Linear Equations

  Complex Numbers
Definitions
Complex Addition and Multiplication
Complex Division
Extending the Square Root

  Linear Equations
Definition
Manipulating Linear Equations
Problem Solving

  More Equations
Variable Denominators
Ratio and Proportion
More Problem Solving

Definitions

A complex number written as a +bi is said to be in standard form.
a is called the real part of the complex number
b is called the imaginary part of the complex number

Examples of Complex Numbers

3 + 5i Already in standard form
10 − 4i 10 + (−4)i
7 7 + 0i
3i 0 + 3i

All real numbers are

Complex Addition and Multiplication

To add complex numbers, just add the real parts and the imaginary
parts separately.

To multiply complex numbers, treat them like binomials and FOIL
the product .

The Conjugate

Recall the special pattern

(a + b)(a − b) = a² − b².

Apply this to complex numbers (replace b with bi):

(a + bi)(a − bi) = a² − (bi)² = a² − b²i² = a² + b².

Changing the sign of the imaginary part is a useful operation: it
can be used to produce a real number!

Note that a + bi is also the conjugate of a − bi.

Dividing by Complex Numbers

To divide by a complex number—that is, when a fraction has a
complex denominator—use the conjugate to transform the
denominator to a real number.

Examples

A Better Definition

Our definition of i (”a number such that i² = −1”) doesn’t really
define i. It just states a property of i.

We can make the definition more specific by stating that i will be
the principal square root of −1:
 

Definition The principal square root of −b is denoted by and defined as

Now we can define square roots of negative numbers :

Complications

Recall the Properties of Radicals :

Properties of Radicals If and are real numbers, then

These do not apply if b or c is negative!

For instance, using the definition of we have

But, if we tried to use the Properties, we’d get

which is wrong.

Linear Equations

Examples

Solving Linear Equations

We may add, subtract, multiply, or divide by the same quantity on
both sides of an equation. The goal is to isolate the variable by
moving everything else to the other side.

It might be necessary to simplify first .

Always check your answers!

Find three consecutive integers whose sum is 21.

We need to represent the three integers with variables. Call the
smallest one n. Then the other two are n + 1 and n + 2.
They add up to 21:

Make sure you answer the question!

The three integers are 6, 7, and 8.

  One number is 5 less than another number. Find the numbers
if five times the smaller number is 11 less than four times the
larger number. 9 and 14

The average of the salaries of Kelly, Renee, and Nina is
$20,000 a year. If Kelly earns $4000 less than Renee, and
Nine’s salary is two-thirds of Renee’s salary, find the salary of
each person. $24,000 for Renee, $20,000 for Kelly,
$16,000 for Nina

Variable Denominators

Beware of Zero Denominators !

If a variable appears in a denominator,
multiply by a common denominator to clear fractions, and
take special care to ensure that the denominator is not zero.
In other words, check your answers!

Oops!

This equation has no solution!