Definition:Complex Number

Informal Definition
A complex number is a number in the form $$a + b \imath$$ or $$a + \imath b$$ where:
 * $$a$$ and $$b$$ are real numbers;
 * $$\imath$$ is the square root of $-1$, i.e. $$\sqrt {-1}$$.

The set of all complex numbers is denoted $$\mathbb{C}$$.

Real Part
The real part of a complex number $$a + \imath b$$ is the coefficient $$a$$.

The real part of a complex number $$z$$ is often denoted $$\Re \left({z}\right)$$ or $$\mathrm {Re} \left({z}\right)$$.

Imaginary Part
The imaginary part of a complex number $$a + \imath b$$ is the coefficient $$b$$ (note: not $$\imath b$$).

The imaginary part of a complex number $$z$$ is often denoted $$\Im \left({z}\right)$$ or $$\mathrm {Im} \left({z}\right)$$.

Wholly Real
The complex number $$z = a + \imath b$$ is called wholly real or completely real, or entirely real, etc. iff $$b = 0$$.

Wholly Imaginary
The complex number $$z = a + \imath b$$ is called wholly imaginary or completely imaginary, or entirely imaginary, etc. iff $$a = 0$$.

Notation
When $$a$$ and $$b$$ are symbols representing variables or constants, the form $$a + \imath b$$ is usually seen.

When $$a$$ and $$b$$ are actual numbers, for example 3 and 4, it usually gets written $$3 + 4 \imath$$.

The symbol $$\imath$$ can also be seen as $$i$$.

When mathematics is applied to engineering, in particular electrical and electronic engineering, the symbol $$\jmath$$ or $$j$$ is usually used, as $$i$$ is the standard symbol used to denote the flow of electric current, and to use it also for $$\sqrt {-1}$$ would cause untold confusion.