Definition:Imaginary Number

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 order to indicate that $$i$$ is "special", the symbol $$\imath$$ is frequently used instead of $$i$$.

In engineering applications, $$j$$ or $$\jmath$$ are usually used instead.

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

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