Properties of a Complex conjugate
- The real part of the complex conjugate is equal to the real part of the complex number, while the imaginary part of the complex conjugate is equal to the negative of the imaginary part of the complex number.
Re(z̅) = Re(z)
Im(z̅) = −Im(z)
- The sum of a complex number z and its complex conjugate z* is a real number.
z + z̅ = (a + ib) + (a − ib) = 2a = 2Re(z)
- The difference between a complex number z and its complex conjugate z* is an imaginary number.
z − z̅ = (a + ib) − (a − ib) = 2ib = 2Im(z)
- The product of the complex number z and its complex conjugate z* is a real number.
z × z̅ = (a + ib)×(a − ib)= a2+b2
- If z and w are two complex numbers, then the complex conjugate of their product is equal to the product of their complex conjugates.
- If z and w are two complex numbers, then the complex conjugate of their quotient is equal to the quotient of their complex conjugates.
- If z and w are two complex numbers, then the complex conjugate of their sum is equal to the sum of their complex conjugates.
- If z and w are two complex numbers, then the complex conjugate of their difference is equal to the difference between their complex conjugates.