Showing posts with label complex. Show all posts
Showing posts with label complex. Show all posts

Complex Rational Expressions

It turns out that we have all the tools necessary to simplify complex algebraic fractions.

Rational Expressions and Equations Playlist on YouTube

The numerator and denominator of these rational expressions contain fractions and look very intimidating.  We will outline two methods for simplifying them.

Method 1: Obtain a common denominator for the numerator and denominator, multiply by the reciprocal of the denominator, then factor and cancel if possible.


Method 2: Multiply the numerator and denominator of the complex fraction by the LCD of all the simple fractions then factor and cancel if possible.


To illustrate what happened after we multiplied by the LCD we could add an extra step.

For the following solved problems, both methods are used. Choose whichever method feels most comfortable for you.

Simplify using method 1. Simplify using method 2.
Video Examples on YouTube:


Complex Numbers

Up to this point, we have been working exclusively with real numbers.  Now we shed this limitation and allow for a much broader range of problems.


With these definitions, we have greatly expanded our space of numbers.  Notice that any real number is also a complex number, for example 5 = 5 + 0i.  Here the real part is 5 and the imaginary part is 0.  Next we consider powers of i.
Simplify.

Adding and subtracting complex numbers is just a matter of adding like terms. Be sure to use the order of operations and add real and imaginary parts separately.

Add or subtract.
We have to use a bit of caution when multiplying complex numbers.  First, we will run into i^2 often. In this case, we will replace them all with -1.  In addition, the property
is true only when either A or B is non-negative; therefore, simplify using the imaginary unit before multiplying.
Tip: Make use of the imaginary unit if the radicand is negative before trying to simplify.

Multiply.
Multiplying complex numbers often requires the distributive property.
Dividing complex numbers requires techniques similar to rationalizing the denominator.

Divide.
When a complex number is in the denominator, multiply numerator and denominator by its conjugate.

Video Examples on YouTube: