2024 JPJC Chapter 13 Complex Numbers

Jurong Pioneer Junior College H2 Mathematics (9758) JC2-2024 Chapter 13 Complex Numbers (Students’ version) / Pg 1 Content Outline 1. Introduction Recall what we have covered at GCE O level: For any quadratic equation ax2 + bx + c = 0, there are 3 cases: (1) b2 – 4ac > 0, there are 2 real and distinct roots (2) b2 – 4ac = 0, there are 2 repeated real roots (3) b2 – 4ac < 0, there is no real root. In this chapter, we are dealing with case 3 for all polynomial equations. Consider the equation 2 1 0.x   We know that the above equation has no real roots, since 1 is not a real number. How then do we solve the equation 2 1 0?x   Chapter 13: Complex Numbers Complex numbers expressed in cartesian form Include: • extension of the number system from real numbers to complex numbers • complex roots of quadratic equations • conjugate of a complex number • four operations of complex numbers • equality of complex numbers • conjugate roots of a polynomial equation with real coefficients Complex numbers expressed in polar form Include: • representation of complex numbers in the Argand diagram • complex numbers expressed in the form , (cos i sin )r   , or ier  where 0r and      and vice versa • calculation of modulus (r) and argument () of a complex number • multiplication and division of two complex numbers expressed in polar form Useful textbooks on applications of Complex Numbers Reference Textbooks: (1) Ho Soo Thong, Tay Yong Chiang & Koh Khee Meng, “College Mathematics Syllabus C Vol 1”, Pan Pacific Publications, Call Number: HO510 (2) H2 Mathematics For 'A' Level, Federick Ho, David Khor, Yui-P'ng Lam, B.S. Ong Volume 1, Call No 510.76 HO

Jurong Pioneer Junior College H2 Mathematics (9758) JC2-2024 Chapter 13 Complex Numbers (Students’ version) / Pg 2 1.1 Definition of Complex Number The quadratic equation 21 0x , i.e. 21x has no real roots since there is no real number whose square is 1. To solve this equation, we need to find a “number” whose square is 1. Let’s suppose such a “number” i exists and define it such that 2i 1, where i 1 . Thus, 21 0x 211 i.xxx     Beyond the real number system, i is called the unit imaginary number in the system of complex numbers. Remark: From the definition of i, we see that 2i 1, 3i i and 4i 1. Can you tell what is the value of i 79 ? With the introduction of ‘i’, all equations have solutions (roots), be they real or complex roots. Example 1 Solve x2 – 2x + 5 = 0. Solution: 2 4 4(1)(5)22 162x   Homework: Verify your answer using the GC. Any number denoted by z of the form z = x + yi, where x, y and i 1 , is called a complex number. x is called the real part of z, denoted by Re(z). y is called the imaginary part of z, denoted by Im(z). The set of all complex numbers is denoted by . So  i: , x y x y   . GC Corner Polynomial Root Finder

Jurong Pioneer Junior College H2 Mathematics (9758) JC2-2024 Chapter 13 Complex Numbers (Students’ version) / Pg 3 Example 2 Write down real and imaginary parts of z given in the table below. z Re(z) Im( z) 3 + 2i 3 2 2 – i 4  5i 2i 5 IMPORTANT NOTE: (1) A complex number can be expressed in various forms. The form ix y is called the cartesian, rectangular or algebraic form. (2) Let z = x + yi. In general, (i) If , the complex number is purely REAL. (ii) If , the complex number is purely IMAGINARY. (iii) x + yi = 0 if and only if x = and y = . (iv) If w is another complex number defined by iw a b  , then z w if and only if x and y . i.e. i i and x y a b x a y b      1.2. The Complex Conjugate The complex conjugate of z is denoted by z* or z . If z = x+ yi is a complex number, then the complex conjugate of z is z* = x − yi. (note the change in sign before i). Example 3 Write down complex conjugate z* of z, given in the table below. Z z * 3 + 2i 3 – 2i 2 – i – 5i 7 If z = x+ yi , it can be proven that z and *z satisfy the following properties: (1) * 2z z x  (2) * 2 iz z y  (3) 2 2*zz x y  , which is a real number [Proof] Let z = x + iy, then *z = x – iy. zz* = (x + iy)(x – iy) = x2 + y2 , which is a real number

Jurong Pioneer Junior College H2 Mathematics (9758) JC2-2024 Chapter 13 Complex Numbers (Students’ version) / Pg 4 2. The Four Operations of Complex Numbers Let 13 2iz  and 21 5iz . Then, i) 1 2 z z  ii) 1 2z z  iii) 1 2z z iv) 12zz Note: The simplification of the division of two complex numbers is similar to the method of rationalising the denominator: For real numbers: Rationalising Denominator For complex numbers: Realising Denominator 11 2 1 1 21 2 1 2  i1 2i i 1 2i1 2i 1 2i   21 2 1 21 211 2      2i 2 2 i 2 1i1 2 3 31 2i      Important Note: Whenever possible, you should use the GC to do your calculations unless exact values are required or No GC is allowed as specified in the question. Homework: Verify your answers above with the use of the GC. Example 4 (Do not use a calculator in answering this question) (a) Simplify the following: ( 5 2i)(1 3i)    (b) It is given that (1 3i)( i) = 2 + ia b  , find the values of a and b. Think! Is there an alternative method for Example 4(b)? Hint: Look at Section 1.1 (1 3i)( i) = 2 + i 2 + ii 1 3i a b a b      

Jurong Pioneer Junior College H2 Mathematics (9758) JC2-2024 Chapter 13 Complex Numbers (Students’ version) / Pg 5 3. The Argand Diagram A complex number iz x y  can be represented by an ordered pair ( , )P x y in a coordinate plane. Such a coordinate plane is called a complex number plane and the resulting diagram is called an Argand diagram. Note: (1) The x-axis is called the real axis and the y-axis the imaginary axis. (2) All points ( , 0)x on the x-axis represent real numbers while all points (0, )y on the y-axis represent purely imaginary numbers. 3.1. Representation of complex numbers in the Argand Diagram. Example 5 Let the point P with coordinates (3, 2) represent the complex number3 2iz . On an Argand diagram, point P is plotted as shown below: Plot on the same diagram, the points P’ and P’’ representing zand –z respectively. z = –z = Note: The point P’ is a reflection of point P about the Re(z) axis. Re(z) Im(z) -5 -4 -3 -2 -1 1 2 3 4 5 -5 -4 -3 -2 -1 1 2 3 4 5 P P’ P’’ O Re(z) Im(z) Assuming x > 0, y > 0

Jurong Pioneer Junior College H2 Mathematics (9758) JC2-2024 Chapter 13 Complex Numbers (Students’ version) / Pg 6 Re(z) 3.2. Modulus and Argument Let point P be represented by the complex numberi, ,z x y x y  , on an Argand diagram. The modulus of the complex number , denoted by | |z z, is the length of OP. By Pythagoras theorem, 2 2| |z r x y  . The angle  made by the positive x-axis and the line OP gives the argument of the complex number z, denoted by arg(z), where arg( )z    (principal argument range and expressed in radian mode) Note: arg(z) is positive if it is measured in anti-clockwise direction and negative if it is measured in clockwise direction. For example, if z is represented by the point P and w is represented by the point Q, as shown in the diagram below, Note: If a and b are positive real numbers, then (i) arg( )a (ii) arg( )a  (iii) arg( i)b (iv) arg( i)b  O  P Im(z) O  P Im(z) Re(z)  Q t