Complex Numbers and Imaginary Numbers

Definition and Structure of Complex Numbers

Complex numbers are numbers that have both a real and an imaginary component. They are written in the form:

• Standard form: , where a is the real part and b is the imaginary part.

• Imaginary unit: is defined as .

Example: In , 2 is the real part and 3 is the imaginary part.

Identifying Real and Imaginary Parts

Each complex number can be separated into its real and imaginary components. The following table classifies several complex numbers:

The Imaginary Number Cycle

Powers of i

The powers of the imaginary unit repeat in a cycle of four:

• (cycle repeats)

For any integer , can be determined by dividing by 4 and using the remainder to find the equivalent power in the cycle.

Example:

Evaluating Expressions with Imaginary Numbers

Sample Problems

1. Evaluate

2. Evaluate

3. Evaluate

4. Evaluate Answer:

5. Evaluate Answer: Additional info: If is negative, the square root would involve further imaginary numbers.

Real and Imaginary Roots of Polynomials

Understanding Roots

Polynomials can have both real and imaginary (complex) roots. Real roots are the x-values where the graph of the polynomial crosses the x-axis. Imaginary roots do not correspond to x-intercepts on the real number line.

• Real roots: Solutions to that are real numbers.

• Imaginary roots: Solutions to that are not real numbers, often involving .

Graphing Example:

The graph of is a parabola opening upwards, with its vertex at (0, 1). It does not cross the x-axis, indicating no real roots. The roots are imaginary:

• Set

Finding Imaginary Roots

To find imaginary roots, use the quadratic formula:

• For ,

• If , the roots are imaginary.

Practice: Graphing Other Polynomials

• Draw This parabola opens upwards and does not cross the x-axis (discriminant ), so both roots are imaginary.

• Draw This parabola opens upwards and crosses the x-axis at two points (discriminant ), so both roots are real and irrational.

Additional info: The discriminant determines the nature of the roots: positive for two real roots, zero for one real root, negative for two imaginary roots.