13 2016 - 2017 H2 Maths Complex Numbers Lecture Notes final

National Junior College Mathematics Department 2017 2016 – 2017 / H2 Maths / Complex Numbers Page 1 of 1 National Junior College 2016 – 2017 H2 Mathematics Complex Numbers (Lecture Notes) Topic 13: Complex Numbers Key questions to answer: 1. What is a complex number? How is it different from a real number? - How is the complex number i defined? - How do we write a complex number in its Cartesian form? - What is the relationship between the set of real numbers and the set of complex numbers? Complex Numbers in Cartesian Form 2. When can we say that two complex numbers given in Cartesian form are equal? - How do we apply this property to solve simple equations involving a complex variable? 3. How do we carry out arithmetic operations (addition, subtraction, multiplication, division and taking square root) on complex numbers in Cartesian form? - How do we relate the addition and subtraction of complex numbers to addition and subtraction of vectors? 4. What is the conjugate of a complex number? What are its properties and applications? 5. What can we say about the roots of polynomial equations with real coefficients? - How do we solve polynomial equations with real coefficients? - Understand that complex roots of a polynomial equation with real coefficients occur in conjugate pairs. 6. How do we find the modulus and argument of a complex number given in Cartesian form? 7. How do we represent a complex number in Cartesian form by a poin t in the Argand diagram? - How do we interpret geometrically, the terms ‘real part’, ‘imaginary part’, ‘modulus’, ‘argument’ and ‘conjugate’ of a complex number? Complex Numbers in Polar & Exponential Form 8. How do we convert a complex number from one of the following forms to another: (a) Cartesian form, (b) polar form and (c) exponential form? 9. How do we multiply and divide two complex numbers given in polar and exponential forms? 10. How do we represent a complex number in polar form by a point in the Argand diagram?

National Junior College Mathematics Department 2017 2016 – 2017 / H2 Maths / Complex Numbers Page 2 of 21 §1 Introduction 1.1 Roots of quadratic equations How many roots do you expect to obtain when solving a quadratic equation? Solve the following quadratic equations: (a) 2 2 3 0xx   Two real and distinct roots 2 16 3 or 12x x x    (b) 2 2 1 0xx   Two real and repeated roots 20 12xx    (c) 2 2 5 0xx   No real roots 2 16 2x    Can we expand our notion of numbers to those that are non-real? How then can we express the roots of equation (c)? 1.2 Definition and Terminology Definition 1.2.1 (Imaginary Unit) The imaginary unit, i, is a number such that 2i1 . Hence, i1 . Thus, the square root of any negative real number can then be written in the form of ai, where a is a positive real number. For example,     3 3 1 3 1 3 i      . Definition 1.2.2 (Complex Number in Cartesian form) A complex number, z , is a number of the form ixy , where ,xy  . The symbol is used to denote the set of complex numbers. Hierarchy of the number system: Complex Numbers, Real Numbers, Purely Imaginary Numbers Rational Numbers, Irrational Numbers Integers, Fractions Negative Integers Zero Positive Integers Let’s be intellectually curious!

National Junior College Mathematics Department 2017 2016 – 2017 / H2 Maths / Complex Numbers Page 3 of 21 Definition 1.2.3 (Real and Imaginary Parts of a Complex Number) If iz x y , ,xy  , then x is the real part of z and is denoted by  Re z . (i.e.  Re zx ) y is the imaginary part of z is denoted by  Im z . (i.e.  Im zy ) Note: 1. If 0x , then izy is a purely imaginary number. 2. If 0y , then zx is a real number. 3.  Im zy is a real number.  Im z is NOT iy. Definition 1.2.4 (Equality of Complex Numbers) Two complex numbers are equal if and only if their real and imaginary parts are equal. That is, given that 1 1 1 iz x y and 2 2 2 iz x y where 1 2 1 2, , ,x x y y  , then 12zz  12xx and 12yy . Note: 1. 0 0 and 0z x y    2. Inequalities do not apply to complex numbers that are not real numbers. For example, we cannot say 2i is larger or smaller than 2i . Example 1.2.5 Find x and y, where ,xy  , if 2 i(3 ) 4 2ix y x y     . Solution: Comparing real part on both sides, we get 2 4 (1)xy   Comparing imaginary part on both sides, we get 3 2 (2)xy   Solving equations (1) and (2) simultaneously, we get 8 7x and 10 7y . Note: We must ensure that the terms on both sides of the simultaneous equations are real.

