A bit more on Complex numbers

We know that, 
i = sqrt (-1)

and therfore, 

i^2 = -1

Also, sqrt (9) = 3

 sqrt (-9) = sqrt(-1).sqrt(9) = i.sqrt(9) = 3i
We can check this. 
(3i)^2 = 3^2.i^2 = 9.-1 = -9

So, 

(3i)^2 = -9

3i = sqrt(-9)

3i is an imaginary number.

Complex number are imaginary and real numbers together.

e.g.

6 + 3i is a complex number.

Addition of complex numbers 

Here we have two complex numbers zi and z2

zi = a + bi
z2 = c + di
we add the real parts then add the imaginary parts.

zi + z2 = (a + c) + (bi + di)

        = (a + c) + (b + d)i

Subtraction of complex numbers 

Here we have two complex numbers zi and z2

zi = a + bi

z2 = c + di

we subtract the real parts then subtract the imaginary parts.

zi - z2 = (a - c) + (bi - di)

        = (a - c) + (b - d)i

Multiplication of complex numbers 

Again we have two complex numbers zi and z2

zi = a + bi

z2 = c + di

zi . z2 = (a + bi) . (c + di)

        = a(c+di) + bi(c+di)

        = ac+adi + cbi + (bi.di)          ....... (eqn 1)

let's sort out the (bi.di)

(bi.di) = bd.i^2

we know that i^2 = -1

so, db.i^2 = -bd

Returning to where we were in eqn (1),

zi . z2 = ac + adi + cbi - (bi.di)

        = ac + adi + cbi - bd

        = (ac - bd) + (adi + cbi)

        = (ac - bd) + (ad + cb).i

Division of complex numbers 

Again we have two complex numbers zi and z2

zi = a + bi

z2 = c + di

zi / z2 = (a+bi)/(c+di)

We can use the rule:

(a+b).(a-b) = a^2-b^2

The Conjugate of a complex number is a reverse of the direction of the imaginary number.

The Conjugate of (a + bi)is (a - bi)

The Conjugate is written with a bar over the top, so the conjugate of z1 is written z1 bar. 

How do I type that here? I don't know!

When we multiply an imaginary number by its conjugate we get a Real number. Here's the trick:

zi     (a+bi)   c-di   ac-adi + bci-bdi^2

--- =  ------ . ---- = -------------------

z2     (c+di)   c-di   c^2 + d^2

      ac-adi + bci-bdi^2         [remember that , i^2 = -1]

    = ------------------- 

         c^2 + d^2

      ac-adi + bci+bd

    = ----------------- 

         c^2 + d^2

       (ac+bd) + (bc-ad)i

    = --------------------- 

         c^2 + d^2

       ac+bd         bc-ad

    =  --------  +  ------- . i

       c^2+d^2      c^2+d^2

example.

1+2i    (1+2i)   2-3i   (1.2) + (1.(-3i)) + (2.2i) + (2i.(-3i))

---- =  ------ . ---- = ---------------------------------------

2+3i    (2+3i)   2-3i   (2.2) + (2.(-3i) + (3i.(2) + (3i.(-3i))

       2 - 3i + 4i - 6i^2

     = ------------------

       4 - 6i + 6i + 9i^2

        2 - 3i + 4i + 6

     =  ------------------

             4 + 9

     

        8 + i

     =  -----

         13  

     

     =  8      1

       ---  + --- i     

       13     13