Problem 16 - Algebra and Trigonometry

Question

\(15-16=\) 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 points: \((-1, 1), (2, -3), (1, -2), (1, 5)\) on the complex plane.

Step 1: Understand the Complex Numbers

The complex numbers are given as \(z_1 = -1 + i\) and \(z_2 = 2 - 3i\). A complex number in the form \(a + bi\) can be plotted as a point on the complex plane, where \(a\) is the real part and \(b\) is the imaginary part.

Step 2: Plot the Complex Numbers

Plot \(z_1 = -1 + i\) on the complex plane at the coordinates (-1, 1). Similarly, plot \(z_2 = 2 - 3i\) at the coordinates (2, -3). These points represent \(z_1\) and \(z_2\) on the plane.

Step 3: Calculate the Sum \(z_1 + z_2\)

Add the complex numbers: \(z_1 + z_2 = (-1 + i) + (2 - 3i) = 1 - 2i\). Plot the sum \(z_1 + z_2\) at the coordinates (1, -2) on the complex plane.

Step 4: Calculate the Product \(z_1 \cdot z_2\)

Multiply the complex numbers: \[z_1 \cdot z_2 = (-1 + i)(2 - 3i) = -2 + 3i + 2i - 3i^2 = -2 + 5i + 3 = 1 + 5i\]. The product is plotted at (1, 5) on the complex plane.

Key Concepts

Imaginary Numbers

Imaginary numbers are an intriguing component of complex numbers that might seem mysterious at first. Unlike the ordinary numbers we often encounter, imaginary numbers are not located on the number line but are part of a broader concept called the complex plane. Imaginary numbers are defined as multiples of the square root of -1, denoted as 'i'. The fundamental rule here is that using the imaginary unit, we can represent numbers that cannot be measured on the traditional number line. For instance, when you hear of a number like 'i', it stands for the square root of -1.

• If you multiply 'i' by itself, you get -1: \( i^2 = -1 \).
• Imaginary units such as '2i' or '-3i' are just multiples of 'i'.

Together, these imaginary numbers are vital in forming complex numbers, which we explore next.

Complex Plane

The complex plane is like a playground for complex numbers, allowing us to see and interact with them visually. Instead of just a single line like the number line, the complex plane consists of two perpendicular axes:

• The horizontal axis, which represents the real part of a complex number.
• The vertical axis, which represents the imaginary part.

Picture it like a graph where each complex number is a point. For example, the complex number \(-1 + i\) is plotted at the coordinates (-1, 1) on this plane. The complex plane not only helps us visualize complex numbers but also makes operations like addition and multiplication more intuitive.

Addition of Complex Numbers

Adding complex numbers is more straightforward than you might imagine. Each complex number has a real part and an imaginary part. To add them, you simply add the corresponding parts together. Given two complex numbers like \( z_1 = -1 + i \) and \( z_2 = 2 - 3i \), the addition involves:

• Adding the real parts: \( -1 + 2 = 1 \)
• Adding the imaginary parts: \( 1 + (-3) = -2 \)

So, the sum \( z_1 + z_2 \) is \( 1 - 2i \). On the complex plane, this resulting complex number is plotted at (1, -2). The key to remember here is to treat the addition of real and imaginary parts separately.

Multiplication of Complex Numbers

Multiplying complex numbers might sound complex, but it follows a pattern that you probably already know: the distributive property. To find the product of two complex numbers like \( z_1 = -1 + i \) and \( z_2 = 2 - 3i \), you will:

• Multiply each part of the first complex number with each part of the second one.
• Apply the formula: \((a + bi)(c + di) = ac + adi + bci + bdi^2 \)

Using this, our product calculation becomes \(-2 + 3i + 2i - 3i^2 \), which simplifies to \( 1 + 5i \) after substituting \( i^2 = -1 \). On the complex plane, this product is represented at (1, 5). Multiplication in the complex plane not only combines the parts but also involves rotating and scaling, which can lead to fascinating insights in fields like signal processing and physics.