# Imaginary number

The **imaginary numbers** are a part of the complex numbers.
Every complex number can be written as
the sum *a*+*b*i of a real number *a* and an imaginary number *b*i
(with real numbers *a* and *b*, and the **imaginary unit** i).
Since the square (*b*i)^{2} = −*b*^{2} of an imaginary number is a negative real number,
the imaginary numbers are just the square roots of the negative real numbers.
In the complex plane the imaginary numbers lie on the imaginary axes.
perpendicular to the real axes,

However, sometimes the term "imaginary" is used more generally
for all non-real complex numbers,
i.e., all numbers with non-vanishing imaginary part (*b* not 0),
are called "imaginary".
In this case, the more specific complex numbers *b*i
(with vanishing real part *a*=0) are called **pure(ly) imaginary**.

**Remark:**

The names "imaginary" and "complex" number are of historical origin,
just as the other names for numbers
— "rational", "irrational", and "real" — are.

Thus they must not be interpreted literally,
and no mathematical or philosophical conclusions may be drawn from them.

## Numbers: complex, real and imaginary

Every complex number is the real linear combination of the **real unit**

and the **imaginary unit**

that is, a complex number can uniquely be written as

In this expression, the real number
*a* is called the **real** part and the real number *b* the **imaginary** part
of the **complex number** *a*+*b*i.

While (real) multiples of 1 (i.e., complex numbera with imaginary part 0) are (identified with) the real numbers:

the (real) multiples of i (i.e., complex numbers with real part 0)

are the **imaginary** numbers, or **pure imaginary** numbers, depending on usage.

### Characterization

The square of an imaginary number

is a negative real number.

Conversely, the (two) square roots of a negative real number

are imaginary numbers.

### Origin

Imaginary numbers first occurred — as square roots of negative numbers —
when the formula for solving cubic equations were found.
It turned out that using square roots of negative numbers
— which "do not exist" and therefore are only "imagined" —
as purely formal expressions may lead to valid "existing" solutions in ("real") numbers.

Moreover, numbers which are "composed" from real and imaginary numbers are not simple but "complex".