Problem 20 - Precalculus Mathematics for Calculus

Question

Sketch \(z_{1}, z_{2}, z_{1}+z_{2},\) and \(z_{1} z_{2}\) on the same complex plane. $$z_{1}=-1+i, \quad z_{2}=2-3 i$$

Step-by-Step Solution

Answer

Plot \((-1, 1)\), \((2, -3)\), \((1, -2)\), and \((1, 5)\) on the complex plane.

Step 1: Plot the Points for \(z_1\) and \(z_2\)

The complex number \(z_1 = -1 + i\) can be plotted on the complex plane as the point \((-1, 1)\). Similarly, \(z_2 = 2 - 3i\) can be plotted as the point \((2, -3)\). Find these points on the plane where the x-axis represents the real part and the y-axis represents the imaginary part.

Step 2: Calculate \(z_1 + z_2\)

To find \(z_1 + z_2\), add the real parts and the imaginary parts separately: \((-1 + 2) + (1 - 3)i = 1 - 2i\). Plot the point \((1, -2)\) on the complex plane.

Step 3: Calculate \(z_1 z_2\)

To find \(z_1 z_2\), use the formula for multiplying complex numbers: \((-1 + i)(2 - 3i) = -1 \cdot 2 + (-1) \cdot (-3i) + i \cdot 2 + i \cdot (-3i)\). Simplify this to get: \(-2 + 3i + 2i - 3i^2\). Recall that \(i^2 = -1\), so this becomes \(-2 + 5i + 3 = 1 + 5i\). Plot the point \((1, 5)\) on the complex plane.

Step 4: Visualize on the Complex Plane

On the complex plane, mark the points \((-1, 1)\), \((2, -3)\), \((1, -2)\), and \((1, 5)\) corresponding to \(z_1\), \(z_2\), \(z_1 + z_2\), and \(z_1 z_2\) respectively. These points will help visualize the interactions and results between the given complex numbers.

Key Concepts

Complex Plane

The complex plane is a fantastic way to visualize complex numbers, which have both a real part and an imaginary part. Think of it like a graph where every point represents a complex number. The horizontal line, or x-axis, stands for the real part of the complex number. The vertical line, or y-axis, represents the imaginary part.

Each complex number is plotted as a point in this plane. For example, the complex number \( -1 + i \) is represented by the point \( (-1, 1) \), where \( -1 \) is the real part and \( 1 \) is the imaginary part. Similarly, \( 2 - 3i \) maps to the point \( (2, -3) \). This makes it easier to spot relationships between complex numbers and to perform operations like addition and multiplication on them.

Addition of Complex Numbers

When adding complex numbers, the process is quite simple—it’s like combining two vectors in a 2D space. You add their real parts together and their imaginary parts together. Consider our given numbers:

• For \( z_1 = -1 + i \), the real part is \( -1 \) and imaginary part is \( i \).
• For \( z_2 = 2 - 3i \), the real part is \( 2 \) and the imaginary part is \( -3i \).

So, to find \( z_1 + z_2 \), just add:

• Real parts: \( -1 + 2 = 1 \)
• Imaginary parts: \( 1 - 3 = -2 \)

Thus, \( z_1 + z_2 = 1 - 2i \). Plotting it on the complex plane results in the point \( (1, -2) \). This point shows where the sum of these complex numbers exists in the plane.

Multiplication of Complex Numbers

Multiplying complex numbers is a bit more complex (no pun intended!). It involves using the distributive property, similar to the multiplication of polynomials. Let's take our numbers \( z_1 = -1 + i \) and \( z_2 = 2 - 3i \). Multiply them by distributing each part:

• First, multiply the real parts: \( -1 \cdot 2 = -2 \).
• Next, multiply the outer parts: \( -1 \cdot (-3i) = 3i \).
• Then, multiply the inner parts: \( i \cdot 2 = 2i \).
• Lastly, multiply the imaginary parts: \( i \cdot (-3i) = -3i^2 \), knowing \( i^2 = -1 \), this becomes \( 3 \).

Combine these results: \(-2 + 3i + 2i + 3 = 1 + 5i\),which places us at the point \( (1, 5) \) on the complex plane. By understanding these operations, we can demystify the process of multiplying complex numbers.