Complex Conjugate

In mathematics, a number written in the from of u + iv is said to be a complex number. Here 'u' and 'v' are real numbers and 'i' is is iota symbol. The value of iota is given as: i = √-1. In this given expression 'u' is real part and 'v' is imaginary part. Now we will discuss different Complex Conjugate rules. The complex number is represented by the symbol 'Z' that contain real part and imaginary part, so we can write it as:

z = Re (z) + i im (z) = u + iv. Let the value of Im z = 0, then the value of z = u that is a real number. If the value of Re (z) = 0 then we get z = iv that is pure imaginary number. The complex conjugate rules are mention below: (know more about Complex Conjugate, here) 

Rule 1: Addition and subtraction:

The addition and subtraction rule of complex number is given as:

z₁ + z₂ = (u₁ + u₂) + i (v₁ + v₂);

Rule 2: Multiplication:

Multiplication rule is given as:

z₁ X z₂ = (u₁ + v2) X (u₂ + i v₂) = (u₁ u₂ – v₁ v₂) + i (u₁ v₂ – u₂ v₁);

Rule 3: Division rule:

Division rule of complex number is given as:

z₁ / z₁ = (u₁ + i v₁) / (u₂ + i v₂) = (u₁ u₂ + v₁ v₂) + i (u₂ v_{1 –} u₁ v₂) / (u₂₂ + v₂₂);

Let's talk about some properties of complex conjugate.

1. (zu)' = s'u'

2. z' = z, it is applicable if and only if the value of 'z' is real.

3. Zn' = z'n for any integer number 'n'.

4. | z' | = | z |;

5. | z |² = zz' = z'z;

6. z'' = z, in which the conjugate of the conjugate of a complex number 'z' is again that number.

7. Z – 1 = z' / | z |², it is applicable if any only if the value of 'z' is not zero.

8. Exp (z') = (exp (z))';

Second Law of Motion is deal with the branch of physics. In this we have studies the different types of motions. Before entering in the 10^th board examination please go through the CBSE sample papers for class x and In the next session we will discuss about solving trigonometric equations.