ACJC 2026 Introduction to Complex Numbers Lecture Notes

1 13 INTRODUCTION TO COMPLEX NUMBERS SYLLABUS Complex numbers expressed in Cartesian form and Argand diagram ▪ Extension of the number system from real numbers to complex numbers ▪ Complex roots of quadratic equations ▪ Modulus, argument and conjugate of a complex number ▪ Four operations of complex numbers ▪ Equality of complex numbers ▪ Conjugate roots of a polynomial equation with real coefficients ▪ Representation of complex numbers in the Argand diagram ▪ Geometrical effects of conjugation, negation, addition, subtraction, and multiplication by i

ACJC 2025/26 H2 Mathematics (9758) 2 CONTENTS 1 Introduction ...................................................................................... 3 2 Algebraic Operations on Complex Numbers ................................... 5 3 Complex Conjugate .......................................................................... 8 4 Complex Roots of a Polynomial Equation ..................................... 11 5 The Argand Diagram ...................................................................... 15 5.1 Modulus and Argument of a Complex Number.................... 16 5.2 Geometrical Effects of Algebraic Operations on Complex Numbers ................................................................................ 21 Annex: Practice Questions on Complex Numbers ............................ 26

13 Introduction to Complex Numbers 3 LECTURE 1 Lesson Outline • Introduction: imaginary number, Cartesian form of complex number, equality • Algebraic operations on complex numbers • Complex conjugate 1 INTRODUCTION The equation 2 10x += does not have a solution in the real number system. An imaginary number i1=− was introduced in the 16 th century to solve equation s such as 2 10x += . With this introduction, t he equation 2 10x += has two roots, i.e. ix = or ix =− . Note i1=− 5ii = 2i1 =− 6i1 =− 32i i i ( 1) i i=  = −  = − 7ii =− 4 2 2i i i ( 1)( 1) 1=  = − − = 8i1 = In general, we have 41ii n+ = , 42i1n+ =− , 43ii n+ =− , 44i1n+ = . Definition The Cartesian form of a complex number z is given by iz x y=+ , where ,xy   and i1=− . x is called the real part of z, denoted by Re(z). y is called the imaginary part of z, denoted by Im(z), which is real. The set of complex numbers is denoted by . The set of all real numbers, , is a proper subset of the set of all complex numbers , i.e.   . In fact,     , as represented in the Venn diagram. Note that =−i1 , so =−2i1 , thus +=2i 1 0 .    

ACJC 2025/26 H2 Mathematics (9758) 4 Note For iz x y=+ , • Im(z) = y, not iy. • If 0y = , then zx= is a real number. • If 0x = , then izy= is a purely imaginary number. Equality of complex numbers Let , , , a b c d   . Let 1 iz a b=+ and 2 iz c d=+ . Two complex numbers 1z and 2z are equal if and only if their real and imaginary parts are equal. 12zz=  iia b c d+ = +  ac= and bd= Note We do not define an ordering on complex numbers. In other words, given any two complex 1z and 2z , the expressions 12zz or 12zz do not mean anything unless 1z and 2z happen to be real numbers. Example 1 Find the real numbers p, q, x and y in the following equations. (a) 2 i 4 7ipq+ = − (b) i( ) 6 2ix y x y+ + − = + Solution (a) 2 i 4 7ipq+ = − 2 4, 7pq = = − 2, 7pq = = − (b) i( ) 6 2ix y x y+ + − = + 6, 2x y x y + = − = Using GC, 4, 2xy== . ■

13 Introduction to Complex Numbers 5 2 ALGEBRAIC OPERATIONS ON COMPLEX NUMBERS Let 1 iz a b=+ and 2 iz c d=+ where , , , a b c d   . Addition 12 ( i ) ( i ) ( ) i( )z z a b c d a c b d+ = + + + = + + + Subtraction 12 ( i ) ( i ) ( ) i( )z z a b c d a c b d− = + − + = − + − Multiplication 1 ( i ) i ,kz k a b ka kb k= + = +   12 2 ( i )( i ) i i i ( ) i( ) z z a b c d ac ad bc bd ac bd ad bc = + + = + + + = − + + Division 1 2 2 2 2 2 22 2 2 2 2 i i ii ii ( i i i ) ( i i i ) ( ) i( ) ( ) ( ) i z ab z c d a b c d c d c d ac ad bc bd c cd dc d ac bd bc ad cd ac bd bc ad c d c d += + +−= +− − + −= − + − + + −= + +−=+ ++ Example 2 Express 5 2i 3 4i + + in Cartesian form ixy+ . Solution ( ) 2 22 5 2i 5 2i 3 4i 3 4i 3 4i 3 4i 15 6i 20i 8i 3 4i 23 14 i25 25 + + −=+ + − + − −= − =− ■ The aim is to turn the denominator into a real number.

