How to calculate the conjugate of a complex number?

To calculate the conjugate of a complex number, simply change the sign of the imaginary part.

A complex number is of the form a + bi, where a and b are real numbers and i is the imaginary unit, which is defined as the square root of -1. The conjugate of a complex number is obtained by changing the sign of the imaginary part, i.e. if z = a + bi, then the conjugate of z, denoted by z*, is given by z* = a - bi.

For example, if z = 3 + 2i, then the conjugate of z is z* = 3 - 2i. Similarly, if w = -4i, then the conjugate of w is w* = 4i.

The conjugate of a complex number has some important properties. Firstly, the product of a complex number and its conjugate is always a real number, i.e. if z = a + bi, then z*z* = (a + bi)(a - bi) = a^2 + b^2, which is a real number. Secondly, the sum and difference of two complex numbers can be expressed in terms of their conjugates, i.e. if z1 = a + bi and z2 = c + di, then z1 + z2 = (a + c) + (b + d)i and z1 - z2 = (a - c) + (b - d)i.

In summary, the conjugate of a complex number is obtained by changing the sign of the imaginary part, and has some important properties that are useful in complex number arithmetic.