# complex number

Figure 1. A complex number represented on the Argand diagram.

Figure 2. The imaginary operator *i* rotates the vector **OP** into **OP"**, and – on a second
application – rotates **OP"** into **OP'**.

A complex number is a real number plus a real number times
the square root of –1; in other words, a number of the form *z* = *a* + *ib*, where *a* and *b* are real and *i* = √(–1).
The term *ib* is known as an imaginary
number or the imaginary part of the complex number *a* + *ib*; *a* is called the real part. The names "complex," "real," and "imaginary,"
which came about historically, are totally misleading because complex numbers
are not particularly complex and imaginary numbers are no less real than
real numbers! Another way to represent a complex number is as an ordered
pair of real numbers (*a*, *b*) together with the operations:
(*a*, *b*) + (*c*, *d*) = (*a* + *c*, *b* + *d*) and (*a*, *b*) × (*c*, *d*) = (*ac *– *bd*, *bc* + *ad*). Alternatively, complex numbers
can be shown as points on an Argand diagram (a representation of the **complex plane**) in which the horizontal
axis is the real number line and the
vertical axis represents all possible purely imaginary numbers. Any point
that appears on the complex plane off-axis has both real and imaginary parts.
On an Argand diagram a complex number can also be shown as a vector,
or directed line segment (a line of a certain length with an arrow), extending
from the origin (0 + 0*i*) to the number (*a* + *bi*). The absolute value or magnitude of a
complex number *z*, thought of as a point on a plane, is its Euclidean
distance from the origin, and is denoted |*z*|; this is always a non-negative
real number. Algebraically, if *z* = *a* + *ib*, we can define
|*z*| = √(*a*^{2} + *b*^{2}). If the
complex number z is written in polar coordinates *z* = *r* e^{iφ},
then |*z*| = *r*.

Complex numbers are a natural extension of the real numbers and form what
is called an **algebraically closed field**. Because of this,
mathematicians sometimes consider the complex numbers to be more "natural"
than the real numbers: all polynomial equations have solutions among the complex numbers, which is not true for
the real numbers. Complex numbers are used in electrical engineering and
other branches of physics as a convenient description for periodically varying
signals. In an expression *z* = *r* e^{iφ} one
may think of *r* as the amplitude and phi as the phase of a sine
wave of given frequency. In special and general relativity theory, some formulas
for the metric on spacetime become simpler
if the time variable is taken to be imaginary.

## Imaginary number

An imaginary number is a number whose square is negative. Every imaginary number can be written
in the form *ib* where *b* is a real
number and *i* is the **imaginary unit,** √–1, with the property that *i*^{ 2} = –1. For example, √–16
is an imaginary number, since √–16 = √–1.√16 can be written
as 4*i*.

Imaginary numbers are complex numbers in which the real part is zero. In the representation of complex numbers on the complex plane (Argand diagram), imaginary numbers lie on the *y*-axis.

The "imaginary" in the term is a vestige from the time when mathematicians had not yet defined a number system to accommodate numbers whose square is less than or equal to zero. Greek mathematician and engineer Hero of Alexandria (c. 10–70 AD) is credited with the concept of imaginary numbers, but it was Rafael Bombelli who first codified and thoroughly described their properties in his treatise L'Algebra (1572). He defined an imaginary number a the square root of minus one and gave it the symbol i.

### Imaginary operator

An imaginary operator, also called an ** i-operator** or

**, is the part of a complex number that defines the magnitude of the part of the complex number at right angles to the real number part. It may be understood as follows: consider a vector**

*j*operator**OP**where O is the origin of a set of Cartesian coordinates and P is the point (

*a*,

*b*) – see diagram. Multiplying

**OP**by -1 will produce a vector

**OP'**; that is, it will rotate

**OP**through 180° such that

**OP'**= –

**OP**. Now consider a number

*i*(called the imaginary operator) such that multiplying

**OP**by

*i*produces

**OP"**, a vector perpendicular to

**OP**. Multiplying

**OP"**by

*i*will produce

**OP'**.

## Argand diagram

An Argand diagram, also known as the **Argand
plane** or the **complex plane**, is a way of representing complex
numbers as points on a coordinate plane using the *x*-axis
as the real axis and the *y*-axis as the imaginary axis. It is named
for the French amateur mathematician Jean Robert Argand (1768–1822)
who described it in a paper in 1806.^{1} A similar method had been
suggested 120 years earlier by John Wallis and had been developed extensively by Casper Wessel.
But Wessel's paper was published in Danish and wasn't circulated in the
languages more common to mathematics at that time. In fact, it wasn't until
1895 that his paper came to the attention of the mathematical community
– long after the name "Argand diagram" had stuck.

In Figure 1, a complex number *z* is shown in terms of
both Cartesian (*x*, *y*) and polar (*r*, *θ*)
coordinates.

**Reference**

1. Argand, R. *Essai sur une manière de représenter les quantités
imaginaires dans les constructions géométriques*. Paris: Albert
Blanchard, 1971. Reprint of the 2nd ed., published by G. J. Hoel in
1874. First edition published Paris, 1806.