Complex Number: Mustansiriya University

‫‪Mustansiriya University‬‬
‫‪complex number‬‬
‫اسم الطالب‪ :‬حـيدر عدي‬

‫بأشراف الدڪتورة‪ :‬أمنة قاسم‬
‫‪2023/12/3‬‬
‫‪1‬‬
The algebraic form of complex numbers is...Z the
set of integers
N the set of positive integers
Q the set of rational numbers
R the set of real numbers
R∗ the set of nonzero real numbers
R2 the set of pairs of real numbers
C the set of complex numbers
C∗ the set of nonzero complex numbers
≤ [a, b] the set of real numbers x such that a
x≤b
< (a, b) the set of real numbers x such that a
x<b
z the conjugate of the complex number z
z| the modulus or absolute value of|
complex number z −→AB the vector AB
(AB) the open segment determined by A and
B
[AB] the closed segment determined by A
and B
AB the open ray of origin A that contains B)
area[F] the area of figure F
Un the set of nth roots of unity
2
z=a+bi Therefore, every complex number is an
ordered pair a and b in the number axis, and
every such pair can have its coordinates
calculated using the angle formed by the
intersection of the axis. x with the straight line
emerging from the origin and passing through
the pair (a,b) and also by the length of the line
between (a,b)} and - (0,0)}. This possibility allows
:the complex number to be formulated as follows
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 has no real solution, since the square
of a real number cannot be negative, but has the
two nonreal complex solutions
3i and 1-3i+1-
3
Addition, subtraction and multiplication of
complex numbers can be naturally defined by
using the rule
,i²=-1 combined with the associative
commutative, and distributive laws. Every
nonzero complex number has a multiplicative
inverse. This makes the complex numbers a field
that has the real numbers as a subfield. The
complex numbers also form a real vector space
of dimension two, with {1, i} as a standard basis
Addition, subtraction and multiplication of

complex numbers can be naturally defined by
using the rule
,i²=-1 combined with the associative
commutative, and distributive laws. Every
nonzero complex number has a multiplicative
inverse. This makes the complex numbers a field
that has the real numbers as a subfield. The
complex numbers also form a real vector space
of dimension two, with {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
expressing in terms of complex numbers some
geometric properties and constructions. For
example, the real numbers form the real line
4
which is identified to the horizontal axis of the
complex plane. The complex numbers of
absolute value one form the unit circle. The
addition of a complex number is a translation in
the complex plane, and the multiplication by a
complex number is a similarity centered at the
origin. The complex conjugation is the reflection
symmetry with respect to the real axis. The
.complex absolute value is a Euclidean norm
We define the set C of complex numbers as the
set of all ordered pairs z=〈 a, b〉 where a and b are
real numbers and where addition and
multiplication are defined. We define the real and
,imaginary parts of z and denote this by a= ℜ (z)
b= ℑ (z). These definitions satisfy all the axioms
for a field. 0C= 0+ 0i and 1C= 1+ 0i are identities for
addition and multiplication respectively, and there
are multiplicative inverses for each non zero
element in C. The difference and division of
complex numbers are also defined. We do not
interpret the set of all real numbers R as a subset
of C. From here on we do not abandon the
ordered pair notation for complex numbers. For
example: i2=(0+ 1i) 2=− 1+ 0i=− 1. We conclude this
article by introducing two operations on C which
are not field operations. We define the absolute
value of z denoted by| z| and the conjugate of z
denoted
5
numbers and shows closure under the basic
arithmetic operations. It is shown that arithmetic
of fuzzy complex numbers may be performed in
terms of α-cuts. Next two special types of fuzzy
complex numbers based on the forms z = x + iy
and z = reiγ for regular complex numbers are
investigated. Introducing a metric on the space
of fuzzy complex numbers then allows us to
discuss continuity and differentiability of fuzzy
complex functions
6
Sources and references
Complex Numbers from A to... Z
Titu Andreescu, Dorin Andric
Birkhäuser Boston, 2006
The complex numbers
Czesław Bylinski
Def 7 (1C), 1, 1990
Complex numbers in geometry
Isaak Moiseevitch Yaglom
Academic Press, 2014
Fuzzy complex numbers
Fuzzy Sets and Systems 338, 1-22, 2018

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‫‪Mustansiriya University‬‬

‫اسم الطالب‪ :‬حـيدر عدي‬

Complex numbers allow solutions to all

Addition, subtraction and multiplication of

This standard basis makes the complex numbers

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