NonReal Complex Numbers:
The “Standard Form” is a+bi
Simplifying Radicals: Always take out a √−𝟏 first
Example: Simplify √−75. First step must be √75 ∙ √−1 → √75 𝑖
Then simplify the square root of the positive number: → √25 √3 𝑖 → 5 √3 𝑖.
Addition and Subtraction are easy Like Terms problems
Example: (5 − 2𝑖 ) − (7 + 6𝑖 ) → 5 − 2𝑖 − 7 − 6𝑖 → −2 − 8𝑖.
Multiplication – works as usual, just remember to change 𝒊𝟐 → (−𝟏) and simplify further
Distributive Property example: 2𝑖 ∙ (5 + 3𝑖) → 10𝑖 + 6𝑖 2 → 10𝑖 + 6(−1) → 10𝑖 − 6 and say it in standard form, −6 + 10𝑖.
FOIL example: (3 + 4𝑖)(5 − 6𝑖) → 15 − 18𝑖 + 20𝑖 − 24𝑖 2 → 15 + 2𝑖 − 24(−1) → 15 + 2𝑖 + 24 → 9 + 2𝑖
Division problems and Rationalize the denominator problems: Don’t leave “i” in the denominator
If there’s One Term in Denominator
How to
do it
Multiply top and bottom by 𝒊.
8 𝒊
8i
8𝑖
8𝑖
∙ → 2→
→
3i 𝒊 3i
3(−1) −3
Example
Put your answer in standard form:
8
− 𝑖
3
If there are Two Terms in Denominator
Multiply top and bottom by the conjugate of the denominator.
Numerator: Distributive Property or FOIL
Denominator: Special pattern shortcut! (𝑨 + 𝑩)(𝑨 − 𝑩) → 𝑨𝟐 − 𝑩𝟐
Distributive or FOIL in the top;
(𝑎 + 𝑏𝑖)(𝑎 − 𝑏𝑖) → 𝐴2 + 𝐵 2 in bottom;
no additional simplification needed.
2
𝟓 − 𝟑𝒊 2(5 − 3𝑖) 2(5 − 3𝑖) 1(5 − 3𝑖)
∙
→
→
→
5 + 3𝑖 𝟓 − 𝟑𝒊
25 + 9
34
17
1(5 − 3𝑖)
5
3
→
− 𝑖
17
17 17
9_i.docx — 4/9/2014 12:20 PM – D.R.S.