Algebra and Trigonometry, 5th Edition

Chapter 3 Summary and Review

Study Guide

Key Terms and Concepts	Example
Section 3.1: The Complex Numbers
The number i is defined such that $i=\sqrt{\mathrm{-1}}$ and $i^2=\mathrm{-1}$.	Express each number in terms of i. $$\sqrt{\mathrm{-5}}=\sqrt{\mathrm{-1}\cdot 5}=\sqrt{\mathrm{-1}}\cdot \sqrt{5}=i\sqrt{5},\text{ or }\sqrt{5}i$$; $$-\sqrt{\mathrm{-36}}=-\sqrt{\mathrm{-1}\cdot 36}=-\sqrt{\mathrm{-1}}\cdot \sqrt{36}=-i\cdot 6=\mathrm{-6}i$$
A complex number is a number of the form $a+bi$, where a and b are real numbers. The number a is said to be the real part of $a+bi$, and the number b is said to be the imaginary part of $a+bi$. To add or subtract complex numbers, we add or subtract the real parts, and we add or subtract the imaginary parts.	Add or subtract. $$\begin{array}{rcl}(\mathrm{-3}+4i)+(5-8i)& =& (\mathrm{-3}+5)+(4i-8i)\\ =& 2-4i;\\ (6-7i)-(10+3i)& =& (6-10)+(\mathrm{-7}i-3i)\\ =& \mathrm{-4}\mathrm{-10}i\end{array}$$
When we multiply complex numbers, we must keep in mind the fact that $i^2=\mathrm{-1}$. Note that $\sqrt{a}\cdot \sqrt{b}\ne \sqrt{ab}$ when $\sqrt{a}$ and ...