Chapter 4: Problem 9
Follow the procedure illustrated in Example 4 to determine the indicated roots
of the given complex number.
Short Answer
Expert verified
Answer: The fourth roots of the complex number 1 are:
1. .
2. .
3. .
4. .
Step by step solution
01
Convert the given complex number to polar form
To find the roots, we first need to convert the given complex number to its polar form. The complex number is 1, which can be represented as . In polar form, we represent a complex number as , where is the magnitude and is the angle.
The magnitude (r) can be calculated using the formula , where and are the real and imaginary parts of the complex number, respectively. In this case, and .
The angle can be calculated using the formula . In this case, since and , the angle is:
So, the polar form of the given complex number is .
02
Use De Moivre's theorem to find the fourth roots
According to De Moivre's theorem, to find the n-th root of a complex number in polar form, we should divide the angle by n and raise the magnitude to the power of . In this case, we want to find the fourth roots, so .
For each root, the angle will be:
Also, the magnitude of the roots will be:
Now we can find the fourth roots:
1. : , so the root is .
2. : , so the root is .
3. : , so the root is .
4. : , so the root is .
03
Writing the roots in rectangular form (optional)
If you need the roots in rectangular form (a + bi), you can use the trigonometric identities:
and
However, since the question doesn't specifically ask for the roots in rectangular form, we'll leave them in polar form.
So, the fourth roots of the given complex number 1 are:
1. .
2. .
3. .
4. .
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Polar Form
To understand complex numbers fully, it's important to know how to convert them into polar form. A complex number like 1 can be expressed as 1 + 0i in its basic rectangular form. In polar terms, it's represented as . Here, is the magnitude and is the angle (often called the argument).
The magnitude is found using the formula , where is the real part, and is the imaginary part of the number.
For the number 1, since and , we get .
The angle is calculated as . Here, that means .
The magnitude
For the number 1, since
The angle
- The polar form of 1 is thus
.
De Moivre's Theorem
De Moivre's theorem is a powerful tool that simplifies many operations on complex numbers when they are in polar form. It's used to find powers and roots of complex numbers.
According to this theorem, if you have a complex number in the form of , its -th root is determined by:
In our example, we find the fourth roots of 1 by dividing the angle by 4 while maintaining a magnitude of 1. We compute four roots by substituting .
According to this theorem, if you have a complex number in the form of
- New magnitude:
- New angles:
, where is an integer.
In our example, we find the fourth roots of 1 by dividing the angle by 4 while maintaining a magnitude of 1. We compute four roots by substituting
Fourth Roots
To understand the fourth roots of a complex number, such as 1, one can use both geometric and algebraic perspectives. The fourth roots, viewed geometrically, are points evenly spaced on a circle in the complex plane.
For a complex number of magnitude 1 like in this example, every root has a magnitude of .
What varies is the angle: we compute four distinct angles by using for .
For a complex number of magnitude 1 like in this example, every root has a magnitude of
What varies is the angle: we compute four distinct angles by using
gives angle . Root is . gives angle . Root is . gives angle . Root is . gives angle . Root is .
Rectangular Form
Complex numbers are often represented in what's called rectangular form, which is . To shift from polar to rectangular form, we apply trigonometric functions to get the Cartesian coordinates and .
For a complex number , we find it in rectangular form by calculating:
For instance, the polar root translates to , or just in rectangular form.
This process helps in visualizing and performing arithmetic on complex roots.
For a complex number
For instance, the polar root
This process helps in visualizing and performing arithmetic on complex roots.