Method of completing Squares in Complex Numbers

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Information about Method of completing Squares in Complex Numbers

Published on January 22, 2009

Author: youmarks

Source: slideshare.net

Description

A beautiful method of completing squares in complex numbers is presented.

Method of completing squares in Complex Numbers The purpose of this slide is to show how do we complete squares in complex numbers. A very beautiful solution for finding square root of (a+ib) is also demonstrated. It is assumed that you know definition of iota and complex numbers Prepared By Parag Arora copyrights © youmarks.com

Problem Find z = One way to find this is by equation z to x + iy, on squaring which gives 7 + 24i = x 2 – y 2 + 2ixy x 2 – y 2 = 7 and xy = 12 which on solving gives the value of x and y. It must be emphasized here that if we visualize 7 = 4 2 – 3 2 and 12 = 4.3, we straight away get value of x and y. This is known as method of completing square. The method of completing squares is shown here for a particular case. We will however generalize this method using a very beautiful approach of completing squares. copyrights © youmarks.com

Find z =

One way to find this is by equation z to x + iy, on squaring which gives

7 + 24i = x 2 – y 2 + 2ixy

x 2 – y 2 = 7 and xy = 12 which on solving gives the value of x and y.

It must be emphasized here that if we visualize 7 = 4 2 – 3 2 and 12 = 4.3, we straight away get value of x and y. This is known as method of completing square.

The method of completing squares is shown here for a particular case. We will however generalize this method using a very beautiful approach of completing squares.

Finding Again as we did, one method is put x + iy = This would as before give us x 2 – y 2 =a and xy = b/2. We can easily solve the above two equations and find x and y in terms of a and b. But we will present a way of completing squares for finding the roots very fast. copyrights © youmarks.com

Again as we did, one method is put x + iy =

This would as before give us

x 2 – y 2 =a and xy = b/2.

We can easily solve the above two equations and find x and y in terms of a and b. But we will present a way of completing squares for finding the roots very fast.

Finding We know that ( c + id ) 2 = c 2 – d 2 + 2icd. To find We see that we need to figure out c and d in such a way that if cd = b/2 then c 2 – d 2 = a. Let us consider the case when b > 0. Now b = √b 2 Or copyrights © youmarks.com

We know that

( c + id ) 2 = c 2 – d 2 + 2icd. To find

We see that we need to figure out c and d in such a way that if cd = b/2 then c 2 – d 2 = a.

Let us consider the case when b > 0. Now

b = √b 2

Or

Finding Also note that a = copyrights © youmarks.com

Finding So we see that complex number a + ib is nothing but copyrights © youmarks.com

Finding So we see a + bi is is denoted by |z| if z = a + ib copyrights © youmarks.com

Finding So we see that is nothing but Repeat the same problem when b < 0. copyrights © youmarks.com

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