Understanding Complex Numbers
A complex number is a number of the form:
where:
- $$ a $$ is the real part
- $$ b $$ is the imaginary part
- $$ i $$ is the imaginary unit, defined as $$ i^2 = -1 $$
Operations with Complex Numbers
Addition and Subtraction
To add or subtract complex numbers:
- Add or subtract the real parts.
- Add or subtract the imaginary parts.
Example:
Multiplication
To multiply complex numbers, use the distributive property and remember that $$ i^2 = -1 $$:
Example:
Expanding:
Since $$ i^2 = -1 $$, we replace $$ -3i^2 $$ with $$ +3 $$:
Division
To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator.
The conjugate of $$ a + bi $$ is $$ a - bi $$, and multiplying by the conjugate eliminates the imaginary part in the denominator.
Example:
Multiply numerator and denominator by $$ 4 + i $$:
Expanding the denominator:
Expanding the numerator:
Since $$ i^2 = -1 $$:
Final result:
Practice Problems
Addition and Subtraction
- $$ (5 + 3i) + (2 - 4i) $$
- $$ (7 - 2i) - (3 + 6i) $$
- $$ (-4 + 5i) + (6 - 2i) $$
- $$ (8 + 3i) - (-2 + 7i) $$
Multiplication
- $$ (2 + i)(3 - 4i) $$
- $$ (-1 + 2i)(5 + 3i) $$
- $$ (6 - i)(4 + 2i) $$
- $$ (-3 + i)(-2 - i) $$
Division
- $$ \frac{5 + 2i}{3 - i} $$
- $$ \frac{4 - i}{2 + 3i} $$
- $$ \frac{6 + 3i}{1 - 2i} $$
- $$ \frac{7 - 5i}{4 + i} $$