**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 z_{1,} z_{2} is the sum of their conjugates

z_{1} = a + ib and z_{2} = c + id

Then z_{1} + z_{2} = (a + ib) + (c + id) = (a + c) + i (b + d)

Conjugate (z_{1} + z_{2) }= (a + c) − i (b + d) ---(1)

z_{1 } = a − ib, z_{2} = c − id

Conjugate (z_{1}) + conjugate (z_{2}) = (a − ib) + (c − id)

= (a + c) − i(b + d) ---(2)

**Property 7 :**

The conjugate of the difference of two complex numbers z_{1,} z_{2} is the difference of their conjugates

z_{1} = a + ib and z_{2} = c + id

Then z_{1} - z_{2} = (a + ib) - (c + id) = (a - c) + i (b - d)

Conjugate (z_{1} - z_{2) }= (a - c) − i (b - d) ---(1)

Conjugate (z_{1}) = a - ib, z_{2} = c - id

Conjugate (z_{1}) - conjugate (z_{2}) = (a − ib) - (c − id)

= (a - c) − i(b - d) ---(2)

**Property 8 :**

The conjugate of the product of two complex numbers z_{1}, z_{2} is the product of their conjugates.

Let z_{1} = a + ib and z_{2} = c + id. Then

z_{1} z_{2} = (a + ib) (c + id) = (ac − bd) + i(ad + bc)

Conjugate (z_{1} z_{2}) = (ac − bd) − i(ad + bc) -----(1)

conjugate (z_{1}) = a − ib, conjugate (z_{2}) = c − id

conjugate (z_{1}) conjugate (z_{2})= (a − ib) (c − id)

= (ac − bd) − i(ad + bc) -----(2)

**Property 9 :**

The conjugate of the quotient of two complex numbers z_{1}, z_{2}, (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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