Imaginary Unit¶

Definition¶

The imaginary unit, symbolized as $\mathrm{i}$ , is characterized by its square being $-1$ , i.e., $\mathrm{i}^2 = -1.$

Interpretation¶

Integer powers mean repeated multiplication, so $\mathrm{i}^2=-1$ implies $\mathrm{i}^3=-\mathrm{i}$ and $\mathrm{i}^4=1$ $1$ . Higher powers therefore repeat in a cycle of length four.

The imaginary unit $\mathrm{i}$ is a solution, not the solution, to the equation $x^2+1=0$ since $-\mathrm{i}$ is also a solution.

The choice of $\mathrm{i}$ over $-\mathrm{i}$ as the "principal" square root of $-1$ is largely a matter of convention, similar to how we choose the positive square root of a positive number as its principal square root.

The unit circle in the complex plane with the points 1, i, -1 and -i, joined by quarter-circle arrows labelled times i: each multiplication by i is a rotation by 90 degrees.

$\mathrm{i}^2=-1$ $- 1$ : multiplying by $\mathrm{i}$ is a rotation by $90^\circ$ , so four multiplications return to the start ( $i^{4} =$ $\mathrm{i}^4=1$ $1$ ).

Drag $z$ : multiplying by $\mathrm{i}$ turns it a quarter turn, and doing it twice gives $\mathrm{i}^2z=-z$ $- z$ .

Key Results¶

Complex numbers extend the real numbers by adding a solution of $x^2=-1$ $- 1$ .
The complex numbers form a field, so they can be used as scalars in complex vector spaces.

Examples¶

$\mathrm{i}^3 = \mathrm{i}^2 \cdot\mathrm{i} = (-1) \cdot \mathrm{i} = -\mathrm{i}, \quad \mathrm{i}^4 = \mathrm{i}^3 \cdot \mathrm{i} = (-\mathrm{i})\cdot \mathrm{i} = 1, \quad \mathrm{i}^5 = \mathrm{i}^4 \cdot \mathrm{i} =\mathrm{i}.$
The roots of $x^2+x+1=0$ are $x_{1,2}=\dfrac{-1\pm\sqrt{3}\mathrm{i}}{2}$ $\frac{- 1 \pm 3 i}{2}$ .
For $z=3+4\mathrm{i}$ $4 i$ , the real part is $3$ , the imaginary part is $4$ , and $|z|=5$ $5$ .

Prerequisites¶

Exponentiation and Power

Sources¶

Math II: Linear Algebra · Week 1

Aliases: unit imaginary number Symbols: square root

Last updated: 20 October 2026 03:02