Tutorial on basic operations such as addition, subtraction, multiplication, division, and equality of complex numbers with online calculators and examples are presented. Practice exercises with detailed step-by-step solutions hidden in collapsible dropdowns are also included.
A complex number \( z \) is a number of the form:
$$ z = a + bi $$where \( a \) and \( b \) are real numbers and \( i \) is the imaginary unit defined by:
$$ i = \sqrt{-1} $$\( a \) is called the real part of \( z \) and \( b \) is the imaginary part of \( z \).
Note that the set \( \mathbb{R} \) of all real numbers is a subset of the complex numbers \( \mathbb{C} \) since any real number may be considered as a complex number having an imaginary part equal to zero.
The conjugate of a complex number \( a + bi \) is a complex number equal to:
$$ a - bi $$Example: Find the conjugate of the following complex numbers:
Addition of two complex numbers \( a + bi \) and \( c + di \) is defined as follows:
$$ (a + bi) + (c + di) = (a + c) + (b + d)i $$This is similar to grouping like terms: real parts are added to real parts and imaginary parts are added to imaginary parts.
Example: Express the following in the form of a complex number \( a + bi \):
An online calculator to add complex numbers for practice is available.
The subtraction of two complex numbers \( a + bi \) and \( c + di \) is defined as follows:
$$ (a + bi) - (c + di) = (a - c) + (b - d)i $$Example: Express in the form of a complex number \( a + bi \):
The multiplication of two complex numbers \( a + bi \) and \( c + di \) is defined as follows:
$$ (a + bi)(c + di) = (ac - bd) + (ad + bc)i $$However, you do not need to memorize the definition as multiplication can be carried out using ordinary algebraic distribution along with the property \( i^2 = -1 \).
Example: Express in the form of a complex number \( a + bi \): \( (3 + 2i)(3 - 3i) \)
Using the distributive law:
$$ (3 + 2i)(3 - 3i) = (3 + 2i)(3) + (3 + 2i)(-3i) = 9 + 6i - 9i - 6i^2 $$Grouping like terms and substituting \( i^2 = -1 \):
$$ = 9 - 3i - 6(-1) = 9 - 3i + 6 = 15 - 3i $$An online calculator to multiply complex numbers for practice is available.
We use the multiplication property of a complex number and its conjugate to divide two complex numbers.
Example: Express \( \dfrac{8 + 4i}{1 - i} \) in the form of a complex number \( a + bi \).
Multiply the numerator and denominator by the complex conjugate of the denominator (\( 1 + i \)):
$$ \dfrac{(8 + 4i)(1 + i)}{(1 - i)(1 + i)} $$Expand and group like terms, noting that \( i^2 = -1 \):
$$ = \dfrac{8 + 8i + 4i + 4i^2}{1 - i + i - i^2} = \dfrac{8 + 12i + 4(-1)}{1 - (-1)} = \dfrac{4 + 12i}{2} = 2 + 6i $$An online calculator to divide complex numbers for practice is available.
Two complex numbers \( a + ib \) and \( x + iy \) are equal if and only if their real parts are equal and their imaginary parts are equal:
$$ a + ib = x + iy \iff a = x \text{ and } b = y $$Example: Find the real numbers \( x \) and \( y \) such that \( 2x + y + i(x - y) = 4 - i \).
Equating the real and imaginary parts yields a system of equations:
Solving the system gives \( x = 1 \) and \( y = 2 \).
Find the complex conjugate of the following complex numbers:
Write the following expressions in the form \( a + bi \):
1) Complex Conjugates:
2) Expressions in Standard Form: