Complex Numbers Part One

Complex Numbers

part 1

Lets start with an example :

Suppose you're told to solve the following equation and find the value of x^2+y^2:

1) (x+y)^2= -25

2) x y = -6        

If you are smart you will say that it is impossible, supposing you don't know about complex numbers. But still you'll be able to solve it. so if it were incorrect it should have been impossible to find its root.

But we have just done it.

Well here is its story:

In late 1500s, the master equation solver Girolamo Cardano was trying to solve polynomial equations. He had no trouble solving equations like x^2−8 x+12=0, because it was easy to find two numbers whose sum was 8 and whose product was 12: namely, 2 and 6. This meant x^2−8 x+12 could be factored as (x−2)(x−6), and expressing this polynomial as a product of two factors made solving the equation x^2−8 x+12=0 easy.

But it wasn’t that easy to do this for equations like x^2-3 x+10=0. Finding two numbers that add to 3 and multiply to 10 seems an impossible challenge. If the product of the two numbers is positive, they must have the same sign, and since their sum is positive, this means they must both be positive. But if two positive numbers add up to 3, they must both be less than 3, which means their product will be less than 3 × 3 = 9. There doesn’t seem to be a way to make this work.

Yet Cardano discovered that he could make it work, if he allowed himself to consider numbers that involved √-1, the square root of –1. It was not a nice discovery. The square root of a number x, or √x, is the number that when multiplied by itself produces x. Now, when you square a real number, the result can never be negative: for example, 3 × 3 = 9, (-1.2) × (-1.2) = 1.44 and 0 × 0 = 0. This means no real number multiplied by itself could equal –1. So it was more like a child without parents which ,no doubt, is impossible.  Cardano was using √-1 to solve his real number equations, but √-1 isn’t itself a real number.

Many mathematician termed this use as evil, useless etc. Later on, in the early 19th century it was widely accepted.

Now lets come to it....

√-1 is termed as i . It is the base of imaginary numbers.

Now you must have known about real numbers. Real numbers and imaginary numbers together make complex numbers.

The usual form of complex numbers is:

z=a+bi

Where, a and b are real. 'i' is imaginary part. So,if a=0, z is imaginary and,if b=0,  z is real number.  

So,now let's do some math..

Add:

(2+3i)+4i+5

Solution:

(2+3i)+4i+5 = 2+5+3i+4i = 7+7i

Just like algebra...

Multiply:

(2+3i)(1+i) 

Solution :

Answer is -1+5i..

What!! Shocked or are you smiling in smartness?????

Look here:

i=√-1; i^2=-1; i^3= -i; i^4= 1;

Actually, i^4n    =1

                 i^4n+1=i

                 i^4n+2=-1

                 i^4n+3=-i