1. The product of a complex number and its conjugate is a real number.

Proof : 

(a + ib) and (a - ib) are two complex numbers conjugate to each other, where a and b are real numbers.   

(a + ib)(a - ib) = a² - (ib)²

= a² - i²b²

= a² - (-1)b²

= a² + b² (real number)

2. The result of finding conjugate for conjugate of any complex number is the given complex number.

Proof : 

Consider the complex number (a + ib).

Conjugate of (a + ib) = a - ib

Conjugate of (a - ib) = a + ib

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

Proof :

Let z = a + ib, where 'a' and 'b' are real numbers.  

If z is real, then b = 0 and also 

conjugate of z = z

Conversely, if z = conjugate of z, that is a + ib = a - ib,  then

b = - b ⇒ 2b = 0 ⇒ b = 0

b = 0 ⇒ z is real.

Therefore, 

z is real ⇔ the imaginary part is 0

4. Sum of a complex number and its conjugate is equal to 2 times the real part of the complex number.

Proof :

Let z = a + ib, conjugate of z = a - ib where a and b are real numbers.  

(a + ib) + (a - ib) = a + ib + a - ib

= 2a

= 2 Re(z)

5. Difference of a complex number and its conjugate is equal to 2i times the imaginary part of the complex number.

Proof :

Let z = a + ib, conjugate of z = a - ib where a and b are real numbers.  

(a + ib) - (a - ib) = a + ib - a + ib

= 2ib

= 2i Im(z)

6. The conjugate of the sum of two complex numbers z₁ and z₂ is the sum of their conjugates. 

Proof :

Let z₁ = a + ib and z₂ = c + id, where a, b, c and d are real numbers. 

conjugate of z₁ + conjugate of z₂ = (a - ib) + (c - id)

= a + c - ib - id

= (a + c) - i(b + d)

conjugate of (z₁ + z₂) = conjugate of [(a + ib) + (c + id)]

= conjugate of [a + c + ib + id]

= conjugate of [(a + c) + i(b + d)]

= (a + c) - i(b + d)

= conjugate of z₁ + conjugate of z₂

7. The difference of two complex numbers z₁ and z₂ is the difference of their conjugates. 

Proof :

Let z₁ = a + ib and z₂ = c + id, where a, b, c and d are real numbers. 

conjugate of z₁ - conjugate of z₂ = (a - ib) - (c - id)

= a - ib - c + id

= a - c - ib + id

= (a - c) - i(b - d)

conjugate of (z₁ - z₂) = conjugate of [(a + ib) - (c + id)]

= conjugate of [a + ib - c - id]

= conjugate of [a - c + ib - id]

= conjugate of [(a - c) + i(b - d)]

= (a - c) - i(b - d)

= conjugate of z₁ - conjugate of z₂

8. The conjugate of the product of two complex numbers z₁ and  z₂ is the product of their conjugates.

Proof : 

Let z₁ = a + ib and z₂ = c + id, where a, b c and d are real numbers. 

conjugate of (z₁) x conjugate of (z₂) = (a - ib)(c - id)

= ac - iad - ibc + i²bd

= ac - iad - ibc + (-1)bd

= ac - bd - iad - ibc

= (ac - bd) - i(ad + bc)

conjugate (z₁z₂) = conjugate of [(a + ib)(c + id)]

= conjugate of [ac + iad + ibc + i²bd]

= conjugate of [ac + iad + ibc + (-1)bd]

= conjugate of [ac - bd + iad + ibc]

= conjugate of [(ac - bd) + i(ad + bc)]

= (ac - bd) - i(ad + bc)

= conjugate of (z₁) x conjugate of (z₂)

9. The conjugate of the quotient of two complex numbers z₁, and z₂ (z₂ ≠ 0) is the quotient of their conjugates.

Conjugate of (z₂/z₂) = conjugate (z₂) / conjugate of (z₂)

10. Let z be a complex number. Then, conjugate of (zⁿ) is equal to conjugate of z raised to the power n. 

conjugate of (zⁿ) = (conjugate of z)ⁿ

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