The properties of addition and multiplication of complex numbers are the same as for real numbers. We list here the basic algebraic properties and verify some of them.

**Basic Algebraic Properties of Complex Numbers**

The properties of
addition and multiplication of complex numbers are the same as for real
numbers. We list here the basic algebraic properties and verify some of them.

**Properties of complex numbers**

Let us now prove some
of the properties.

**Property**

The commutative
property under addition

For any two complex
numbers *z*_{1} and *z*_{2} , we have *z*_{1} + *z*_{2} = *z*_{2} + *z*_{1}.

Let *z*_{1} = *x*_{1} + *iy*_{1} , *z*_{2} = *x*_{2} + *iy*_{2} , and *x*_{1} , *x*_{2} , *y*_{1} , and *y*_{2} âˆˆ **R** ,

*z*_{1} + *z*_{2} = ( *x*_{1} + *iy*_{1} ) + ( *x*_{2} + *iy*_{2} )

= ( *x*_{1} + *x*_{2} ) + *i *( *y*_{1} + *y*_{2} )

= ( *x*_{2} + *x*_{1} ) + *i *( *y*_{2} + *y*_{1} )
(since *x*_{1} , *x*_{2} , *y*_{1} , and *y*_{2} âˆˆ **R** )

= ( *x*_{2} + *iy*_{2} ) + ( *x*_{1} + *iy*_{1} )

= *z*_{2} + *z*_{1}.

Prove that the multiplicative inverse of a nonzero complex
number *z *= *x *+ *iy *is

The multiplicative inverse is less obvious than the additive
one.

Let *z*^{âˆ’1} = *u *+ *iv *be the inverse of *z *= *x *+ *iy*

We have *z z*^{-1}* = 1*

That is ( *x *+ *iy *)(*u *+ *iv*) = 1

( *xu *- *yv*) + *i*(*xv *+ *uy*) = 1+ *i*0

Equating real and imaginary parts we get

*xu *- *yv *= 1and *xv *+ *uy *= 0 .

Solving the above system of simultaneous equations in *u *and
*v*

Note that the above example shows the existence of *z*^{-1 }of the complex number *z
*. To compute the inverse of a given complex number, we conveniently use z^{-1} = 1/*z*. If z_{1} and z_{2} are two complex
numbers where z_{2} â‰ 0, then the product of z_{1} and 1/ z_{1} is denoted by z_{1}/z_{2}. Other properties can
be verified in a similar manner. In the next section, we define the conjugate
of a complex number. It would help us to find the inverse of a complex number
easily.

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12th Mathematics : UNIT 2 : Complex Numbers : Basic Algebraic Properties of Complex Numbers |

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