Complex Numbers

The Need for a New Number

You already know that the equation $x^2 + 1 = 0$ has no real solution — no real number, when squared, gives $-1$. To solve such equations, we extend the real number system by introducing a new symbol.

We denote $\sqrt{-1}$ by the symbol $i$. This gives us the defining property:

So $i$ is a solution of $x^2 + 1 = 0$. With this single new number, we can now build an entirely new system of numbers.

Definition of a Complex Number

A complex number is any number of the form $a + ib$, where $a$ and $b$ are real numbers, and $i = \sqrt{-1}$.

For example, each of the following is a complex number:

• $2 + i3$
• $(-1) + i\sqrt{3}$
• $\frac{1}{4} - \frac{11}{4}i$

Real and Imaginary Parts

For a complex number $z = a + ib$:

• $a$ is called the real part of $z$, written as $\text{Re}(z)$
• $b$ is called the imaginary part of $z$, written as $\text{Im}(z)$

Example: If $z = 2 + i5$, then $\text{Re}(z) = 2$ and $\text{Im}(z) = 5$.

Equality of Two Complex Numbers

Two complex numbers $z_1 = a + ib$ and $z_2 = c + id$ are equal if and only if their real parts are equal and their imaginary parts are equal. That is:

This is a crucial property — a single equation in complex numbers gives two separate equations in real numbers.

Worked Example (from the textbook)

Example 1: If $4x + i(3x - y) = 3 + i(-6)$, where $x$ and $y$ are real numbers, find the values of $x$ and $y$.

Solution:

We have:

Equating the real parts:

Equating the imaginary parts:

From the first equation:

Substitute into the second equation:

$3\left(\frac{3}{4}\right) - y = -6$ …