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Complex Number Arithmetic,
Conjugate and Inverse of Complex Numbers


Before looking at the conjugate and inverse of a complex number, it’s a good idea to look at some standard examples of adding and subtracting complex numbers.

When faced with complex numbers examples requiring common arithmetic, there are some standard approaches to use for such situations.




Adding and Subtracting Complex Numbers

Adding and subtracting complex numbers in Math is often fairly straight forward.

( a + bi ) + ( c + di )   =   ( a + c ) + ( b + d )i

( a + bi ) − ( c + di )   =   ( a − c ) + ( b − d )i



Examples    


(1.1) 

( 3 + 4i ) + ( 2 + 3i )  =  ( 3 + 2 ) + ( 4 + 3 )i

=   5 + 8i


(1.2) 

( 3 + 4i ) − ( 2 + 3i )  =  ( 3 − 2 ) + ( 4 − 3i )

=   1 + 1i  =  1 + i






Multiplying Complex Numbers


Not entirely dissimilar to adding and subtracting complex numbers.

Multiplying complex numbers follows standard patterns.

( a + bi ) × ( c + di )   =   ( a × c ) + ( a × di ) + ( bi × c ) + ( bi × di )

=   ac + adi + bci + bdi2

ac + adi + bci + bdi2     =     ac + adi + bci − bd        ( as i2 = -1 )

Now rewriting slightly:

ac bd + adi + bci   =   ( ac bd ) + ( ad + bc )i

In summary:

( a + bi )( c + di )   =   ( ac − bd ) + ( ad + bc )i



Examples    


(2.1) 

( 3 + 5i )( 1 + 2i )   =   ( 3 − 10 ) + ( 6 + 5 )i

=   -7 + 10i


(2.2) 

( 4 − 2i )( 3 + 3i )   =   ( 12 − (-6) ) + ( 12 + (-6) )i   =   18 + 6i


(2.3) 

3 × ( 2 + 2i )   =   6 + 6i






Division of Complex Numbers,
Conjugate of a Complex Number


Division with complex numbers is very similar to how we rationalise surds.
By which we multiply the top and bottom of the division sum by the conjugate of the denominator.

The conjugate of a complex number has the same real part, but the imaginary part will have the opposite sign.


We do this multiplication between the two because multiplying a complex number by its conjugate has the following result:

( a + bi ) × ( abi )   =   a2 + b2



Example    


(3.1) 


\bf{\frac{3 \space + \space 5i}{2 \space – \space 3i}}          [ CONJUGATE OF  2 − 3i  IS ( 2 + 3i ) ]


=>   \bf{\frac{3 \space + \space 5i}{2 \space – \space 3i}}  ×  \bf{\frac{2 \space + \space 3i}{2 \space + \space 3i}}   =   \bf{\frac{(6 – 15) \space + \space (9 + 10)i}{4 \space + \space 9}}


=   \bf{\frac{{\text{-}}9 \space + \space 19i}{13}}   =   \bf{\frac{{\text{-}}9}{13}}\bf{\frac{19}{13}}i






Inverse of Complex Numbers

The inverse of complex numbers is sometimes referred to as the multiplicative inverse of complex numbers, but they mean the same thing.

The inverse of a complex number is simply the reciprocal of the number, which is the fraction of  1  over the complex number.

But usually it’s good practice for this fraction to be rationalized, which is where we again make use of the complex conjugate.



Example    


(4.1) 

Find the multiplicative inverse of   2 + 5i.

Solution   

Reciprocal of   2 + 5i   is   \bf{\frac{1}{2 \space + \space 5i}}.

Now we look to rationalize.

[ CONJUGATE OF  2 + 5i  IS ( 2 5i ) ]

\bf{\frac{1}{2 \space + \space 5i}} × \bf{\frac{2 \space – \space 5i}{2 \space – \space 5i}}   =   \bf{\frac{2 \space – \space 5i}{4 \space – \space 10i \space + \space 10i \space – \space 25i^2}}   =   \bf{\frac{2 \space – \space 5i}{4 \space – \space ({\text{-}}25)}}   =   \bf{\frac{2 \space – \space 5i}{29}}


Now   \bf{\frac{2 \space – \space 5i}{29}}   can be left as is, or the multiplicative inverse can also be written as   \bf{\frac{2}{29}} + \bf{\frac{5}{29}}i.





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