ACJC 2026 Introduction to Complex Numbers Summary

1 SUMMARY: INTRODUCTION TO COMPLEX NUMBERS Representations of Complex Numbers • Cartesian form of complex number: iz x y=+ o x and y are real numbers o i1=− o Real part of z, Re( )zx= o Imaginary part of z, Im( )zy= • Argand diagram representation: o As point ( , )P x y o As position vector ijOP x y=+  • Modulus of z: 22z r x y= = + • Argument of z: arg( )z = (1) Find basic angle 1tan y x −= (2) Find argument based on quadrant that complex number lies in . (Refer to lecture notes page 17 for more details on finding argument for complex numbers that lies in 4 different quadrants) • Find Re(z) and Im(z) given modulus and argument: o Re( ) cosz x r == o Im( ) sinz y r == Solving Polynomial Equations • Fundamental Theorem of Arithmetic: o Any polynomial equation P( ) 0z = , where P( )z is a polynomial of degree n, has exactly n roots. These roots may be real or complex.  It is possible to express any polynomial P( )z of degree n as a product of n linear factors ()z − , where  is a root of the equation P( ) 0z = . • Complex conjugate roots: o If a polynomial P( )z only contains real coefficients, and the complex number iz a b=+ is a root of the polynomial equation P( ) 0z = , then the conjugate *iz a b=− is also a root of the equation. Re Im O

ACJC 2025/26 H2 Mathematics (9758) 2 Geometrical Interpretations of Algebraic Operations on Complex Numbers Let iz a b=+ be denoted by point ( , )P a b and iw c d=+ be denoted by point ( , )Q c d on the Argand diagram. Conjugation • *iz a b=− represented by point ( , )P a b − • Point P is a reflection of point P in the real axis • *z z r== • arg( *) arg( )zz =− =− Negation • iz a b− =− − represented by point ( , )P a b−− • Point P is a reflection of point P about the origin • z z r− = = • arg( ) arg( )zz − = −= − Addition and Subtraction • ( ) i( )z w a c b d+ = + + + represented by point ( , )R a c b d++ • ( ) i( )z w a c b d− = − + − represented by point ( , )R a c b d −− Multiplication by i • iiz b a=− + represented by point ( , )S b a− • Point S is obtained by rotating point P anticlockwise about the origin by 2  radians • iz z r== • arg(i ) arg( ) 22zz = + = + Re Im O  r r − Re Im O  r r Re Im O Re Im O  r r