Definition:Imaginary Number
From ProofWiki
Informal Definition
The quadratic equation $ax^2 + bx + c$ 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 in the real number plane, 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:
- $a i = i a$
- $a + i = i + a$
- $i a + i b = i \left({a + b}\right) = \left({a + b}\right) i = a i + b i$ 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.
