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Chapter 6: Problem 20

Graph each complex number, and find its absolute value. \(-4-4 i\)

Short Answer

Expert verified
The complex number \(-4-4i\) is plotted at \(-4, -4\) and its absolute value is \4\sqrt{2}\.

Step by step solution

01

- Identify the Real and Imaginary Parts

For the complex number \(-4-4i\), the real part is \(-4\) and the imaginary part is \(-4\).
02

- Plot the Complex Number on the Complex Plane

Draw a complex plane with the real axis (horizontal) and the imaginary axis (vertical). Plot the point at \(-4\) on the horizontal axis and \(-4\) on the vertical axis. This is the location of the complex number \(-4-4i\) on the complex plane.
03

- Apply the Absolute Value Formula

The absolute value of a complex number \(a+bi\) is given by \(|a+bi| = \sqrt{a^2 + b^2}\). For the complex number \(-4 - 4i\), we substitute \(a = -4\) and \(b = -4\) into the formula.
04

- Calculate the Absolute Value

Using the formula \(|a+bi| = \sqrt{a^2 + b^2}\): \(|-4 - 4i| = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}\). The absolute value of \(-4-4i\) is \4\sqrt{2}\.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

complex plane
When dealing with complex numbers, we use a special type of graph called the complex plane.
The complex plane has two axes:
  • The horizontal axis is called the real axis.
  • The vertical axis is called the imaginary axis.
Each point on this plane represents a complex number. The position of a complex number is determined by its real and imaginary parts.
For example, the complex number \(-4-4i\) has:
  • A real part of -4, which we plot on the real axis.
  • An imaginary part of -4, which we plot on the imaginary axis.
So the point \(-4-4i\) is located at the coordinates (-4, -4) on the complex plane.
absolute value of complex numbers
The absolute value (or modulus) of a complex number \((a+bi)\) measures its distance from the origin (0,0) on the complex plane.
This concept is similar to finding the length of a line segment in geometry.
The formula to calculate the absolute value of a complex number is: \[|a+bi| = \sqrt{a^2 + b^2}\].
Let's apply this to our example \(-4-4i\):
  • Real part \(a = -4\)
  • Imaginary part \(b = -4\)
The absolute value is: \[ |-4-4i| = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2}\].
This tells us that the complex number \(-4-4i\) is \4\sqrt{2}\ units away from the origin.
plotting complex numbers
Plotting complex numbers on the complex plane helps us visualize and understand them better.
To plot a complex number:
  • First, draw a set of axes with the horizontal axis labeled as the real axis and the vertical axis as the imaginary axis.
  • Next, locate the real part of the number on the real axis.
  • Then, locate the imaginary part on the imaginary axis.
  • Plot the point where these two values intersect.
Take for instance the complex number \(-4-4i\).
Start at the origin (0,0):
  • Move 4 units to the left on the real axis to -4.
  • Then move 4 units down on the imaginary axis to -4.
  • The point where these two moves meet is (-4, -4), which is our complex number.
By plotting complex numbers, we can better see patterns, relationships, and even solve complex mathematical problems more easily.

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Most popular questions from this chapter

Find all solutions to the equation \(2 \sin ^{2}(2 \pi x)-1=0\) in the interval \((0, \pi / 2)\),

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Write each complex number in the form \(a+b i\) $$ 6.5\left(\cos 33.1^{\circ}+i \sin 33.1^{\circ}\right) $$
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