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Magazine Chapter 8: Problem 42
Rewrite each complex number into polar \(r e^{i \theta}\) form. $$ -4-4 i $$
Short Answer
Expert verified
The polar form is \(4\sqrt{2} e^{i \frac{5\pi}{4}}\).
Step by step solution
01
Identify the Cartesian Coordinates
The complex number is given in the form of \(-4 - 4i\), where the real part \(a = -4\) and the imaginary part \(b = -4\).
02
Calculate the Magnitude (r)
The magnitude \(r\) of a complex number \(a + bi\) is calculated using the formula: \[ r = \sqrt{a^2 + b^2} \] Substituting the values, we get: \[ r = \sqrt{(-4)^2 + (-4)^2} = \sqrt{16 + 16} = \sqrt{32} = 4\sqrt{2} \]
03
Determine the Argument (胃)
The argument \(\theta\) is found using the formula \(\tan \theta = \frac{b}{a}\). So, \(\tan \theta = \frac{-4}{-4} = 1\). Thus, \(\theta = \text{arctan}(1)\). Since the point \(-4, -4\) is in the third quadrant, \(\theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4}\).
04
Write the Polar Form
The polar form of a complex number is expressed as \(r e^{i \theta}\). Substituting \(r = 4\sqrt{2}\) and \(\theta = \frac{5\pi}{4}\), the polar form is: \[ 4\sqrt{2} e^{i \frac{5\pi}{4}} \].
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Cartesian Coordinates
When dealing with complex numbers, Cartesian coordinates provide a way of expressing them in terms of their real and imaginary parts. For any complex number expressed as \( a + bi \), \( a \) represents the real part and \( b \) represents the imaginary part. These correspond to coordinates on a plane:
For example, in the complex number \(-4 - 4i\), the Cartesian coordinates are \(-4\) (real) and \(-4\) (imaginary). Knowing these coordinates helps to graph the number on the complex plane, which is an essential step in converting it to polar form.
- \( a \) is the x-coordinate (horizontal axis).
- \( b \) is the y-coordinate (vertical axis).
For example, in the complex number \(-4 - 4i\), the Cartesian coordinates are \(-4\) (real) and \(-4\) (imaginary). Knowing these coordinates helps to graph the number on the complex plane, which is an essential step in converting it to polar form.
Magnitude of Complex Number
The magnitude of a complex number is akin to its distance from the origin of the complex plane. Calculating the magnitude requires Pythagoras' theorem as complex numbers are considered as points in a two-dimensional space. For the complex number \( a + bi \), the magnitude \( r \) can be calculated as:
This formula is derived from the Pythagorean theorem. It computes the hypotenuse of a right-angled triangle formed by the real and imaginary components. In our example of \(-4 - 4i\), the magnitude is:
The magnitude gives us the "length" of the complex number from the origin in the plane.
- \( r = \sqrt{a^2 + b^2} \)
This formula is derived from the Pythagorean theorem. It computes the hypotenuse of a right-angled triangle formed by the real and imaginary components. In our example of \(-4 - 4i\), the magnitude is:
- \( r = \sqrt{(-4)^2 + (-4)^2} = \sqrt{32} = 4\sqrt{2} \)
The magnitude gives us the "length" of the complex number from the origin in the plane.
Argument of Complex Number
The argument of a complex number is the angle it makes with the positive direction of the x-axis on the complex plane. This angle helps express complex numbers in polar form. For a complex number \( a + bi \), the argument \( \theta \) is calculated by:
In the case of \(-4 - 4i\), we find the tangent of the angle:
Solving \( \theta = \text{arctan}(1) \), initially, we might think \( \theta = \frac{\pi}{4} \), but we must account for the quadrant it lies in.
Since the complex number is located in the third quadrant, we adjust the angle:
The adjustment is critical for accurately determining the direction and not just the size of the angle.
- \( \tan \theta = \frac{b}{a} \)
In the case of \(-4 - 4i\), we find the tangent of the angle:
- \( \tan \theta = \frac{-4}{-4} = 1 \)
Solving \( \theta = \text{arctan}(1) \), initially, we might think \( \theta = \frac{\pi}{4} \), but we must account for the quadrant it lies in.
Since the complex number is located in the third quadrant, we adjust the angle:
- \( \theta = \pi + \frac{\pi}{4} = \frac{5\pi}{4} \)
The adjustment is critical for accurately determining the direction and not just the size of the angle.
Third Quadrant
Understanding the position of a complex number in terms of quadrants is crucial, especially when determining the argument. The complex plane is divided into four quadrants:
In our example, \(-4 - 4i\) resides in the third quadrant, where both the real and imaginary parts are negative. Because of its placement:
This adjustment ensures that the angle properly reflects the direction from the positive x-axis, encapsulating both the magnitude and direction of the complex number.
- First Quadrant (positive real and imaginary parts)
- Second Quadrant (negative real and positive imaginary parts)
- Third Quadrant (negative real and imaginary parts)
- Fourth Quadrant (positive real and negative imaginary parts)
In our example, \(-4 - 4i\) resides in the third quadrant, where both the real and imaginary parts are negative. Because of its placement:
- We add \( \pi \) (180 degrees) to the initial argument derived from the inverse tangent.
This adjustment ensures that the angle properly reflects the direction from the positive x-axis, encapsulating both the magnitude and direction of the complex number.
