MATH 118 - Section 1.6 Hints

Exercise 45 in Section 1.6

Write the expression below in the standard form for a complex number:

\frac{1}{1 + i} - \frac{1}{1 - i} .

We have two options for this one:

We can first write the two separate expressions in standard form and then simplify the answer; or
We can combine the two fractions into one fraction and then express that single fraction in standard form.

Let's try the second method. We first put the two expressions over a common denominator, then combine the fractions and simplify:

\begin{aligned} \frac{1}{1 + i} - \frac{1}{1 - i} & = \frac{1}{1 + i} \cdot \frac{1 - i}{1 - i} - \frac{1}{1 - i} \cdot \frac{1 + i}{1 + i} \\ = \frac{1 - i}{(1 + i) (1 - i)} - \frac{1 + i}{(1 - i) (1 + i)} \\ = \frac{(1 - i) - (1 + i)}{(1 + i) (1 - i)} \\ = \frac{1 - i - 1 - i}{1 - i^{2}} \\ = \frac{- 2 i}{1 - (- 1)} \\ = \frac{- 2 i}{2} \\ = - i \\ = 0 + (- 1) i . \end{aligned}

In this case, we got lucky that the denominator simplified to a nice number.

Exercise 73 in Section 1.6

Given that $z = 3 - 4 i$ and $w = 5 + 2 i$ , simplify the expression $\overset{―}{z} + \overset{―}{w}$ .

Recall that $\overset{―}{z}$ denotes the complex conjugate of $z$ , which is the complex number with the same real part and opposite imaginary part as $z$ . So in this case $\overset{―}{z} = 3 + 4 i$ and $\overset{―}{w} = 5 - 2 i$ . So we have $\overset{―}{z} + \overset{―}{w} = (3 + 4 i) + (5 - 2 i) = 8 + 2 i$ .

Domain of a ratio of two functions

What is the domain of a ratio of two functions, say $\frac{f (x)}{g (x)}$ ?

The answer to this one is that the domain of such an expression is the set of all $x$ with the following properties:

$f (x)$ is defined, i.e., $x$ is in the domain of $f$ ;
$g (x)$ is defined, i.e., $x$ is in the domain of $g$ ; and
$g (x) \neq 0$ .

So the domain of that ratio is the set of values $x$ where both $f$ and $g$ are defined, and $g$ is nonzero. (It's okay if $f (x) = 0$ .)