Complex conjugation

• Complex conjugation refers to the operation that changes the sign of the imaginary part of a complex number. If a complex number is represented as a + bi, its conjugate is a - bi.
• The complex conjugate of a real number is the number itself, as the imaginary part is zero. Therefore, it has no effect on real numbers.
• The complex conjugate of an imaginary number a + bi is a - bi. For instance, the conjugate of 2 + 3i is 2 - 3i, and the conjugate of -1 + 4i is -1 - 4i.
• Complex conjugation plays significant roles in mathematics and engineering due to its properties. It is widely used in the calculation and simplification of complex numbers.
• Taking the conjugate twice gives the original complex number back. If z is your complex number, then the conjugate of the conjugate (i.e., (z)) is equal to z.
• The product of a complex number and its conjugate gives a real number. Specifically, it gives the square of the magnitude (or modulus) of the complex number. If z = a + bi, then zz* = a^2 + b^2.
• If you add or subtract two complex numbers and then take the conjugate of the result, it’s the same as taking the conjugate of each original number first and then adding or subtracting them. That is, (z + w)* = z* + w* and (z - w)* = z* - w*.
• If you multiply or divide two complex numbers and then take the conjugate of the result, it’s the same as taking the conjugate of each original number first and then multiplying or dividing them. That is, (zw)* = zw and (z/w)* = z/w.
• It’s important to remember that complex conjugation affects the sign of the imaginary part only. The real part remains the same. No other changes are made to the structure of the complex number.
• In polar form, taking the complex conjugate amounts to negating the argument while keeping the magnitude the same.
• Knowing complex conjugation is pivotal to understanding further concepts like Hermitian matrices in linear algebra, inner product spaces, and Fourier transforms in Physics and Engineering.