RI Complex numbers C5 Lect Notes

RAFFLES INSTITUTION H2 Mathematics (9758) 2025 Year 5 _____________________________________ Chapter 5: Introduction to Complex Numbers Page 1 of 26 Chapter 5: Introduction to Complex Numbers SYLLABUS INCLUDES Complex numbers expressed in Cartesian form and Argand diagrams ● extension of the number system from real numbers to complex numbers ● complex roots of quadratic equations ● modulus, argument and conjugate of a complex number ● four operations of complex numbers ● equality of complex numbers ● conjugate roots of a polynomial equation with real coefficients ● representation of complex numbers in the Argand diagram ● geometrical effects of conjugation, negation, addition, subtraction, and multiplication by i PRE-REQUISITES ● Trigonometry ● Coordinate Geometry ● Vectors ● Algebraic manipulation CONTENT 1 Introduction to the Imaginary Number i 2 Complex Numbers 2.1 Definition of a Complex Number 2.2 Operations on Complex Numbers 2.3 Complex Conjugates 2.4 Some Properties of Complex Conjugates 3 Roots of Polynomials 4 Geometrical Representation of a Complex Number 4.1 Argand Diagram 4.2 Geometrical Representation of Addition and Subtraction of Complex Numbers 4.3 Modulus and Argument of a Complex Number 4.4 Geometrical Effects of Conjugation and Negation 4.5 Geometrical Effect of Multiplying a Complex Number by i 4.6 Modulus and Argument of zw and z w Appendix:  Proof of Result that Non-Real Roots of a Polynomial with Real Coefficients occur in Conjugate Pairs

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________ Chapter 5: Introduction to Complex Numbers Page 2 of 26 1 Introduction to the Imaginary Number i We know that the solution to the equation 2 1 0x   cannot be a real number, as the square of a real number cannot be negative. We say 2 1x  has no real roots. In order to solve the above equation, we need to find a “number” whose square is 1 . Let’s suppose such a “number” exists. Since we imagined it, let’s call this number the imaginary number i. We define i as Hence the solutions to 2 1 0x   are 1 ix   . Example 1 (a) If i 1  , simplify 2 3 4 2009 2010 2011 2012i , i , i , i , i , i , i . Solution i i 2i 1 3 2i (i )(i) i  4 2 2i (i )(i ) 1  2009 4 502i (i ) (i) i  2010 4 502 2i (i ) (i ) 1  2011 4 502 3i (i ) (i ) i  2012 4 503i (i ) 1  Let’s generalize: If k is a positive integer, then 4 4 1 4 2 4 3i 1, i i, i 1, i ik k k k       (b) Perform the four basic operations on i: (i) i i  2i (ii) 5i i  4i (iii) 5i 3i  215i 15 (iv) 6i 3i  2 (c) Solve for x if 2 2 2 0x x   . Solution 2 2 2 0x x   OR: 2 2 2 0x x     2 2 2 1 1 1 1 1 i 1 i x x x x x           2( 2) ( 2) 4(1)(2) 2(1) 2 4 2 2 2i 1 i2 x           i 1 

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________ Chapter 5: Introduction to Complex Numbers Page 3 of 26 2 Complex Numbers Notice that the solution to Example 1(c) is a combination of real numbers and imaginary numbers. Such numbers are called complex numbers. 2.1 Definition of a Complex Number A complex number is of the form ix y where x and y are real numbers and i 1  . The set of complex numbers is denoted by { : i , , }z z x y x y     . x is known as the real part of z , denoted by Re( )z , and y is known as the imaginary part of z , denoted by Im( )z . ix y is known as the cartesian form of the complex number z. Example 2 Write down the real and imaginary parts of the following complex numbers: z Re(z) Im(z) –2 + 3i –2 3 1 – i 1 –1 – 4 –4 0 5i 0 5 Remarks: ●       ● If y  0, then z x is a real number. ● If x 0, then iz y is a purely imaginary number. ● Complex numbers cannot be ordered, i.e. given any two complex numbers z1 and z2 , we cannot compare whether z z1 2 unless they are real numbers. Note that Im( )z does not include i .

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________ Chapter 5: Introduction to Complex Numbers Page 4 of 26 2.2 Operations on Complex Numbers In this section, let , , , a b c d  and 1 2 3, , z z z   . (a) Equality of Two Complex Numbers 2 complex numbers are equal if and only if their corresponding real and imaginary parts are equal, i.e. i i and a b c d a c b d      . For example, if ,x y   and x + iy = 5 – 3i, we have x = 5 and y = –3 (b) Addition of Complex Numbers ( i ) ( i ) ( ) i( )a b c d a c b d       Addition of complex numbers is commutative: z z z z1 2 2 1   Addition of complex numbers is associative: 1 2 3 1 2 3( ) ( )z z z z z z     For example, (3 4i) (1 i)    4 3i (c) Subtraction of Complex Numbers ( i ) ( i ) ( ) i( )a b c d a c b d       For example, (3 4i) (1 i)    2 5i (d) Multiplication of Complex Numbers 2( i )( i ) i i i i i ( 1) ( ) i( ) a b c d ac ad bc bd ac ad bc bd ac bd ad bc                Multiplication of complex numbers is commutative: 1 2 2 1z z z z Multiplication of complex numbers is associative: 1 2 3 1 2 3( ) ( )z z z z z z Multiplication of complex numbers is distributive over addition: 1 2 3 1 2 1 3( )z z z z z z z   For example, (3 4i)(1 i)   23 3i 4i 4i   7 i  (note that 2i 1 ) 2(3 4i) (3 4i)(3 4i)    29 12i 12i 16i 7 24i      Remarks: The above manipulations of complex numbers are the same as algebraic manipulations of expressions such as       ,a b c d a b c d     and so on. The only additional consideration is 2i 1 .

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________ Chapter 5: Introduction to Complex Numbers Page 5 of 26 We can use the GC to perform operations on complex numbers Operation Example GC Screen Multiplication of complex numbers (Press 2nd . to get i ) (3 4i)(1 i)  Square of a complex number 2(3 4i) Division of complex numbers 3 4i 1 i   Question: How did the GC obtain 3 4i 1 7 i1 i 2 2    ? (e) Division of Complex Numbers Recall when we tried to simplify 2 3 1 2   , we multiply it by 1 2 1 2   so that we could rationalize the denominator. For 3 4i 1 i   , we will multiply it by 1 i 1 i   , so that 3 4i 1 i   23 4i 1 i 3 3i 4i 4i 1 7i 1 7 i1 i 1 i 1 1 2 2 2               In general,       2 2 2 2 2 i i i ii i i i i i i + a b c d a b c da b a b c d c d c d c d c d c d             Remark: Note that ic d is chosen based on the denominator of i i a b c d   . We call ic d , the complex conjugate of ic d .

Raffles Institution H2 Mathematics 2025 Year 5 _____________________________________ Chapter 5: Introduction to Complex Numbers Page 6 of 26 Example 3 [RJC Prelim 9233/2005/01/Q1(i)] The complex numbers z and w are such that 1 2iz  and 1 iw b  , where b . Given that the imaginary part of w z is 3 5 , find the value of b . Solution 1 i 1 2i w b z    2 2 2 1 2i 1 2i i 2 i 1 2 ( 2 ) i1 2i ( 1) 2 5 5 b b b b                Given 3Im 5 w z      2 3 15 5 b b     Example 4 Find the square roots of 24 – 10i. By completing the square, or otherwise, solve z2 – 6z = 15 – 10i. Solution: