Algebra II Recipe: Complex Numbers

A. Definitions

1. Imaginary unit "i" is equal to
2. Complex number in standard form is a + bi.
   • "a" and "b" are real numbers.
3. Pure imaginary number is in the form bi.

B. Simplifying "i" Raised to Some Number

1. Divide the exponent by 4 and determine the remainder
2. Raise "i" to that power to simplify.
• i = i
• i² = -1
• i³ = -i
• i⁴ = 1

C. Simplifying the Square Root of a Negative Number

1. Factor out the square root of -1, which equals "i".
2. Simplify the radical of the positive number if possible.

D. Graphing a Complex Number in Standard Form a + bi

1. Rename the axes
• The traditional x-axis is the real axis or "a".
• The traditional y-axis is the imaginary axis or "b".
2. Start at the origin.
3. Use the values of "a" and "b" to graph the point.
• The value of "a" determines whether to move left or right along the real axis.
• The value of "b" determines whether to move up or down along the imaginary axis.
4. Graph the point after making these two moves.

E. Operations with Complex Numbers

1. Perform the operations as though "i" were any variable.
2. Simplify "i" when possible. Refer to Part B above.
3. Write answers in standard form a + bi.

Practice Problems