Definition

The imaginary unit $i$ is defined as the square root of $-1$, satisfying $i^2 = -1$: since no real number squared equals $-1$, $i$ is a new kind of number extending the real number system, even though the name "imaginary" is misleading, $i$ is just as valid a mathematical object as any other number, historically distrusted before complex numbers were fully understood. Its powers cycle: $i^0=1$, $i^1=i$, $i^2=-1$, $i^3=-i$, $i^4=1$, and repeat, with $i^n$ found from $n \bmod 4$; it lets us express square roots of negatives, $\sqrt{-a} = i\sqrt{a}$ for $a > 0$. Algebraically, $i$ is a root of the irreducible polynomial $x^2 + 1$ over $\mathbb{R}$, and the field $\mathbb{C} = \mathbb{R}[x]/(x^2+1)$ is the field extension obtained by adjoining that root; equivalently, $\mathbb{C}$ can be represented as $2\times 2$ real matrices of the form $\begin{bmatrix}a & -b\\b & a\end{bmatrix}$, where $i$ corresponds to $\begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}$.