# Number System : Imaginary and Complex Number

**Imaginary Number:** Since no real number satisfies the equation , a number of the form , where b is a real number, b 0 and is called an imaginary number. Since real numbers does not contain the square root of -ve number, an imaginary number was introduced to overcome this.

**Quotlly**

**For e.g. **

Imaginary numbers when squared gives a negative number.

**Complex Numbers: **A number that can be expressed in the form of where a and b are real numbers and is a solution of the equation

In the complex number, a is called the real part, b is called the imaginary part and .**For e.g.** :

Either a or b or both can be **0**, so all Real Numbers and Imaginary Numbers are also Complex Numbers.

If a = 0 and b 0, then the number become a imaginary number. Hence, all Imaginary numbers are complex numbers.

If b = 0 , then the number becomes a real number. Hence, all real numbers are complex numbers.

**Conjugate of a Complex Number :** If we change the sign of the imaginary part, we get the conjugate of a complex number.

So, if (a-bi) is a complex number, its conjugate will be (a+bi)

and, if (a+bi) is a complex number, its conjugate will be (a-bi)**For e.g. **: 3 – 5i is the conjugate of 3 + 5i.

5+3i is the conjugate of 5-3i.

**Addition of a Complex Numbers : **To add two complex numbers,

~ Add the real parts to get the final real part.

~ Add the imaginary parts to get the final imaginary part.**For e.g. **Add 3+5i & 4+7i

**Multiplication of complex numbers :** To multiply two complex numbers,

~ multiply each part of first complex number with each part of second complex number.**For e.g. Multiply 3+5i and 4+7i**

**Division of complex numbers : **To divide one complex number by another, we actually use conjugate of the number.

~ Multiply the numerator and denominator with the conjugate of the denominator**For e.g. Divide 3+4i by 2+3i**

**Multiplication of Conjugates : **In the previous example, what we did was multiplication of conjugate.

Lets see what multiplication of conjugates results in.

So,