The modulus-argument form, also known as polar form, is an alternative method of representing complex numbers. For a complex number $z = a + bi$, its polar form is given by:
$$z = r(\cos \theta + i \sin \theta) = r \text{ cis } \theta$$
Where:
Consider the complex number $z = 1 + \sqrt{3}i$. To convert it to polar form:
Therefore, $z = 2(\cos\frac{\pi}3 + i \sin\frac\pi3) = 2 \text{ cis } \frac\pi3$
The argument $\theta$ is not unique; it can be expressed as any angle that differs by multiples of $2\pi$ radians or $360°$. Always consider the quadrant of the complex number when determining $\theta$.
The Euler form of a complex number is closely related to the polar form and is based on Euler's formula:
$$e^{ix} = \cos x + i \sin x$$
Using this, we can express a complex number in Euler form as:
$$z = re^{i\theta}$$
Where $r$ and $\theta$ are the same as in the polar form.
For the complex number $z = 1+\sqrt{3}i$, which we found earlier to have $r = 2$ and $\theta =\frac\pi3$, the Euler form would be:
$z = 2e^{\frac{\pi}3i}$
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