PROPERTIES OF COMPLEX NUMBERS

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 zand 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 zand  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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