Problem Analysis:
The task involves solving a series of problems related to
complex numbers. The problems are structured in a way that requires understanding the properties and operations of complex numbers, such as addition, multiplication, conjugates, magnitudes, and roots of unity.
#### Key Concepts:
1.
Complex Numbers: A complex number is of the form \( z = a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit with \( i^2 = -1 \).
2.
Magnitude (Modulus): The magnitude of \( z = a + bi \) is given by \( |z| = \sqrt{a^2 + b^2} \).
3.
Conjugate: The conjugate of \( z = a + bi \) is \( \overline{z} = a - bi \).
4.
Roots of Unity: The \( n \)-th roots of unity are the solutions to the equation \( z^n = 1 \). They are evenly spaced on the unit circle in the complex plane.
5.
De Moivre's Theorem: For any complex number \( z = r(\cos \theta + i \sin \theta) \) and integer \( n \), \( z^n = r^n (\cos(n\theta) + i \sin(n\theta)) \).
Solution to Each Problem:
#### Problem 1:
Question: If \( z_1 = 1 + i \) and \( z_2 = 2 - i \), find \( z_1 + z_2 \) and \( z_1 \cdot z_2 \).
-
Addition:
\[
z_1 + z_2 = (1 + i) + (2 - i) = (1 + 2) + (i - i) = 3
\]
-
Multiplication:
\[
z_1 \cdot z_2 = (1 + i)(2 - i) = 1 \cdot 2 + 1 \cdot (-i) + i \cdot 2 + i \cdot (-i) = 2 - i + 2i - i^2
\]
Since \( i^2 = -1 \):
\[
z_1 \cdot z_2 = 2 - i + 2i + 1 = 3 + i
\]
Answer:
\[
z_1 + z_2 = 3, \quad z_1 \cdot z_2 = 3 + i
\]
#### Problem 2:
Question: Find the magnitude of \( z = 3 - 4i \).
-
Magnitude:
\[
|z| = \sqrt{3^2 + (-4)^2} = \sqrt{9 + 16} = \sqrt{25} = 5
\]
Answer:
\[
|z| = 5
\]
#### Problem 3:
Question: If \( z = 2 + 3i \), find the conjugate of \( z \) and the product \( z \cdot \overline{z} \).
-
Conjugate:
\[
\overline{z} = 2 - 3i
\]
-
Product:
\[
z \cdot \overline{z} = (2 + 3i)(2 - 3i) = 2^2 - (3i)^2 = 4 - 9(-1) = 4 + 9 = 13
\]
Answer:
\[
\overline{z} = 2 - 3i, \quad z \cdot \overline{z} = 13
\]
#### Problem 4:
Question: Solve the equation \( z^2 + 4z + 5 = 0 \) for \( z \).
-
Quadratic Formula:
\[
z = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
Here, \( a = 1 \), \( b = 4 \), and \( c = 5 \):
\[
z = \frac{-4 \pm \sqrt{4^2 - 4 \cdot 1 \cdot 5}}{2 \cdot 1} = \frac{-4 \pm \sqrt{16 - 20}}{2} = \frac{-4 \pm \sqrt{-4}}{2} = \frac{-4 \pm 2i}{2}
\]
Simplify:
\[
z = -2 \pm i
\]
Answer:
\[
z = -2 + i \quad \text{or} \quad z = -2 - i
\]
#### Problem 5:
Question: Find the cube roots of unity.
-
Cube Roots of Unity:
The cube roots of unity are the solutions to \( z^3 = 1 \). These are:
\[
z = 1, \quad z = \omega, \quad z = \omega^2
\]
where \( \omega = e^{2\pi i / 3} = -\frac{1}{2} + i\frac{\sqrt{3}}{2} \) and \( \omega^2 = e^{-2\pi i / 3} = -\frac{1}{2} - i\frac{\sqrt{3}}{2} \).
Answer:
\[
1, \quad -\frac{1}{2} + i\frac{\sqrt{3}}{2}, \quad -\frac{1}{2} - i\frac{\sqrt{3}}{2}
\]
Final Answer:
\[
\boxed{3, 3+i, 5, 2-3i, 13, -2+i, -2-i, 1, -\frac{1}{2} + i\frac{\sqrt{3}}{2}, -\frac{1}{2} - i\frac{\sqrt{3}}{2}}
\]
Parent Tip: Review the logic above to help your child master the concept of dot to dot mindfulness printable.
