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) = a^{2} - (ib)^{2 }
= a^{2} - i^{2}b^{2}
= a^{2} - (-1)b^{2}
= a^{2} + b^{2} (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 z_{1 }and z_{2} is the sum of their conjugates.
Proof :
Let z_{1} = a + ib and z_{2} = c + id, where a, b, c and d are real numbers.
conjugate of z_{1} + conjugate of z_{2} = (a - ib) + (c - id)
= a + c - ib - id
= (a + c) - i(b + d)
conjugate of (z_{1} + z_{2}) = 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 z_{1} + conjugate of z_{2}
7. The difference of two complex numbers z_{1 }and z_{2} is the difference of their conjugates.
Proof :
Let z_{1} = a + ib and z_{2} = c + id, where a, b, c and d are real numbers.
conjugate of z_{1} - conjugate of z_{2} = (a - ib) - (c - id)
= a - ib - c + id
= a - c - ib + id
= (a - c) - i(b - d)
conjugate of (z_{1} - z_{2}) = 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 z_{1} - conjugate of z_{2}
8. The conjugate of the product of two complex numbers z_{1 }and z_{2} is the product of their conjugates.
Proof :
Let z_{1} = a + ib and z_{2} = c + id, where a, b c and d are real numbers.
conjugate of (z_{1}) x conjugate of (z_{2}) = (a - ib)(c - id)
= ac - iad - ibc + i^{2}bd
= ac - iad - ibc + (-1)bd
= ac - bd - iad - ibc
= (ac - bd) - i(ad + bc)
conjugate (z_{1}z_{2}) = conjugate of [(a + ib)(c + id)]
= conjugate of [ac + iad + ibc + i^{2}bd]
= 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 (z_{1}) x conjugate of (z_{2})
9. The conjugate of the quotient of two complex numbers z_{1}, and z_{2} (z_{2} ≠ 0) is the quotient of their conjugates.
Conjugate of (z_{2}/z_{2}) = conjugate (z_{2}) / conjugate of (z_{2})
10. Let z be a complex number. Then, conjugate of (z^{n}) is equal to conjugate of z raised to the power n.
conjugate of (z^{n}) = (conjugate of z)^{n}
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