13_2016_-_2017_H2_Maths_Complex_Numbers_Lecture_Questions_(Stu)

National Junior College Mathematics Department 2017 Complex Numbers Page 1 of 14 National Junior College 2016 – 2017 H2 Mathematics Complex Numbers Lecture Questions Key questions to answer:  What is a complex number? o How is the complex number i defined? o How do we write a complex number in its Cartesian form? o What do we mean when we say that two complex numbers are equal? Lecture Reading: Section 1 Definition (Imaginary Unit) The imaginary unit, i, is a number such that 2i1 . Hence, i1 . Definition (Complex Number in Cartesian form) A complex number, z , is a number of the form ixy , where ,xy  . Question 1 (Real and Imaginary Parts of a Complex Number) Write down the real and imaginary parts of the following complex numbers: (a) 3i (b) 2i (c) 7 (d) i6 Solution: (a)    Re 3 i 3, Im 3 i 1    (b)    Re 2i 0, Im 2i 2    (c) (d)    Re 7 7, Im 7 0    Re i 6 6, Im i 6 1    Note: 1. The imaginary part of a complex number does not include ‘i'. 2. A complex number having only the imaginary part (i.e. real part equals zero) is known as a purely imaginary number.

National Junior College Mathematics Department 2017 Complex Numbers Page 2 of 14 Question 2 (Equality of Complex Numbers) Find x and y, where ,xy  , if  2 3i 2i 1 ix y x y     . Solution:      2 3i 2i 1 i 2 3i 2 ix y x y x y x x y           Comparing real part on both sides, we get 2 0 (1)x y x x y      Comparing imaginary part on both sides, we get 3 2 (2)xy   Solving equations (1) and (2) simultaneously, we get 1x and 1y . Alternatively, we may group all the real and imaginary parts respectively on one side of the equation, giving us    2 2 3 2 i 0x y x y     . Comparing real and imaginary parts on both sides, we get the equations 2 2 0 and 3 2 0x y x y     , which we can solve simultaneously. Question 3 Given that i , where , ,z x y x y   f ind the value of x such that  2 i i 1 .x z z x    Identify the mistake in the following proposed solution and correct it. Comparing real part on both sides: 2xz Comparing imaginary part on both sides: 1zx  (*) Substitute 2zx into equation (*): 121 3x x x    . Correct solution:        2 i i i i 1 2 i i 1x x y x y x x y x x y x             Comparing real part on both sides: 2x y x x y    Comparing imaginary part on both sides: 1 2 1x y x x y      Solving both equations simultaneously, we get 1x .

National Junior College Mathematics Department 2017 Complex Numbers Page 3 of 14 Key questions to answer:  How do we carry out arithmetic operations (addition, subtraction, multiplication, division and taking square root) on complex numbers in Cartesian form? o How do we relate the addition and subtraction of co