Complex Numbers, Imaginary Numbers, and Roots in Precalculus

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: $a + bi$, where a is the real part and b is the imaginary part.

• Imaginary unit: $i$ is defined as $i = \sqrt{-1}$.

Example: In $2 + 3i$, 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:

z	Real	Imaginary
2 + 3i	2	3
1 - 4i	1	-4
-6 + i	-6	1
-1 - 5i	-1	-5
2i	0	2
3	3	0
0	0	0

The Imaginary Number Cycle

Powers of i

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

• $i^1 = i$

• $i^2 = -1$

• $i^3 = -i$

• $i^4 = 1$

• $i^5 = i$ (cycle repeats)

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

Example: $i^7 = i^{4 \times 1 + 3} = i^3 = -i$

Evaluating Expressions with Imaginary Numbers

Sample Problems

1. Evaluate $33i^{10}$ $i^{10} = i^{4 \times 2 + 2} = (i^4)^2 \cdot i^2 = 1^2 \cdot (-1) = -1$ $33i^{10} = 33 \times (-1) = -33$

2. Evaluate $-3i^{25}$ $i^{25} = i^{4 \times 6 + 1} = (i^4)^6 \cdot i^1 = 1^6 \cdot i = i$ $-3i^{25} = -3i$

3. Evaluate $11i^{47}$ $i^{47} = i^{4 \times 11 + 3} = (i^4)^{11} \cdot i^3 = 1^{11} \cdot (-i) = -i$ $11i^{47} = 11 \times (-i) = -11i$

4. Evaluate $\sqrt{-44x^2}$ $\sqrt{-44x^2} = \sqrt{-1} \cdot \sqrt{44x^2} = i \cdot 2x\sqrt{11}$ Answer: $2x\sqrt{11}i$

5. Evaluate $\sqrt{-100y^2z}$ $\sqrt{-100y^2z} = \sqrt{-1} \cdot \sqrt{100y^2z} = i \cdot 10y\sqrt{z}$ Answer: $10y\sqrt{z}i$ Additional info: If $z$ 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 $P(x) = 0$ that are real numbers.

• Imaginary roots: Solutions to $P(x) = 0$ that are not real numbers, often involving $i$.

Graphing Example: $P(x) = x^2 + 1$

The graph of $y = x^2 + 1$ 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 $x^2 + 1 = 0$

• $x^2 = -1$

• $x = \pm i$

Finding Imaginary Roots

To find imaginary roots, use the quadratic formula:

• For $ax^2 + bx + c = 0$, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$

• If $b^2 - 4ac < 0$, the roots are imaginary.

Practice: Graphing Other Polynomials

• Draw $P(x) = x^2 + 2x + 3$ This parabola opens upwards and does not cross the x-axis (discriminant $= 4 - 12 = -8$), so both roots are imaginary.

• Draw $P(x) = x^2 - x - 1$ This parabola opens upwards and crosses the x-axis at two points (discriminant $= 1 + 4 = 5$), so both roots are real and irrational.

Additional info: The discriminant $b^2 - 4ac$ determines the nature of the roots: positive for two real roots, zero for one real root, negative for two imaginary roots.