chtan FYHS-Kulai81 (3)(3) If 4 points P1, P2, P3, P4 are concyclic, then 1/31/2016By Chtan FYHS-Kulai82 0 y x P1, P2, P3, P4 are concyclic 1/31/2016By Chtan FYHS-Kulai83 (4)(4) If 3 points P1, P2, P3 formed an equilateral triangle, 1/31/2016By Chtan FYHS-Kulai84 Do pg.280 Ex 20d & Misc 20 1/31/2016By Chtan FYHS-Kulai85 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 1/31/2016By Chtan FYHS-Kulai86 The end