Polar form

Definition

Polar form is a way of expressing complex numbers in terms of their magnitude (or modulus) and angle (or argument) instead of the traditional rectangular coordinate system. This representation connects the complex number's position on the complex plane to trigonometric concepts, making it easier to perform multiplication and division operations. Polar form is particularly useful in various applications, including electrical engineering and signal processing, where rotations and oscillations are involved.

5 Must Know Facts For Your Next Test

1. In polar form, a complex number is expressed as $$r( ext{cos} \theta + i\text{sin} \theta)$$ or more compactly as $$re^{i\theta}$$, where $$r$$ is the magnitude and $$\theta$$ is the argument.
2. To convert from rectangular form to polar form, you can use the formulas $$r = \sqrt{x^2 + y^2}$$ for magnitude and $$\theta = \text{tan}^{-1}\left(\frac{y}{x}\right)$$ for the argument.
3. Multiplication of complex numbers in polar form involves multiplying their magnitudes and adding their arguments: if $$z_1 = r_1 e^{i\theta_1}$$ and $$z_2 = r_2 e^{i\theta_2}$$, then $$z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}$$.
4. Division of complex numbers in polar form involves dividing their magnitudes and subtracting their arguments: if $$z_1 = r_1 e^{i\theta_1}$$ and $$z_2 = r_2 e^{i\theta_2}$$, then $$\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}$$.
5. Polar form simplifies the calculation of powers of complex numbers; using De Moivre's Theorem, if $$z = re^{i\theta}$$, then $$z^n = r^n e^{in\theta}$$.

Review Questions

• How do you convert a complex number from rectangular form to polar form?
  • To convert a complex number from rectangular form, represented as $$z = x + yi$$, to polar form, you first calculate its magnitude using the formula $$r = \sqrt{x^2 + y^2}$$. Next, determine the argument by finding the angle with $$\theta = \text{tan}^{-1}\left(\frac{y}{x}\right)$$. The polar form is then expressed as $$r( ext{cos} \theta + i\text{sin} \theta)$$ or simply as $$re^{i\theta}$$.
• Discuss how multiplication and division of complex numbers are simplified using polar form.
  • In polar form, multiplication of complex numbers becomes more straightforward because you only need to multiply their magnitudes and add their angles. For instance, if you have two complex numbers $z_1$ and $z_2$, represented as $$z_1 = r_1 e^{i\theta_1}$$ and $$z_2 = r_2 e^{i\theta_2}$$, their product is given by $$z_1 z_2 = r_1 r_2 e^{i(\theta_1 + \theta_2)}$$. On the other hand, division requires dividing their magnitudes and subtracting their angles: $$\frac{z_1}{z_2} = \frac{r_1}{r_2} e^{i(\theta_1 - \theta_2)}$$. This simplification makes calculations involving complex numbers much easier.
• Evaluate how understanding polar form impacts the application of De Moivre's Theorem in solving powers of complex numbers.
  • Understanding polar form is crucial when applying De Moivre's Theorem because it allows us to easily compute powers and roots of complex numbers. According to this theorem, for a complex number expressed as $$z = re^{i\theta}$$, raising it to the power of n results in $$z^n = r^n e^{in\theta}$$. This means that instead of multiplying out the number repeatedly in rectangular form, we can simply raise the magnitude to the power n and multiply the angle by n. This method significantly simplifies calculations for higher powers or roots.

Related terms

Magnitude: The distance of a complex number from the origin in the complex plane, calculated as the square root of the sum of the squares of its real and imaginary parts.

Argument:

The angle formed between the positive x-axis and the line representing the complex number in the complex plane, usually measured in radians.

Euler's Formula:

A mathematical formula that establishes a deep relationship between trigonometric functions and exponential functions, stated as $$e^{ix} = ext{cos}(x) + i ext{sin}(x)$$.