The polar form of a complex number expresses it as $z = r(\cos(\theta) + i\sin(\theta))$ or $z = re^{i\theta}$, where $r$ is the magnitude and $\theta$ is the argument (angle). This form emphasizes the geometric interpretation of complex numbers.
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The magnitude $r$ of a complex number $z = a + bi$ is given by $r = \sqrt{a^2 + b^2}$.
The argument $\theta$ can be found using $\tan^{-1}(b/a)$, with considerations for the quadrant in which the complex number lies.
Multiplying two complex numbers in polar form involves multiplying their magnitudes and adding their arguments: $(r_1 e^{i \theta_1})(r_2 e^{i \theta_2}) = r_1 r_2 e^{i (\theta_1 + \theta_2)}$.
To convert from rectangular to polar form, use $a = r \cos(\theta)$ and $b = r \sin(\theta)$.
Euler's formula states that $e^{i \theta} = \cos(\theta) + i \sin(\theta)$, making it useful for expressing complex numbers in polar form.
Review Questions
How do you calculate the magnitude of a complex number?
What is Euler's formula and how does it relate to the polar form?
Describe how you would multiply two complex numbers in polar form.
Related terms
Magnitude: The distance from the origin to the point representing the complex number in the complex plane, calculated as $r = \sqrt{a^2 + b^2}$.
Argument: The angle formed with the positive real axis by a line passing through the origin and the point representing the complex number. It is denoted as $\theta$.
Rectangular Form: $z = a + bi$, where 'a' represents the real part and 'b' represents the imaginary part of a complex number.