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Chapter 8: Problem 47

Rewrite each complex number into polar \(r e^{1 \theta}\) form. $$ -1-4 i $$

Short Answer

Expert verified
The polar form is \(\sqrt{17}e^{i(\tan^{-1}(4) + \pi)}\).

Step by step solution

01

Identify the real and imaginary parts

The given complex number is \(-1-4i\). Here, the real part is \(-1\) and the imaginary part is \(-4\).
02

Calculate the modulus \(r\)

The modulus \(r\) of a complex number \(a+b i\) is given by \(r=\sqrt{a^2+b^2}\). Here, \(a=-1\) and \(b=-4\), so \\[r=\sqrt{(-1)^2+(-4)^2}=\sqrt{1+16}=\sqrt{17}.\]
03

Find the argument \(\theta\)

The argument \(\theta\) is the angle made with the positive real axis. It can be calculated using \(\theta = \tan^{-1}\left(\frac{b}{a}\right)\).Substituting the values, \[\theta = \tan^{-1}\left(\frac{-4}{-1}\right) = \tan^{-1}(4).\] Since the point is in the third quadrant (both \(a\) and \(b\) are negative), we add \(\pi\) to the principal value, \[\theta = \tan^{-1}(4) + \pi\].
04

Write in polar form \(re^{i\theta}\)

Using the modulus \(r\) and the argument \(\theta\), the polar form is \[re^{i\theta} = \sqrt{17}e^{i(\tan^{-1}(4) + \pi)}.\]

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Key Concepts

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

Polar Form
Every complex number can be represented in a different way called the polar form. The polar form expresses a complex number using a modulus and an argument. It looks like this: \(re^{i \theta}\). Here, \(r\) is the distance from the origin to the point, and \(\theta\) is the angle between the positive real axis and the line connecting the origin to the point. This form is particularly useful when multiplying or dividing complex numbers as it simplifies the processes significantly.
  • In the polar form, complex numbers make use of Euler鈥檚 formula which links trigonometry with exponential functions.
  • Euler's formula states: \(e^{i\theta} = \cos(\theta) + i\sin(\theta)\).
Modulus
The modulus of a complex number is its absolute value, representing its distance from the origin in the complex plane. It is calculated as \(r = \sqrt{a^2 + b^2}\), where \(a\) and \(b\) are the real and imaginary parts of the number, respectively.
  • For the complex number \(-1 - 4i\), \(a = -1\) and \(b = -4\).
  • Plug these values into the modulus formula to find \(r = \sqrt{(-1)^2 + (-4)^2} = \sqrt{1 + 16} = \sqrt{17}\).
  • The modulus provides the non-negative value of the distance from the complex number to the origin (0,0).
Understanding the modulus gives a clear geometric meaning of the complex number鈥檚 size or magnitude.
Argument of a Complex Number
The argument of a complex number is the angle \(\theta\) made with the positive real axis in the complex plane. It's essentially the direction of the complex number originating from the origin.
Calculating the argument requires trigonometry, specifically the arctangent function. Using \(\tan^{-1}(\frac{b}{a})\), we can find the argument, but be mindful of the quadrant in which your complex number lies. Each quadrant affects the final computation of \(\theta\):
  • For \(-1 - 4i\), both \(a\) and \(b\) are negative, placing our number in the third quadrant.
  • This means we need to add \(\pi\) to the principal value resulting from \(\tan^{-1}(4)\) to adjust the angle properly.
  • The full formula becomes \(\theta = \tan^{-1}(4) + \pi\).
Once you know the argument, you can locate exactly where the complex number resides in the complex plane, aiding in its geometric interpretation.

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