Complex Numbers Practice

Practice Objectives

Practice Working with Complex Numbers

Instructions:

Answer 7/10 questions correctly to pass.

Simplify each, write your answer as:

$$a + bi$$

a is the real part
b is the imaginary part
Fractions can be written using the "/" key
- Negative fractions can be written as -a/b or a/-b
- All fractions must be simplified
Use 0 when needed:
- $$\text{Ex:} \, 3 = 3 + 0i$$
- $$\text{Ex:} \, 7i = 0 + 7i$$

Find each power of i and then choose the simplified form as 1, i, -1, or -i.

Note: Since i represents the square root of -1, you must algebraically eliminate i from any denominator.

Simplify each.

Note: Since i represents the square root of -1, you must algebraically eliminate i from any denominator.

Problem:

Correct!

Not Correct!

Your answer was: 0

The correct answer was: 0

Operations with Complex Numbers:

To add two or more complex numbers, we add their real parts and add their imaginary parts
Subtraction can be performed using addition of the opposite
- Alternatively, we can subtract their real parts and subtract their imaginary parts
We multiply complex numbers using the commutative, associative, and distributive properties
- Replace any occurrence of i² with -1
- Simplify by combining all real parts and all imaginary parts separately
When we divide complex numbers, we need to eliminate i in the denominator
- For a denominator of the form a + bi, (a ≠ 0 and b ≠ 0):
  - We multiply both numerator and denominator by the conjugate of the denominator
  - The conjugate of a + bi is a - bi
  - (a + bi)(a - bi) = a² + b²
- For a denominator of the form bi, (b ≠ 0):
  - We multiply both numerator and denominator by i if b < 0
  - We multiply both numerator and denominator by -i if b > 0
- The denominator will become a positive real number

Simplifying Powers of i:

If the exponent on i is negative, rewrite the power of i using the rule for negative exponents:
- $$i^{-a} = \frac{1}{i^{a}}, \, a > 0$$
- After rewriting, simplify the denominator
If the exponent on i is 0-4:
- $$i^0 = 1$$
- $$i^1 = i$$
- $$i^2 = -1$$
- $$i^3 = -i$$
- $$i^4 = 1$$
If the exponent on i is larger than 4, divide the exponent by 4 and record the remainder
Use the remainder to find the simplified value:
- The remainder will be either 0, 1, 2, or 3
- Match up the remainder with the corresponding exponent from the cycle (listed above)
Algebraically eliminate i from any denominator:
- Multiply both numerator and denominator by i to eliminate i from the denominator

Simplifying Square Roots of Negative Numbers:

For any positive real number b:
- $$\sqrt{-b} = i\sqrt{b}$$
When multiplying or dividing square roots of negative numbers:
- Change the form using the imaginary unit before performing any multiplication or division operations
- Use the product/quotient rule for radicals to simplify
- Use the definition i² = -1 to simplify

Step-by-Step:

You Have Missed 4 Questions...