Method of completing squares in Complex Numbers The purpose of this slide is to show how do we complete squares in complex...
Problem <ul><li>Find z =  </li></ul><ul><li>One way to find this is by equation z to x + iy, on squaring which gives  </li...
Finding  <ul><li>Again as we did, one method is put x + iy = </li></ul><ul><li>This would as before give us  </li></ul><ul...
Finding  <ul><li>We know that  </li></ul><ul><li>( c + id ) 2  = c 2  – d 2  + 2icd. To find </li></ul><ul><li>We see that...
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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Method of completing Squares in Complex Numbers

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

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  • in slide number 6/8, the ...+2(root((root a^2+b^2)+a)/2 . root((root a^2+b^2)-a)/2) part of calculation, could it be an error of dividing both the root((root a^2+b^2)+a) by 2 and the root((root a^2+b^2)-a) by 2 inside the bracket, is mean to be only divide by 2 on one of the root, not both, so that this is then canceled by the 2 outside the bracket? otherwise the 2(root((root a^2+b^2)+a)/2 . root((root a^2+b^2)-a)/2) will just be the cd, which is b/2, not b, ...or am i misunderstood this?
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Method of completing Squares in Complex Numbers

  1. 1. 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
  2. 2. Problem <ul><li>Find z = </li></ul><ul><li>One way to find this is by equation z to x + iy, on squaring which gives </li></ul><ul><li>7 + 24i = x 2 – y 2 + 2ixy </li></ul><ul><li>x 2 – y 2 = 7 and xy = 12 which on solving gives the value of x and y. </li></ul><ul><li>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. </li></ul><ul><li>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. </li></ul>copyrights © youmarks.com
  3. 3. Finding <ul><li>Again as we did, one method is put x + iy = </li></ul><ul><li>This would as before give us </li></ul><ul><li>x 2 – y 2 =a and xy = b/2. </li></ul><ul><li>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. </li></ul>copyrights © youmarks.com
  4. 4. Finding <ul><li>We know that </li></ul><ul><li>( c + id ) 2 = c 2 – d 2 + 2icd. To find </li></ul><ul><li>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. </li></ul><ul><li>Let us consider the case when b > 0. Now </li></ul><ul><li>b = √b 2 </li></ul><ul><li>Or </li></ul>copyrights © youmarks.com
  5. 5. Finding Also note that a = copyrights © youmarks.com
  6. 6. Finding So we see that complex number a + ib is nothing but copyrights © youmarks.com
  7. 7. Finding So we see a + bi is is denoted by |z| if z = a + ib copyrights © youmarks.com
  8. 8. Finding So we see that is nothing but Repeat the same problem when b < 0. copyrights © youmarks.com

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