Section 1.4 - Complex Numbers

Section Objectives

1. Identify and simplify complex numbers.
2. Add, subtract, multiply, and divide complex numbers.
3. Simplify powers of $i$.

Imaginary Numbers

We've been solving lots of equations, but not every equation involving real numbers is solvable with real numbers. For example, there is no real number, $x$, with the property that $x^2+1=0$. To handle these kinds of equations, we must extend the real number system.

An imaginary number is a "number" whose square is a negative real number. The imaginary unit is $i=\sqrt{-1}$, and imaginary numbers can be written in the form $bi$, where $b$ is a real number.

$i$ has the property that $i^2=-1$. No real number has this property!

Every imaginary number can be written in terms of $i$ by using $\quad \sqrt{-k} = i \, \sqrt{k}$.

Examples

• [1] Write in terms of $i$ and simplify: $\quad \sqrt{-40}$ (Solution)

• [2] Write in terms of $i$ and simplify: $\quad \displaystyle \frac{\sqrt{-49}}{\sqrt{-7}}$ (Solution)

• [3] Write in terms of $i$ and simplify: $\quad \sqrt{-3} \cdot \sqrt{-12}$ (Solution)

Complex Numbers

A complex number is an expression of the form $a+bi$, where $a$ and $b$ are real numbers.

In the complex number $a+bi$, $a$ is called the real part and $b$ is called the imaginary part. The $a$ and $bi$ terms cannot be combined---they are not like terms. But every complex number can be simplified to the standard form $a+bi$.

Examples

• [4] Write as a complex number in standard form: $\quad 3+\sqrt{-9}+7$ (Solution)

• [5] Write as a complex number in standard form: $\quad (3-2i)+(4+2i)$ (Solution)

• [6] Write as a complex number in standard form: $\quad -3i \, (-2+5i)$ (Solution)

• [7] Write as a complex number in standard form: $\quad (5-i)(-3+3i)$ (Solution)

• [8] Write $i^5$ as a complex number in standard form. (Solution)

Conjugates and Dividing

The complex conjugate of the complex number $a+bi$ is the complex number $a-bi$. Notice that in forming the complex conjugate, we negate the imaginary part, but not the real part.

An important property of conjugates is the following:

$\displaystyle (a+bi)(a-bi) = a^2+b^2 \mbox{ (a real number!)}$

Example

• [9] Simplify: $\quad (3-2i)(3+2i)$ (Solution)

In order to divide by a complex number, we really just use the conjugate (and multiplication) to make the denominator real:

$\displaystyle$$\displaystyle \frac{a+bi}{c+di} = \frac{a+bi}{c+di} \cdot \frac{c-di}{c-di} = \frac{(a+bi)(c-di)}{c^2+d^2}=\frac{ac+bd}{c^2+d^2}+\frac{bc-ad}{c^2+d^2}i$

Examples

• [10] Write as a complex number in standard form: $\quad \displaystyle \frac{1-4i}{2+i}$ (Solution)

• [11] Write as a complex number in standard form: $\quad \displaystyle \frac{5+6i}{i}$ (Solution)

For further help...

1. Khan Academy has some excellent videos on introductions to complex numbers. Start at this page.