Section 10.5 (a) - Intro to 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

The imaginary unit is $i=\sqrt{-1}$, where $i^2=-1$.

The square roots of negative numbers can be written in terms of $i$.

Examples

• Write in terms of $i$ and simplify: $\quad \sqrt{-40}$

• Write in terms of $i$ and simplify: $\quad \sqrt{-49}$

• Write in terms of $i$ and simplify: $\quad \sqrt{-3} \cdot \sqrt{-12}$

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

• Write as a complex number in standard form: $\quad 3+\sqrt{-9}+7$

• Write as a complex number in standard form: $\quad (5+7i)-(4+2i)$

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

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

• Write $i^5$ as a complex number in standard form.

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

• Simplify: $\quad (4-3i)(4+3i)$

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)+i(bc-ad)}{c^2+d^2}$

Examples

• Write as a complex number in standard form: $\quad \displaystyle \frac{7-i}{3-2i}$

• Write as a complex number in standard form: $\quad \displaystyle \frac{5+6i}{i+3}$