#math/linear-algebra #flashcards/math Complex Numbers are simply a way to represent $\sqrt{-1}$, denoted as $i$. Formally, the set of all Complex Numbers, $\mathbb{C}$, is defined by $\mathbb{C} = \{ a + bi : a, b \in \mathbb{R}\}$. You can do addition and multiplication on $\mathbb{C}$ and it has all the arithmetic properties of a [[1A2 - Fields| Field]], since it is one. Complex numbers are introduced because many theorems are defined on fields which naturally covers both $\mathbb{C} \text{ and } \mathbb{R}$. Division is the inverse of multiplication it does achieved by multiplying the term by the [[Complex Conjugate]] of the denominator $ \frac{a+bi}{c + di} = \frac{a+bi}{c + di} \cdot \frac{c-di}{c-di} $ which turns the denominator into a real scalar. Division is not $ \frac{a+bi}{c + di} \neq \frac{a}{c} + \frac{bi}{ci} $