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 Polyn