Definition of Complex NumbersA complex number z is a number of the formNote that the set R of all real numbers is a subset of the complex number C since any real number may be considered as having the imaginary part equal to zero.. Complex ConjugateThe conjugate of a complex number a + b i is a complex number equal toExamples: Find the conjugate of the following complex numbers. a) 2  i , b) 3 + 4i , c) 5 , d) 5i Solution to above example a) 2 + i b) 3  4i c) 5 d) 5i Addition of Complex NumbersAddition of two complex numbers a + b i and c + d i is defined as follows.(a + b i) + (c + d i) = (a + c) + (b + d) i This is similar to grouping like terms: real parts are added to real parts and imaginary parts are added to imaginary parts. Example: Express in the form of a complex number a + b i.
Solution to above example.
(2 + 3i) + (4 + 5i) = 2 + 3 i  4 + 5 i = 2 + 8 i Calculator to add complex numbers for practice is available. Subtraction of Complex NumbersThe subtraction of two complex numbers a + b i and c + d i is defined as follows.(a + b i)  (c + d i) = (a  b) + (b  d) i Example: Express in the form of a complex number a + b i.
Solution to above example
Multiply Complex NumbersThe multiplication of two complex numbers a + b i and c + d i is defined as follows.(a + b i)(c + d i) = (a c  b d) + (a d + b c) i However you do not need to memorize the above definition as the multiplication can be carried out using properties similar to those of the real numbers and the added property i ^{ 2} = 1. (see the example below) Example: Express in the form of a complex number a + b i. (3 + 2 i)(3  3i) Solution to above example (3 + 2 i)(3  3i) Using the distributive law, (3 + 2 i)(3  3 i) can be written as (3 + 2 i)(3  3 i) = (3 + 2 i)(3) + (3 + 2 i)(3 i) = 9 + 6 i  9 i 6 i ^{ 2} Group like terms and use i ^{ 2} = 1 to simplify (3 + 2 i)(3  3 i) (3 + 2 i)(3  3 i) = 15  3 i Calculator to multiply complex numbers for practice is available. Divide two Complex NumbersWe use the multiplication property of complex number and its conjugate to divide two complex numbers.Example: Express in the form of a complex number a + b i.
\( \dfrac{(8 + 4 i)\color{red}{(1+i)}}{(1i)\color{red}{(1+i)}} \) Multiply and group like terms \( = \dfrac{8 + 4 i + 8 i + 4 i^2}{1  i + i  i^2} = \dfrac{4 + 12i}{2} \) \( = 2 + 6 i \) Calculator to divide complex numbers for practice is available. Equality of two Complex NumbersThe complex numbers a + i b and x + i y are equal if their real parts are equal and their imaginary parts are equal.a + i b = x + i y if and only if a = x and b = y Example: Find the real numbers x and y such that 2x + y + i(x  y) = 4  i. For the two complex numbers to be equal their real parts and their imaginary parts has to be equal. Hence 2x + y = 4 and x  y =  1 Solve the above system of equations in x and y to find x = 1 and y = 2.
Exercises
Solutions to above exercises
