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Magazine Chapter 7: Problem 4
Graph each complex number in the complex plane. $$-3-2 i$$
Short Answer
Expert verified
Plot the point at (-3, -2) on the complex plane.
Step by step solution
01
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.
02
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.
03
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.
04
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.
05
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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:
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.
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:
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.
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 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.
- Locating \(-3\) on the real axis.
- Finding \(-2\) on the imaginary axis.
- Drawing a point where these two values intersect, specifically at \((-3, -2)\).
