Arithmetic of ℂ

A complex number has the form $a+bi$, where $a$ and $b$ are real numbers and $i^2=-1$. Add or subtract complex numbers by combining real parts and imaginary parts separately. To multiply, expand as with binomials and replace every $i^2$ with $-1$. To divide, multiply the numerator and denominator by the complex conjugate of the denominator, $a-bi$, which makes the denominator a real number. The modulus $|a+bi|$ is $\sqrt{a^2+b^2}$, and $|zw|=|z||w|$. Powers of $i$ repeat in a cycle of four: $i^1=i$, $i^2=-1$, $i^3=-i$, $i^4=1$.