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Complex Numbers
In mathematics, the complex numbers are an extension of the real numbers obtained by
adjoining an imaginary unit, denoted i, which satisfies:
i2 = −1.
Every complex number can be written in the form a + bi, where a and b are real numbers
called the real part and the imaginary part of the complex number, respectively.
Complex numbers are a field, and thus have addition, subtraction, multiplication, and
division operations. These operations extend the corresponding operations on real
numbers, although with a number of additional elegant and useful properties, e.g.,
negative real numbers can be obtained by squaring complex (imaginary) numbers.
Complex numbers were first conceived and defined by the Italian mathematician
Girolamo Cardano, who called them "fictitious", during his attempts to find solutions to
cubic equations. The solution of a general cubic equation may require intermediate
calculations containing the square roots of negative numbers, even when the final
solutions are real numbers, a situation known as casus irreducibilis. This ultimately led to
the fundamental theorem of algebra, which shows that with complex numbers, it is
always possible to find solutions to polynomial equations of degree one or higher.
The rules for addition, subtraction, multiplication, and division of complex numbers were
developed by the Italian mathematician Rafael Bombelli. A more abstract formalism for
the complex numbers was further developed by the Irish mathematician William Rowan
Hamilton, who extended this abstraction to the theory of quaternions.
Complex numbers are used in many different fields including applications in engineering,
electromagnetism, quantum physics, applied mathematics, and chaos theory .When the
underlying field of numbers for a mathematical construct is the field of complex
numbers, the name usually reflects that fact. Examples are complex analysis, complex
matrix, complex polynomial and complex Lie algebra.
The set of all complex numbers is usually denoted by C, or in blackboard bold by C.
Although other notations can be used, complex numbers are very often written in the
form
a+bi
where a and b are real numbers, and i is the imaginary unit, which has the property
i2 = −1
The real number a is called the real part of the complex number, and the real number
b is the imaginary part.
For example, 3 + 2i is a complex number, with real part 3 and imaginary part 2. If z = a +
ib, the real part a is denoted Re(z) or R(z), and the imaginary part b is denoted Im(z) or
J(z).
The real numbers, R, may be regarded as a subset of C by considering every real number
a complex number with an imaginary part of zero; that is, the real number a is identified
with the complex number a + Oi. Complex numbers with areal part of zero are called
imaginary numbers; instead of writing O + bi, that imaginary number is usually denoted
as just bi. If b equals 1, instead of using 0+ li or li, the number is denoted as i.
In some disciplines (in particular, electrical engineering, where i is a symbol for current),
the imaginary unit i is instead written as j, so complex numbers are sometimes written as
a+bj.
Two complex numbers are equal if and only if their real parts are equal and their
imaginary parts are equal. In other words, if the two complex numbers are written as
a + hi and c + di with a, h, c, and d real, then they are equal if and only if a = c and h = d.
Complex numbers are added, subtracted, multiplied, and divided by formally applying
the associative, commutative and distributive laws of algebra, together with the equation
i2=-I:
.Addition: ( a, + bi ) + ( c + di) = ( a + c) + ( b + d)i
.Subtraction: (a + bi) -(c+ di) = (a -c) + (b -d)i
.Multiplication:
(a, + bi)(c+ di) = ac + bci + adi + bdi2 = (ac -bd) + (bc+ ad)i
(a + bi)
(ac + bd
)(bc -ad
).
.Division:~= ~ + ~ I,
where c and d are not both zero.
It is also possible to represent complex numbers as ordered pairs of real numbers, so that
the complex number a + ih corresponds to ( a, h ). In this representation, the algebraic
operations have the following formulas:
(a, h) + (c, d) = (a + c, h + d)
(a, h)(c, d) = (ac -hd, hc + ad)
Since the complex number a + hi is uniquely specified by the ordered pair (a, h), the
complex numbers are in one-to-one correspondence with points on a plane. This complex
plane is described below.
Afield is an algebraic structure with addition, subtraction, multiplication, and division
operations that satisfy certain algebraic laws. The complex numbers are a field, known as
the complex number field, denoted by C. In particular, this means that the complex
numbers possess:
.An additive identity ("zero"), 0 + 0i.
.A multiplicative identity ("one"), 1 + 0i.
.An additive inverse of every complex number. The additive inverse of a + hi is
-a- hi.
.A multiplicative inverse (reciprocal) of every nonzero complex number. The
a,
(-b)2 + 2 2 i.
multiplicative inverse of a + hi is a, + b2 a + b
Other fields include the real numbers and the rational numbers. When each real number a
is identified with the complex number a + Oi, the field of real numbers R becomes a
sub field of C.
The complex numbers C can also be characterized as the topological closure of the
algebraic numbers or as the algebraic closure of R.