## About "Properties of complex numbers"

Properties of complex numbers :

Here we are going to the list of properties used in complex numbers.

Property 1 :

The product of a complex number and its conjugate is a real number. Property 2 :

The result of finding conjugate for conjugate of any complex number is the given complex number. Property 3 :

If the conjugate of complex number is the same complex number, the imaginary part will be zero.

If z is real, i.e., b = 0 then z = conjugate of z.

Conversely, if z = conjugate of z

i.e., if a + ib = a − ib then b = − b ⇒ 2b = 0 ⇒ b = 0

(2 ≠ 0 in the real number system).

b = 0 ⇒ z is real.

From this we come to know that,

z is real ⇔ the imaginary part is 0

Let us look into the next property on "Properties of complex numbers".

Property 4 :

Sum of complex number and its conjugate is equal to 2 times real part of the given complex number. Let z  =  a + ib

conjugate of z  =  a - ib

z + conjugate of z  =  a + ib + a - ib

=  2 a

=  2 Re(z)

Property 5 :

the difference of a complex number and its conjugate  equal to 2 times imaginary part of the given complex number. Let z  =  a + ib

conjugate of z  =  a - ib

z + conjugate of z  =  a + ib - (a - ib)

=  a + ib - a + ib

=  2 ib

=  2 im(z)

Property 6 :

The conjugate of the sum of two complex numbers z1, z2 is the sum of their conjugates z1 = a + ib and z2 = c + id

Then z1 + z2 = (a + ib) + (c + id) = (a + c) + i (b + d)

Conjugate (z1 + z2)  =  (a + c) − i (b + d)  ---(1)

z =  a − ib, z2  =  c − id

Conjugate (z1) + conjugate (z2) = (a − ib) + (c − id)

= (a + c) − i(b + d) ---(2)

Property 7 :

The conjugate of the difference of two complex numbers z1, z2 is the difference of their conjugates z1 = a + ib and z2 = c + id

Then z1 - z2 = (a + ib) - (c + id) = (a - c) + i (b - d)

Conjugate (z1 - z2)  =  (a - c) − i (b - d)  ---(1)

Conjugate (z1)  =  a - ib, z2  =  c - id

Conjugate (z1) - conjugate (z2) = (a − ib) - (c − id)

= (a - c) − i(b - d) ---(2)

Property 8 :

The conjugate of the product of two complex numbers z1, z2 is the product of their conjugates. Let z1 = a + ib and z2 = c + id. Then

z1 z2 = (a + ib) (c + id) = (ac − bd) + i(ad + bc)

Conjugate (z1 z2)  =  (ac − bd) − i(ad + bc)  -----(1)

conjugate (z1)  =  a − ib, conjugate (z2)  =  c − id

conjugate (z1) conjugate (z2)= (a − ib) (c − id)

= (ac − bd) − i(ad + bc)  -----(2)

Property 9 :

The conjugate of the quotient of two complex numbers z1, z2, (z2 ≠ 0) is the quotient of their conjugates. Property 10 : After having gone through the stuff given above, we hope that the students would have understood "Properties of complex numbers".

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