Complex Numbers on the SAT - Complete Interactive Lesson
Part 1: Imaginary Unit
๐ข The Imaginary Unit
Part 1 of 7 โ Definition, Powers of , and the Cycle
The imaginary unit was invented to give meaning to the square root of negative numbers:
| Expression | Value |
|---|---|
General rule: for any positive number .
A complex number has the form , where is the real part and is the imaginary part.
Powers of โ The 4-Step Cycle
The powers of repeat every 4:
| Power | Value | Why |
|---|---|---|
| Definition | ||
Practice โ Powers of ๐
Simplifying Square Roots of Negatives
Always extract first, then simplify the radical:
Example 1:
Simplify these expressions. ๐งฎ
Write answers using (e.g. type "5i" or "-3i" or "1").
Match each expression to its simplified value. ๐
SAT-Style Questions ๐
Part 2: Complex Arithmetic
โ Adding & Subtracting Complex Numbers
Part 2 of 7 โ Combine Real with Real, Imaginary with Imaginary
Complex numbers have the form . When adding or subtracting, treat like a variable โ combine like terms.
Part 3: Complex Conjugates
โ๏ธ Multiplying Complex Numbers
Part 3 of 7 โ FOIL Method, , and Special Products
To multiply two complex numbers, use FOIL just like with binomials, then replace with :
Part 4: Quadratics & Complex Roots
โ Complex Conjugates & Division
Part 4 of 7 โ Conjugates, Rationalizing, Division
The complex conjugate of is . The product of a number and its conjugate is always real:
Part 5: Powers of i
๐ Solving Equations with Complex Solutions
Part 5 of 7 โ Negative Discriminants, , Quadratic Formula
Not every quadratic equation has real solutions. When the discriminant , the solutions are .
Part 6: Problem-Solving Workshop
๐ฏ Complex Numbers on the SAT
Part 6 of 7 โ Pattern Recognition, Common Traps, Cycles
Complex number questions appear in the Calculator and No Calculator sections. They test three main skills:
| Skill | Frequency |
|---|---|
| Powers of | โ โ โ Very common |
| Add/subtract/multiply | โ โ โ Very common |
| Division (conjugates) | โ โ Common |
| Discriminant / non-real solutions | โ โ Common |
| Writing in form |
Part 7: Review & Applications
๐ Review & Mixed Practice
Part 7 of 7 โ Cheat Sheet, Mixed Problems, Strategies
Complex Numbers Cheat Sheet
| Concept | Formula / Rule |
|---|---|
| Imaginary unit |