Problem 4 - Precalculus

Question

Graph each complex number in the complex plane. $$-3-2 i$$

Step-by-Step Solution

Answer

Plot the point at (-3, -2) on the complex plane.

Step 1: Understand the complex number

The given complex number is \(-3 - 2i\). This number is in the form \(a + bi\), where \(a\) is the real part and \(b\) is the imaginary part.

Step 2: Identify the real and imaginary parts

In the complex number \(-3 - 2i\): - The real part, \(a\), is \(-3\). - The imaginary part, \(b\), is \(-2\).These values determine the position on the complex plane.

Step 3: Plot the real part on the horizontal axis

On the complex plane: - The horizontal axis represents the real part. - Locate \(-3\) on the horizontal axis, which is the negative side of the axis.

Step 4: Plot the imaginary part on the vertical axis

On the complex plane: - The vertical axis represents the imaginary part. - Locate \(-2\) on the vertical axis, which is below the origin since it's a negative value.

Step 5: Plot the complex number on the complex plane

Combine the positions from the real and imaginary parts: - From \(-3\) on the horizontal axis, move down to \(-2\) on the vertical axis. - Plot the point where these positions intersect, which is at \((-3, -2)\) in the complex plane.

Key Concepts

Understanding the Real Part of a Complex Number

When dealing with complex numbers, the real part refers to the component that doesn't involve the imaginary unit, represented by the letter \( i \). Complex numbers are expressed in the form \( a + bi \), where \( a \) is the real part. For example, in the complex number \(-3 - 2i\), the real part is \(-3\).
Understanding the real part is crucial because it determines the position of the complex number along the horizontal axis on the complex plane:

• The real part is a standard integer or decimal number.
• It tells you how far left or right the number is from the origin, along the real axis.
• If the real part is negative, as in this exercise, it appears on the left side of the vertical axis.

Observing the real part helps in visualizing where the complex number lies along the horizontal dimension of the plane. By recognizing this component, you can graph the point accurately on the complex plane.

Understanding the Imaginary Part of a Complex Number

The imaginary part of a complex number is the component that is accompanied by the imaginary unit \( i \), such as in \( a + bi \). This part is crucial because it defines the position of the number along the vertical axis in the complex plane. Taking the example \(-3 - 2i\), the imaginary part is \(-2\).
The imaginary part on the complex plane is plotted on the vertical axis, known as the imaginary axis:

• This component dictates how far up or down the number is from the horizontal axis.
• In the exercise, the imaginary part is negative, placing it below the horizontal real axis.
• The vertical positioning reflects the influence of the imaginary unit, \( i \), which represents a 90-degree rotation.

The existence of the imaginary part allows for a broader number system, extending beyond real numbers. Recognizing and plotting this component is vital to accurately locating the complex number on the complex plane.

Exploring the Complex Plane

The complex plane is a two-dimensional plane used to graph complex numbers. It is similar to the Cartesian coordinate system but specifically designed to handle complex numbers. Each complex number corresponds to a unique point in this plane.
The complex plane consists of two intersecting axes:

• The horizontal axis (real axis) represents the real part of complex numbers.
• The vertical axis (imaginary axis) represents the imaginary part.

In the case of the complex number \(-3 - 2i\), plotting it involves:

• Locating \(-3\) on the real axis.
• Finding \(-2\) on the imaginary axis.
• Drawing a point where these two values intersect, specifically at \((-3, -2)\).

Understanding the complex plane is crucial for visualizing complex numbers and seeing the relationship between their real and imaginary components. This visual representation helps in various fields of mathematics and engineering, offering insights into how complex systems behave.