Imaginary Unit
Functions & Advanced AlgebraThe imaginary unit i is defined as the square root of negative one, satisfying i^2 = -1, and is the foundation of complex numbers.
Formula
i = \sqrt{-1}, i^2 = -1
Definition
The imaginary unit $i$ is defined as the square root of $-1$. Since no real number squared equals $-1$, $i$ is a new kind of number that extends our number system.
Example
$i^1 = i$, $i^2 = -1$, $i^3 = -i$, $i^4 = 1$. Then the pattern repeats: $i^5 = i$, $i^6 = -1$, and so on. The powers of $i$ cycle through four values.
Key Insight
The word "imaginary" is misleading: i is just as valid a mathematical object as any other number. It was named "imaginary" historically because mathematicians distrusted it before they understood complex numbers fully.
Definition
The imaginary unit $i$ satisfies $i^2 = -1$. Powers cycle: $i^0=1$, $i^1=i$, $i^2=-1$, $i^3=-i$, $i^4=1$. To find $i^n$, compute $n \bmod 4$. Used to express square roots of negatives: $\sqrt{-a} = i\sqrt{a}$ for $a > 0$.
Example
$\sqrt{-25} = i\sqrt{25} = 5i$. $i^{47}$: $47 \bmod 4 = 3$, so $i^{47} = i^3 = -i$. Simplify $(3i)^2 = 9i^2 = 9(-1) = -9$.
Key Insight
The cycle $i, -1, -i, 1$ corresponds to rotation by $90^\circ$ in the complex plane. Multiplying by $i$ rotates a complex number $90^\circ$ counterclockwise. This geometric interpretation makes complex multiplication intuitive.
Definition
The imaginary unit $i$ is a root of the irreducible polynomial $x^2 + 1$ over $\mathbb{R}$. The field $\mathbb{C} = \mathbb{R}[x]/(x^2+1)$ is the field extension obtained by adjoining a root of $x^2 + 1$. Equivalently, $\mathbb{C}$ can be represented as the ring of $2\times 2$ real matrices of the form $\begin{bmatrix}a & -b\\b & a\end{bmatrix}$, where $i$ corresponds to $\begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}$.
Example
The matrix representation $\begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}$ satisfies $\begin{bmatrix}0 & -1\\1 & 0\end{bmatrix}^2 = \begin{bmatrix}-1 & 0\\0 & -1\end{bmatrix} = -I$, confirming $i^2 = -1$. Complex multiplication corresponds to matrix multiplication in this representation.
Key Insight
The matrix representation of $\mathbb{C}$ as $2\times 2$ real matrices reveals that complex multiplication is a rotation and scaling, making the geometric interpretation of complex arithmetic rigorous and algebraically inevitable.