## Complex numbers

Complex numbers are one form of numbers. It is a combination of real and imaginary numbers. It can be generally expressed as a+ib where a is the real number and 'i' is the imaginary number.

History:

Till now we know that we can not take the square root for a negative number. How to take a square root of a negative number? For that a new form of number was introduced by the Italian mathematician Girolamo Cardano. Even though the first mention of imaginary number started in the 1st century. But for a long time no one tried to give the correct explanation of imaginary numbers. In 1545 Girolamo Cardano he solved an equation x(10-x)=40 and got one real number solution and another one square root of a negative number. That gave him the idea to use the imaginary number (whose square is a negative number). In 1833  William Rowan Hamilton expressed this number as a pair of real numbers, for example a+ib is expressed as (a,b) making this form of numbers more believable.

Imaginary number 'i':

The letter 'i' stands for the imaginary number, because we know that it is not real. So 'i' is defined as

i= √-1

Then i2 = -1

The powers and signs of 'i' and 1 is a cycle.

Combination of real and imaginary number:

Examples:

2+3i, 148-75i, 0.28+0.75i

Note: As we know this number is a combination of real and imaginary part. But either part can be 0. So all the real numbers are complex numbers.

Here are some problems involving 'i'.

Simplify sqrt(-7)

Solution:

Simplify -sqrt(-7)

solution:

While adding two of this numbers we have to add the real parts together and imaginary parts together. 'i' should be treated as the variable x, and we have to add as how we add terms involving x.

Examples:

1.     Simplify: 2i+5i

Solution: 2i+5i = (2+5)i = 7i

2.     Simplify: (3+4i)+(2+8i)

solution: We have to add the real part together and imaginary part together.

= (3+2) + (4+8)i

=    5   +    12i

3.      Simplify: (24+3i) + 5i

Solution: Here in the second number we don't have the real part. In this case we have to add the imaginary part of the first and second numbers.

=  24  +  (3i+5i)

=  24  +  (3+5)i

=  24  +     8i

4.      Simplify: 92 + (8+3i)

Solution: Here we have only real part in the first part. So we have to add the real parts together and leave the imaginary part as it is.

=  (92+8)  +  3i

=     100    +  3i

Subtraction:

Subtraction is also like addition. We have to subtract the real parts and imaginary numbers separately.

Examples:

1.         Simpliy:  (17+5i)  - (3+2i)

Solution:         =  (17-3)  +  (5-2)i

=     14     +   3i

2.         Simplify:  (19-2i) - (7+4i)

Solution:           =  (19-7) +  (-2i-4i)

=     12   +   (-6i)

=     12 - 6i