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Magazine Chapter 3: Problem 16
Plot the complex numbers on the complex plane. \(-3-4 i\)
Short Answer
Expert verified
Plot the point at (-3, -4) on the complex plane.
Step by step solution
01
Understanding Complex Numbers
A complex number has the form \(a + bi\), where \(a\) is the real part and \(b\) the imaginary part. Here, \(-3-4i\) means that the real part is \(-3\) and the imaginary part is \(-4\).
02
Identifying the Axes
The complex plane uses two axes: the horizontal axis (real axis) and the vertical axis (imaginary axis). Plot points by moving on these axes similar to coordinates in a Cartesian plane.
03
Plotting the Real Part
Locate the real part, \(-3\), on the horizontal axis (real axis). From the origin (0,0), move 3 units to the left, as \(-3\) is negative.
04
Plotting the Imaginary Part
Locate the imaginary part, \(-4\), on the vertical axis (imaginary axis). From the current location \(-3\) on the horizontal axis, move 4 units down, since \(-4\) is negative.
05
Marking the Point
Mark the point at the intersection of \(-3\) on the real axis and \(-4\) on the imaginary axis. This point represents the complex number \(-3-4i\) on the complex plane.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Complex Plane
The complex plane is a two-dimensional plane used to visually represent complex numbers. Unlike the typical Cartesian coordinate plane, the complex plane integrates an imaginary component. There are two key axes:
- The horizontal axis, also known as the real axis, represents the real part of complex numbers.
- The vertical axis, known as the imaginary axis, stands for the imaginary component.
Real Part
The real part of a complex number is the component that does not involve the imaginary unit
"i". It is represented by the coefficient of the "a" in the expression "a + bi". For example, in
"-3-4i", the real part is
"-3". This value is plotted on the horizontal axis of the complex plane.
When plotting the real part, the sign of the coefficient determines its position on the real axis:
When plotting the real part, the sign of the coefficient determines its position on the real axis:
- A positive value moves to the right from the origin.
- A negative value, like our example "-3," shifts leftwards.
Imaginary Part
The imaginary part of a complex number involves the coefficient multiplying
"i", the imaginary unit. Think of it as the "b" in the format
"a + bi". In the example
"-3 - 4i", the imaginary part
is
"-4". This is represented on the vertical axis of the complex plane.
Here's a simple breakdown of how to plot the imaginary component:
Here's a simple breakdown of how to plot the imaginary component:
- Positive values shift the point upward from the position on the real axis.
- Negative values, such as our "-4", require a downward movement.
Plotting Points
Plotting points on the complex plane follows a straightforward approach similar to Cartesian coordinates. To accurately represent a complex number like
e
"-3 - 4i", follow these steps:
- Start by identifying the real and imaginary parts.
- Next, locate the real part on the horizontal axis.
- From there, move according to the imaginary part on the vertical axis.
- The intersection of these moves marks the location of the complex number.
