Conjugate of a Complex Number

The conjugate of a complex number is obtained by changing the sign of the imaginary part. In other words, for a complex number a + ib, its conjugate is given by a - ib.

Explanation:
To understand why a-ib is called the conjugate of a+ib, let's consider a complex number z = a + ib, where a and b are real numbers.

The conjugate of z, denoted by z*, is obtained by changing the sign of the imaginary part of z. So, if z = a + ib, then z* = a - ib.

Example:
Let's take an example to illustrate the concept. Consider the complex number z = 3 + 4i.

The conjugate of z, denoted by z*, is obtained by changing the sign of the imaginary part. So, z* = 3 - 4i.

Properties of Conjugate:
1. The sum of a complex number and its conjugate is always a real number. For example, if z = a + ib, then z + z* = (a + ib) + (a - ib) = 2a.

2. The product of a complex number and its conjugate is always a real number. For example, if z = a + ib, then z * z* = (a + ib) * (a - ib) = a^2 + b^2.

3. The conjugate of a conjugate is the original complex number. In other words, if z = a + ib, then (z*)* = z.

4. The complex number and its conjugate have the same real part but opposite signs for the imaginary part. For example, if z = a + ib, then the real part of z is a and the real part of z* is also a. However, the imaginary part of z is b and the imaginary part of z* is -b.

Conclusion:
The conjugate of a complex number a + ib is obtained by changing the sign of the imaginary part, resulting in a - ib. The conjugate has several important properties and is often used in various mathematical operations involving complex numbers.

This is called conjugate of complex no... z=a+ib. conjugate of z=a-ib - sign are put before i