Complex numbers

Add or subtract complex numbers by combining real parts and imaginary parts separately. Multiply by expanding brackets and replacing every $i^2$ with $-1$. The conjugate of $z=a+bi$ is $\overline{z}=a-bi$, which reflects the point across the real axis. Multiplying a number by its conjugate gives $z\overline{z}=a^2+b^2$, a real value. To divide by $c+di$, multiply the numerator and denominator by $c-di$. The denominator becomes $c^2+d^2$, so the result can be separated into its real and imaginary parts, provided the denominator is not zero.

Plot $z=a+bi$ at coordinates $(a,b)$, with the real axis horizontal and the imaginary axis vertical. The modulus $r=|z|$ is its distance from the origin. Its argument $\theta$ is the angle measured anticlockwise from the positive real axis. Determine the quadrant before using an inverse tangent: $\theta=\operatorname{atan2}(b,a)$ is a reliable calculator expression. A principal argument is commonly chosen in $(-\pi,\pi]$. Numbers in different quadrants can have the same ordinary tangent value but different arguments, so simply entering $\tan^{-1}(b/a)$ can produce a wrong angle.

For $z\ne0$, write $z=r(\cos\theta+i\sin\theta)$, where $r=|z|$ and $\theta=\arg z$. Polar form makes products easier: multiply moduli and add arguments. For quotients, divide moduli and subtract arguments. For integer powers, de Moivre's result gives $z^n=r^n(\cos(n\theta)+i\sin(n\theta))$. Angles that differ by an integer multiple of $2\pi$ describe the same complex number. When a question requests the principal argument, convert the angle into the specified principal range. The number zero has modulus zero but no defined argument.