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Complex Numbers Representation

Let

i.e.

Surely,

Let

Let us now show that the mapping

i.e.

Hence under the mapping

Let

Clearly,

Let us denote

and from

Hence

which is the usual notion of a complex number.

In this complex number,

**C**_{R}be the set of complex numbers of the type**(a, 0)**i.e.

**CR = {(a, 0) | a****R}**Surely,

**C**. Now, it is possible to set up a one-one correspondence between the elements of_{R}⊂ C**C**_{R}and the elements of**R**, the set of real numbers.Let

**ƒ : R C**_{R}be defined as**ƒ(a) = (a, 0), ∀ a R**Let us now show that the mapping

**ƒ**has a composition preserving property.i.e.

**(i) ƒ(a + b) = (a + b, 0) = (a, 0) + (b, 0) = ƒ(a) + ƒ(b), ∀ a, b R**

(ii) ƒ(ab) = (ab, 0) = (a, 0) (b, 0) = ƒ(a). ƒ(b), ∀ a, b R(ii) ƒ(ab) = (ab, 0) = (a, 0) (b, 0) = ƒ(a). ƒ(b), ∀ a, b R

Hence under the mapping

**ƒ**, the addition and multiplication composition are preserved and also it associates to each real number**R**, whose first coordinate is a and the second coordinate is zero.Let

**z = (a, b) C**; then**(a, b) = (a, 0) + (0, b)**

= (a, 0) + (0, 1) (b, 0) (1)= (a, 0) + (0, 1) (b, 0) (1)

Clearly,

**(0, 1) (0, 1) = (-1, 0) - -1**[by the above form**ƒ : C**]_{R}- RLet us denote

**(0, 1)**by**i**; then**i**,^{2}= -1and from

**(1)**, we get**z = (a, b) = (a, 0) + (b, 0) = a + ib**Hence

**z = a + ib**,which is the usual notion of a complex number.

In this complex number,

**“a”**is called the real part of complex number (**a, b**, and**“b”**is called the imaginary part of the complex number**(a, b)**. A complex number is called purely real if its imaginary part is zero and it is called purely imaginary if its real part is zero but its imaginary part is not zero.**Services: -**Complex Numbers Representation Homework | Complex Numbers Representation Homework Help | Complex Numbers Representation Homework Help Services | Live Complex Numbers Representation Homework Help | Complex Numbers Representation Homework Tutors | Online Complex Numbers Representation Homework Help | Complex Numbers Representation Tutors | Online Complex Numbers Representation Tutors | Complex Numbers Representation Homework Services | Complex Numbers RepresentationSubmit Your Query ???

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Real Number Absolute Value
Addition Of Matrices
Square Matrix Adjoint
Algebraic Structures
Alternating Series
Linear Equations Determinants
Archimedean Real Numbers
Binary Operation
Binary Relation In A Set
Bounded, Unbounded Sets
Cauchy Root Test
Caylay Hamiltion Theorem
Circular Permutation
Common Roots
Complex Numbers
Complex Number Conjugate
Conjugate Of A Matrix
Constant Sequences
Convergence Of A Sequence
Cosets
Cubic, Biquadratic Equations
De Moivre Theorem
Real Number Denseness
Order 3 Determinants
Differences Of Matrices
Direct Sum Of Vector Subspaces
Eigen Vector
Elementary Matrices
Matrix Elem. Transformations
Equal Matrices
Equal Roots
Two Permutations Equity
Equivalent Matrices
Trigonometry Function Expansion
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Algebra Fundamental Theorem
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Infinite Series Convergent
Integers
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Linear Dependence, Independence
Linear Homogeneous Equations
Two Subspaces Linear Sum
Matric Polynomial
Matrix
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Matrix Multiplication
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Method Of Difference
Minors And Co-factors
Multiplication Modulo P
Normal Sub-Group
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Orbit Of Permutation
Peano Axioms
Permutation Function
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Matrices Integral Powers
Mathematical Induction Principal
Two Determinants Product
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Properties Of Modulus
Rank Of A Matrix
Rational Numbers
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Synthetic Division
Transformation In General
Transformations Of Equations
Transpose Of A Matrix
Matrix Transposed Conjugate
Transposition
Complex Numbers Representation
Vector Space
Vector Sub-spaces
Whole Numbers

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