National Junior College Mathematics Department 2017 2016 – 2017 / H2 Maths / Complex Numbers Page 4 of 21 §2 Arithmetic Operations on Complex Numbers 2.1 ‘i’ follows all arithmetic operations on real numbers (i) Addition: 3i + 4i = 7i (ii) Subtraction: 10i – 3i = 7i (iii) Multiplication: iiaa ;  i i ia b a b   ; 2i , wher i ei ,a abb aab b   . In particular,   2 32 242 54 i 1 (by definition) i ii i i i 1 i ii i      Hence, i to any power can be reduced to one of i, 1, i or 1 . (iv) Division: 1 2 1 1 i iii i i i i       . For S ections 2.2 to 2.3, consider two complex numbers 1 1 1 iz x y and 2 2 2 iz x y where 1 2 1 2, , ,x x y y  . 2.2 Addition, Subtraction & Multiplication of Complex Numbers (a)    1 2 1 1 2 2 1 2 1 2 i i ( ) i ( )z z x y x y x x y y         (b)      1 2 1 1 2 2 1 2 1 2 i ( i ) iz z x y x y x x y y         (c)  1 1 1 1 1 iikz k x y kx ky    , k (d) 2 1 2 1 1 2 2 1 2 1 2 1 2 1 2( i )( i ) i i iz z x y x y x x y x x y y y        1 2 1 2 1 2 2 1( ) i( )x x y y x y x y    2.3 Division of Complex Numbers This is do ne by realising the denominator, which is multiplying the complex conjugate (refer to page 6) of the denominator to the numerator and denominator. 1 1 1 2 2 1 2 1 2 2 1 1 2 1 2 1 2 2 1 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 i i ( ) i( ) iii z x y x y x x y y x y x y x x y y x y x y z x y x y x y x y x y                 

National Junior College Mathematics Department 2017 2016 – 2017 / H2 Maths / Complex Numbers Page 5 of 21 Example 2.3.1 Express the following complex numbers in the form i , where ,x y x y . (a) (2 + 3i) − (−1 + 2i) (b) (2 + 4i)(1 – i) (c) 5 10i 4 3i   Solution: (a) (2 + 3i) − (−1 + 2i) = 2 + 3i + 1 – 2i = 3 + i (b) (2 + 4i)(1 – i) = 2 – 2i + 4i – 4i2 = 2 + 2i + 4 = 6 + 2i (c) 5 10i 5 10i 4 3i 20 15i 40i 30 50 25i. 2 i4 3i 4 3i 4 3i 16 9 25               Example 2.3.2 Solve the simultaneous equations (1 i) 2i 0 and 3i (1 i) iz w z w      . Solution: (1 i) 2i 0 ----- (1)zw   ; 3i (1 i) i ----- (2)zw   From (1), we have 2i 1i wz   . Substituting into (2), we get   2 2i3i (1 i) i 1i 6 (1 i) i1i 6 (1 i ) i 1 i 8 i 1 11 i88 w w w w ww w w               Example 2.3.3 Find the square roots of 3 4i . Solution: The question requires us to evaluate 3 4i . Let 3 4i i xy   . Then   2 22i 3 4i 2i 3 4ix y x y xy        . Comparing the real and imaginary parts on both sides, we have: 22 3 ----- (1)xy ; 2 4 ----- (2)xy Solving simultaneously, we have 2x , 1y or 2x , 1y . The square roots of 3 + 4i are 2i and 2i . Therefore,   11 12i i 1i 188 4 1 i 1 i 4z     

National Junior College Mathematics Department 2017 2016 – 2017 / H2 Maths / Complex Numbers Page 6 o