Introduction to Complex Numbers

For a long time in mathematics, it was believed that you couldn't take the square root of a negative number. Complex numbers were introduced to solve this exact problem.

The Imaginary Unit

The foundation of complex numbers is the imaginary unit, denoted by $i$. It is defined as the square root of $-1$:

$i = \sqrt{-1}$

Because of this definition, squaring $i$ gives us $-1$:

$i^2 = -1$

The Standard Form: $a + bi$

A complex number is any number that can be written in the form $a + bi$, where:

• $a$ is the real part.
• $b$ is the imaginary part.
• Both $a$ and $b$ are real numbers.

For example, in the complex number $3 + 2i$, the real part is $3$ and the imaginary part is $2$.

Operations with Complex Numbers

You can perform standard arithmetic operations with complex numbers just like you do with polynomials, keeping in mind that $i^2 = -1$.

Addition and Subtraction

To add or subtract complex numbers, simply combine the real parts together and the imaginary parts together.

Example: $(4 + 3i) + (2 - i) = (4 + 2) + (3i - i) = 6 + 2i$

Multiplication

To multiply complex numbers, use the distributive property (or FOIL method), and remember to replace any $i^2$ with $-1$.

Example: Simplify $(3 + 2i)(4 - 5i)$

$(3 + 2i)(4 - 5i) = 12 - 15i + 8i - 10i^2$ $= 12 - 7i - 10(-1)$ $= 12 - 7i + 10$ $= 22 - 7i$

Division

To divide complex numbers, you need to eliminate $i$ from the denominator. You do this by multiplying both the numerator and the denominator by the complex conjugate of the denominator. The conjugate of $a + bi$ is $a - bi$.

Example: Find the quotient $\frac{2 + 3i}{1 - i}$

The complex conjugate of the denominator $1 - i$ is $1 + i$. Multiply the top and bottom by $1 + i$:

$\frac{2 + 3i}{1 - i} \cdot \frac{1 + i}{1 + i} = \frac{(2 + 3i)(1 + i)}{(1 - i)(1 + i)}$

Expand the numerator and the denominator:

• Numerator: $2(1) + 2(i) + 3i(1) + 3i(i) = 2 + 2i + 3i + 3i^2 = 2 + 5i + 3(-1) = -1 + 5i$
• Denominator: $1(1) + 1(i) - i(1) - i(i) = 1 - i^2 = 1 - (-1) = 2$

Put them back together:

$\frac{-1 + 5i}{2} = -\frac{1}{2} + \frac{5}{2}i$