Definition:Imaginary Number

Informal Definition
The quadratic equation $ax^2 + bx + c = 0$ has no solutions in the real number space $\R$ when $b^2 - 4 a c < 0$.

In particular, this applies to the equation $x^2 + 1 = 0$.

In order to be able to allow such equations to have solutions, the concept $i = \sqrt {-1}$ is introduced.

$i$ does not exist on the real number line, but is a completely separate concept.

It can be treated as a number, and combined with real numbers in algebraic expressions.

When $a, b$ are real numbers, we have:


 * $a i = i a$
 * $a + i = i + a$
 * $i a + i b = i \paren {a + b} = \paren {a + b} i = a i + b i$

etc.

In engineering applications, $j$ is usually used instead.

Numbers of the form $a i$ (or $i a$), where $a \in \R$, are known as imaginary numbers.

Numbers of the form $a + b i$ are known as complex numbers.