Concept
Complex number
Overview
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted i, called the imaginary unit and satisfying the equation
i
2
=
−
1
{\displaystyle i^{2}=-1}
. Since no real number satisfies the above equation, i was called an imaginary number by René Descartes. Every complex number can be expressed in the form
a
+
b
i
{\displaystyle a+bi}
, where a and b are real numbers, a is called the real part, and b is called the imaginary part. The set of complex numbers is denoted by either of the symbols
C
{\displaystyle \mathbb {C} }
or C. Despite the historical nomenclature, "imaginary" numbers are not fictitious: they are no less real mathematically than real numbers, and they are essential to the scientific description of the physical world.
Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficients has a solution which is a complex number. For example, the equation
(
x
+
1
)
2
=
−
9
{\displaystyle (x+1)^{2}=-9}
has no real solution, because the square of a real number cannot be negative, but has the two nonreal complex solutions
x
=
−
1
+
3
i
{\displaystyle x=-1+3i}
and
x
=
−
1
−
3
i
{\displaystyle x=-1-3i}
.
Addition, subtraction and multiplication of complex numbers are defined, taking advantage of the rule
i
2
=
−
1
{\displaystyle i^{2}=-1}
, along with the associative, commutative, and distributive laws. Every nonzero complex number has a multiplicative inverse, allowing division by complex numbers other than zero. This makes the complex numbers a field with the real numbers as a subfield. Because of these properties,
a
+
b
i
=
a
+
i
b
{\displaystyle a+bi=a+ib}
, and which form is written depends upon convention and style considerations.
The complex numbers also form a real vector space of dimension two, with
{
1
,
i
}
{\displaystyle \{1,i\}}
as a standard basis. This standard basis makes the complex numbers a Cartesian plane, called the complex plane. This allows a geometric interpretation of the complex numbers and their operations, and conversely some geometric objects and operations can be expressed in terms of complex numbers. For example, the real numbers form the real line, which is pictured as the horizontal axis of the complex plane, while real multiples of
i
{\displaystyle i}
are the vertical axis. A complex number can also be defined by its geometric polar coordinates: the radius is called the absolute value of the complex number, while the angle from the positive real axis is called the argument of the complex number. The complex numbers of absolute value one form the unit circle. Adding a fixed complex number to all complex numbers defines a translation in the complex plane, and multiplying by a fixed complex number is a similarity centered at the origin (dilating by the absolute value, and rotating by the argument). The operation of complex conjugation is the reflection symmetry with respect to the real axis.
The complex numbers form a rich structure that is simultaneously an algebraically closed field, a commutative algebra over the reals, and a Euclidean vector space of dimension two.
Overview adapted from Wikipedia (CC BY-SA 4.0; see the article history for authors).
Possible passages
Found by search and not yet checked by a person; they may only mention the idea in passing.
_Ten_ must mean ten bodies, or ten sounds, or ten beatings of the pulse. But though numbers must be numbers of something, they may be numbers of anything. Propositions, therefore, concerning numbers, have the remarkable peculiarity that they are propositions concerning all things whatever; all objects, all existences of every kind, known to our experience. All things possess quantity; consist of parts which can be…
John Stuart Mill, A System of Logic
States what Complex number is.
Thus there arises a classification of modes of formation, or in the language commonly used by mathematicians, a classification of Functions. Any number, considered as formed from any other number, is called a function of it; and there are as many kinds of functions as there are modes of formation. The simple functions are by no means numerous, most functions being formed by the combination of several of the…
John Stuart Mill, A System of Logic, Ratiocinative and Inductive
States what Complex number is.
After this examination we will find that nothing remains in the idea of body, except that it is something extended in length, breadth, and depth; and this something is comprised in our idea of space, not only of that which is full of body, but even of what is called void space.
René Descartes, Selections from the Principles of Philosophy
States what Complex number is; the author is tied to Complex number in the knowledge graph.
Philosophers who discussed Complex number
Related ideas
- Addison-Wesley
- General relativity
- Princeton University Press
- Gerolamo Cardano
- Differential equation
- Imaginary number
- Isomorphism
- If and only if
- Mathematics
- Rational number
- Real number
- Norway
- Number theory
- Ordered pair
- Richard Dedekind
- Applied mathematics
- Quantum mechanics
- Quantum field theory
- René Descartes
- Limit (mathematics)
- Set (mathematics)
- Special relativity
- Spacetime
- Topology
Place in the tradition
Part of Philosophy of mathematics.