H2 maths complex numbers

• quailmaster

  22 Aug 11, 00:34

  Find the roots of z^7 = 1. Show that the sum of the roots equal to zero I managed to find the roots, they are 1, e^2/7 pi , e^ 4/7pi , e^6/7 pi e^-2/7 pi , e^ -4/7pi , e^-6/7 pi How to show that the sum of all the roots equal to zero? Just use GC? Thanks in advance
• jayh272416

  22 Aug 11, 00:57

  express everything in components

  something magical will happen.

• wee_ws

  22 Aug 11, 07:41

  Hi,

    GC cannot be used to show a result. These steps may be useful:

    1. Use the fact that e^(θi) + e^(-θi) = 2 cos θ.

    2. Use the factor formulae in MF15.

    Another question that is easier to show is:

    Show that the product of all the roots is 1.

    Thanks.

  Cheers,
  Wen Shih

• eagle

  22 Aug 11, 08:46

  Another way to sum is to use a GP with first term e^-6/7 pi, common ratio e^2/7 pi and 7 terms

  Hence, the sum of all roots =
  e^-6/7 pi   ( e^2pi - 1)  /   (e^2/7 pi  - 1)

  Since e^2pi - 1  = 0, hence the sum of all roots = 0

   

• quailmaster

  22 Aug 11, 08:58

  using the trigo method i ended up with 2cos 4/7pi ( 1 + cos 2/7pi ) .. how do i simplify from there? the GP method is really neat :) thanks
• wee_ws

  22 Aug 11, 17:20

  Dear eagle,

    Your method is most direct and elegant, thanks :)

    I realised, upon closer scrutiny, that my method is not feasible.

  Cheers,
  Wen Shih

• frekiwang

  22 Aug 11, 18:39

  This is a method that does not require any calculation:

  Draw a unit circle, the seventh roots of unity are unit vectors with 2pi/7 in between. 

  Note that each interior angle of a regular 7-side polygon is 5pi/7 

  Do some parallel-shift of the 7 unit vectors, you can easily show them to form a closed 7-side polygon. So according to the rule of adding vectors, the sum of these 7 unit vectors is 0.

• wee_ws

  22 Aug 11, 18:49

  Hi frekiwang,

    This geometrical method is elegant, thanks for sharing!

  Cheers,
  Wen Shih

• wee_ws

  23 Aug 11, 08:00

  Hi,

    Here is a diagram to clarify frekiwang's geometrical approach:

    http://wenshih.files.wordpress.com/2010/02/sum-of-roots-explanation1.pdf

    Since the roots sum to 1, we can show that

    cos (2pi/7) + cos (4pi/7) + cos (6pi/7) = -1/2

    Thanks.

  Cheers,
  Wen Shih