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Algebraic Structure of Complex Numbers

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Abstract algebra is the main branch in pure mathematics. It is basically concerned of algebraic structures. An algebraic structure is said to be a set along with at least one binary operation that satisfies particular axioms. The groups, fields, rings etc are the common examples of algebraic structures. In this article, we shall focus on the algebraic structures built from the set of complex numbers. Recall that a complex number is a number attaining the form $a + ib$; where $a$ is the real part and b denotes the imaginary part. Let us go ahead and learn about algebraic structure of complex numbers.

Definition

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The algebraic structure on the set of complex numbers, denoted by $C$, is an example of a field. It means that it satisfy certain conditions of an algebraic structure. A complex number $z$ = $x + iy$ is denoted by a vector $(x,\ y)$ defined on the real numbers field. A set $R^{2}$ = ${(x, y)\ |\ x,\ y \in\ R}$ is said to form an algebraic structure, if it holds the following basic axioms :
(a) When two complex numbers are multiplied or added and multiplied, they yield a (usually new) complex number.

(b) The additive inverse of a complex number $z$ is -$z$.

(c) The multiplicative inverse of reciprocal of a complex number $z$ is $z^{-1}$ or $\frac{1}{z}$.
(d) The law of commutativity for multiplication as well as addition over two complex numbers $z_{1}$ and $z_{2}$ must be satisfied. i.e.

      $z_{1}.\ z_{2}$ = $z_{2}\ z_{1}$

      $z_{1}\ +\ z_{2}$ = $z_{2}\ +\ z_{1}$
(e) Law of distributivity should be satisfied.

     $u\ (v + w)$ = $uv + uw$

     and

     $(u + v)\ w$ = $uw + vw$
(f) Law of associativity of addition and multiplication should hold.

    $u\ +\ (v + w)$ = $(u + v)\ +\ w$

    and

    $u\ (v w)$ = $(u v)\ w$

Addition and Multiplication of Complex Numbers

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Let $z_{1}$ = $u_{1}\ +\ iv_{1}$ and $z_{2}$ = $u_{2}\ +\ iv_{2}$ be two complex numbers denoted by $(u_{1},\ v_{1})$ and $(u_{2},\ v_{2})$ respectively.
Addition:

Addition of complex number in algebraic structures is defined as :

$(u_{1},\ v_{1})$ + $(u_{2},\ v_{2})$ = $(u_{1}\ +\ u_{2},\ v_{1}\ +\ v_{2})$
Multiplication:

Multiplication is performed as under.

$(u_{1},\ v_{1}).(u_{2},\ v_{2})$ = $(u_{1}\ u_{2}\ -\ v_{1}\ v_{2},\ u_{1}\ v_{2}\ +\ v_{1}\ u_{2})$

Properties

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Few complex numbers follow certain properties that are discussed below.
(i) Complex number $(0,\ 0)$ inherits the properties of zero.
    
    $(u, v)\ +\ (0, 0)$ = $(u + 0,\ v + 0)$ = $(u, v)$

    and

    $(u, v)\ (0, 0)$ = $(u.0\ -\ v.0,\ u.0\ +\ v.0)$ = $(0, 0)$

Thus, $(0, 0)$ additive and multiplicative identity.
(ii) The complex number $(1, 0)$ has the property:

     $(u, v)\ (1, 0)$ = $(u.1\ -\ v.0,\ u.0\ +\ v.1)$ = $(u, v)$
(iii) The complex number $(0, 1)$ when multiplied by any complex number, gives the following effect :

      $(u, v)\ (0, 1)$ = $(u.0\ -\ v.1,\ u.1\ +\ v.0)$ = $(-v,\ u)$
(iv) If $k$ be a constant (real number), then

     $k\ (u, v)$ = $(k, 0)\ (u, v)$ = $(ku, kv)$
(v) $i^{2}$ can be defined in the following way.

    Since $i$ = $0 + i$ = $(0, 1)$

    $i^{2}$ = $(0, 1)\ .(0, 1)$ = $(0.0\ -\ 1.1,\ 0.\ 1\ +\ 1\ .0)$ = $(-1,\ 0)$ = $-1$