Complex Numbers: Polar Form

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Complex Numbers in Cartesian Form A complex number $z$ is typically written as $z = x + iy$, where: $x$ is the real part , $Re(z)$ $y$ is the imaginary part , $Im(z)$ $i$ is the imaginary unit, $i = \sqrt{-1}$ This form represents a point $(x, y)$ in the complex plane. Complex Numbers in Polar Form A complex number $z = x + iy$ can also be expressed in polar coordinates $(r, \theta)$. The polar form is $z = r(\cos\theta + i\sin\theta)$. This is often abbreviated using Euler's formula: $z = re^{i\theta}$. Modulus (Magnitude) Definition: The modulus of a complex number $z = x + iy$ is its distance from the origin in the complex plane. Notation: $|z|$ or $r$. Formula: $r = |z| = \sqrt{x^2 + y^2}$. Properties: $|z| \ge 0$ $|z_1 z_2| = |z_1| |z_2|$ $|z_1 / z_2| = |z_1| / |z_2|$ (if $z_2 \ne 0$) $|z^n| = |z|^n$ $|z| = |\bar{z}|$ (where $\bar{z}$ is the complex conjugate) Argument (Angle) Definition: The argument of a complex number $z = x + iy$ is the angle $\theta$ (in radians) that the line segment from the origin to $z$ makes with the positive real axis. Notation: $\arg(z)$ or $\theta$. Formula: $\theta = \arctan\left(\frac{y}{x}\right)$, but careful with quadrants! Principal Argument: The unique value of $\arg(z)$ in the interval $(-\pi, \pi]$ (or $[0, 2\pi)$). Denoted as $Arg(z)$. Calculating $\theta$ based on Quadrant: If $x > 0$: $\theta = \arctan\left(\frac{y}{x}\right)$ If $x If $x If $x = 0, y > 0$: $\theta = \frac{\pi}{2}$ If $x = 0, y If $x = 0, y = 0$: $\theta$ is undefined (modulus is 0) Properties: $\arg(z_1 z_2) = \arg(z_1) + \arg(z_2) + 2k\pi$ for some integer $k$ $\arg(z_1 / z_2) = \arg(z_1) - \arg(z_2) + 2k\pi$ for some integer $k$ $\arg(z^n) = n \arg(z) + 2k\pi$ for some integer $k$ Conversion Steps (Cartesian to Polar) Identify $x$ and $y$ from $z = x + iy$. Calculate the modulus: $r = \sqrt{x^2 + y^2}$. Calculate the argument $\theta$: Determine the quadrant of $(x, y)$. Use $\theta = \arctan(y/x)$ and adjust based on the quadrant. Ensure $\theta$ is in radians. Write in polar form: $z = r(\cos\theta + i\sin\theta)$ or $z = re^{i\theta}$. Example Convert $z = -1 + i\sqrt{3}$ to polar form. $x = -1$, $y = \sqrt{3}$ Modulus: $r = \sqrt{(-1)^2 + (\sqrt{3})^2} = \sqrt{1 + 3} = \sqrt{4} = 2$. Argument: The point $(-1, \sqrt{3})$ is in the second quadrant. $\arctan\left(\frac{\sqrt{3}}{-1}\right) = \arctan(-\sqrt{3}) = -\frac{\pi}{3}$. Since it's in the second quadrant, add $\pi$: $\theta = -\frac{\pi}{3} + \pi = \frac{2\pi}{3}$. Polar Form: $z = 2\left(\cos\left(\frac{2\pi}{3}\right) + i\sin\left(\frac{2\pi}{3}\right)\right)$ or $z = 2e^{i\frac{2\pi}{3}}$.