ACJC 2025/26 H2 Mathematics (9758) 6 To carry out algebraic operations on complex numbers using the GC, we need to press the MATH button and select the complex numbers menu CMPLX. Example 3 Simplify the following: (a) ( ) ( )4 2i 3 i+ + − (b) ( )( )4 2i 3 i+− (c) ( ) 3 4 2i+ (d) 4 2i 3i + − Solution Since the same complex numbers, 4 2i+ and 3i− are used, store them into the variables A and B. Alternatively, they can be typed in with brackets. The imaginary number i is found above the decimal point, i.e. [2nd][.] The variables A and B are accessed using [alpha]. For A, key [alpha] [math], for B [alpha][apps]. 1) Key in 4 2i+ [sto>] A [enter] 2) Key in 3i− [sto>] B [enter] Then, perform the computations accordingly. ■ Extra Practice Annex: Practice Question 1

13 Introduction to Complex Numbers 7 Example 4 Two complex numbers w and z are such that 6 2iwz+ = + and 203 2iwz−= − . Find w and z, giving your answer in the form ixy+ . Solution 6 2i (1)wz+ = + − − − 20 2 i3 2 i 2 i 40 20i 41 8 4i (2) wz +  −=   −+   += + = + − − − (1) – (2): 4 2 2iz = − − 11 i22z = − − Substitute 11 i22z = − − into (1): 6 2i 116 2i i 22 wz= + − = + + + 13 5 i22=+ ■ Extra Practice Annex: Practice Question 2

ACJC 2025/26 H2 Mathematics (9758) 8 3 COMPLEX CONJUGATE If iz x y=+ , then the complex number ixy− is called the complex conjugate of z. We denote the complex conjugate of z by *z (or z ). Note that ixy+ and ixy− are conjugates of each other. For example, 1 2iz =+ and * 1 2iz =− are conjugates of each other. Properties of complex conjugates Property Proof Let i , * iz x y z x y= + = − Example 1 2i, * 1 2izz= + = − 1. ( )**zz = ( ) ( )* * i * i z x y xy z =− =+ = ( )* * (1 2i)* 1 2i z z =− =+ = 2. ( )* *, kz kz k=  ( )* [ ( i )]* ( i )* i ( i ) * kz k x y kx ky kx ky k x y kz =+ =+ =− =− = (3 )* [3(1 2i)]* (3 6i)* 3 6i 3(1 2i) 3* z z =+ =+ =− =− = 3. * 2Re( )z z z+= * ( i ) ( i ) 2 2 Re( ) z z x y x y x z + = + + − = = * (1 2i) (1 2i) 2 2 Re( ) zz z + = + + − = = 4. * 2i Im( )z z z−= * ( i ) ( i ) 2i 2i Im( ) z z x y x y y z − = + − − = = * (1 2i) (1 2i) 4i 2i Im( ) zz z − = + − − = = 5. 22*zz x y=+ 22 22 * ( i )( i ) (i ) zz x y x y xy xy = + − =− =+ 2 * (1 2i)(1 2i) 12 5 zz = + − =+ =

13 Introduction to Complex Numbers 9 Property Proof Let i , iz x y w a b= + = + Example 1 2i, 3 4izw= + = + 6. ( )* * *z w z w =      ( )* ( i ) ( i ) * ( ) i( ) * ( ) i( ) ( i ) ( i ) ** z w x y a b x a y b x a y b x y a b zw  = +  + =  +  =  −  = −  − =  ( )* (1 2i) (3 4i) * (4 6i)* 4 6i zw+ = + + + =+ =− * * (1 2i) (3 4i) 4 6i zw+ = − + − =− 7. ( )* ( *) ( *)z w z w =  In general, ( ) ( )** nnzz = , where n +    2 2 ( )* ( i ) ( i ) * ( i ) i( ) * ( i ) i( ) ( i ) ( i ) ( *) ( *) z w x y a b xa yb ya xb xa yb ya xb x y a b zw  = +  + = + + + = + − + = −  − =  ( )* (1 2i)(3 4i) * ( 5 10i)* 5 10i zw = + + = − + = − − ( *) ( *) (1 2i)(3 4i) 5 10i zw  = − − = − − 8. (a) *11 *ww  = (b) * * * zz ww  = (a) See https://bit.ly/conjprop8a (b) *z w   *1z w = *1( *)z w =  (property 7) 1( *) *z w =  (by (a)) * * z w= (b) *z w   *1 2i 3 4i +=  + (0.44 0.08i)* 0.44 0.08i =+ =− * 1 2i * 3 4i 0.44 0.08i z w −= − =− 9. *zz=  z is real *zz= i ix y x y + = −