Recall: Within the real number system, we can solve equation of the form and b 2 4ac 0. ax 2 + bx + c =0, where a, b, c R What is R? They are real numbers on the number line e.g: 2, 4, π, 3.167, 2 3 Therefore, solve the quadratic equation below: Equation Working Real Solution x 2 3x 4=0 x 2 + 16 = 0 x 2 +3x 5=0 x 2 6x =0 x 2 9=0 2x 2 2x + 15 = 0 Some of the equations has. It doesn t mean that they cannot be solve. They can t be solve using the real number system. But they can be solve using the complex number system. 1
HOW? We introduce an imaginary number, i defined by Now that you know about i, we can solve the ones on the previous page using this complex number system. (1) Note: We denote a complex number by the symbol C (2) (3) x 2 +2x +2=0 2
SO WHAT IS A COMPLEX NUMBER? A complex number is a number of the form a + ib where a and b are real numbers and i 2 = 1 For example, 3+5i is a complex number with a = and b =. If a = 0, the number is wholly imaginary If b = 0, the number is real If the complex number is 0, then a = and b =. Complex Number a State the values of a and b b 6 3i 3+6i 5i 3 2i We usually use x + iy to represent an unknown complex number and we denote: Therefore, when we have an unknown in an equation is a complex number we denote it by z, for example Solve the complex equation. z 2 + z +1=0 If we write z = x + iy, then the real part of the complex number z is, written as and the imaginary part of the complex number z is, written as 3
OPERATIONS OF COMPLEX NUMBERS (1) Equality Two complex numbers are equal if and only if their real parts are equal and imaginary parts are equal. Then, For example: 3+qi = x +6i. Find the value of x and q. (2) Addition and Subtraction Complex numbers are added (or subtracted) by adding (or subtracting) on their real parts and also their imaginary parts. The sum of two complex numbers is a complex number. For example: (a) (3 + 2i) (6 5i) (b) (3 2i)+6 3i (3) Multiplication by a real number Let k R. (i.e. k is any real number) Then kz = For example: (a) 3( 3+6i) (b) 2(2 i) (4) Multiplication of two complex numbers The product of two complex numbers is itself a complex number. For example: (a) (2 + 3i)(4 8i) (b) (3 i)(2 + i) 4
CONJUGATE OF COMPLEX NUMBERS Let z = x + iy. We define the conjugate of z to be denoted by Properties: (1) For example: (4 3i)+(4+3i) Hence, sum of a conjugate pair of complex number is a real number. (2) For example: ( 3+i) ( 3 i) Hence, difference of a conjugate pair of complex number is number of the form Often, this is called purely imaginary number since they have no real part. where (3) For example: (2 + 3i)(2 3i) Hence, product of the conjugate pair of complex numbers is a real number. (4) Hence, conjugate of the conjugate pair of complex number is z itself. 5
Example 1: 1 Express 1+3i in the form of a + ib where a, b R. Example 2: 4+i Express 3 i in the form of a + ib where a, b R. 6
SOLVING AND SIMPLIFYING COMPLEX EQUATIONS / EXPRESSIONS Example 3: Solve for z in the equation z 2 + z +1=0 Example 4: Factorize z 2 6z + 13 Example 5: Find the equation having roots of (1 + 5i) and (1 5i) Example 6: Solve 2x 3 12x 2 + 25x 21 = 0 7
SQUARE ROOTS OF COMPLEX NUMBERS Within the real number system, the roots 2 and -2 of the equation x 2 =4are called the square roots of 4. In general, the roots of the equation x 2 = a where a R are the square roots of a. Similarly, within the complex number system, the roots of the equation z 2 = a + bi where a, b R are called the square roots of a+bi. Example 7: Find the square roots of: (a) 5+2i (b) 8i 8
THE ARGAND DIAGRAM Complex numbers can be represented geometrically using the x and y-axes as the Real (Re) and Imaginary (Im) axes. The plane of the axes is then referred to as the complex plane and a diagram showing complex numbers is said to be an Argand Diagram. On the Argand diagram, each complex number z = x+iy may be represented by: (1) the point P with coordinates (x,y) (2) the vector OP, where O is the origin (3) any vector equal to OP in magnitude and direction Addition and Subtraction on the Argand Diagram The sum and the difference of two complex numbers can be shown on an Argand Diagram in the same way as we show vectors which are added or subtracted. The Graph of z and its conjugate z* Let the complex number z = x + iy and its conjugate z* = x iy be represented by OP and OP respectively. Since the coordinates of P and P are (x,y) and (x,-y) respectively, P must be the reflection of P in the real axis. 9
Example 8: If z 1 =2+i and z 2 = 1+i, show on Argand Diagram the points representing: z 1,z 2,z1,z 2,z 1 + z 2,z1 + z2 Is (z 1 + z 2 ) = z1 + z2? Multiplication of a Complex Number by i If z = x + iy, then iz = Argand diagram? Therefore, from the Argand diagram, we can see the effect of multiplying a complex number z by i results in an anti-clockwise rotation of the vector OP through 90 degrees about the origin. Example 9: If z =2+3i, locate the points representing the complex numbers iz, i 2 z,i 3 z and i 4 z on the complex plane. What can you conclude from this? 10
Distance between Two Points If complex numbers z 1 and z 2 are represented by the vectors OP1 and OP2 respectively, then z 1 z 2 is represented by P 1 P 2 Example 10: Given that the complex numbers 2+5i and 3 2i are represented in an Argand Diagram by the points A and B respectively, find the length of AB. 11