# Help me find the complex roots of... **URL:** \ **Category:** Factual Questions\ **Created:** [June 17, 2002, 7:29am UTC](https://boards.straightdope.com/t/help-me-find-the-complex-roots-of/114490 "2002-06-17T07:29:53Z")\ **Posts on this page:** 4\ **Page:** 1
**Author:** ![Phlip](https://avatars.discourse-cdn.com/v4/letter/p/ed8c4c/32.png) [@Phlip](https://boards.straightdope.com/u/Phlip)\ **Post date:** [June 17, 2002, 7:29am UTC](https://boards.straightdope.com/t/help-me-find-the-complex-roots-of/114490/1 "2002-06-17T07:29:53Z")
(-7+24I)^(1/4) . So that is the 4th root of (-7+24I) . The book I have does not explain in very good detail how to do this. I need to find all 4 roots. How I tried to do it was by setting it equal to x. Then I made x^4=-7+24I . Next I try to pump this into the quadratic equation. To to this I make a=1, b=0, and c=7-24I . This gives me the roots of (-7+24I)^(1/2) but I am not sure how to use the quadratic to solve for powers higher than 2. I am thinking that a= 1 and -1 but I am not sure. When I do this I get 4 seperate answers, but they do not multiply out to -7+24I . Any ideas here? ---
**Author:** ![zut](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zut/32/2875_2.png) [@zut](https://boards.straightdope.com/u/zut)\ **Post date:** [June 17, 2002, 12:18pm UTC](https://boards.straightdope.com/t/help-me-find-the-complex-roots-of/114490/2 "2002-06-17T12:18:35Z")
The easiest way of taking roots of complex numbers is to represent them in a polar form. If you’re unfamiliar with that notion, think of a 2D plane where the x-axis is a real number line, and the y-axis is an imaginary number line. All complex numbers can be represented as a point on this 2D plane: for example, (-7+24I) is the point (-7, 24), using cartesian coordinates. Switch the cartesian representation (X + YI) into a polar form (Rcis[symbol]q[/symbol]) [“cis” = shorthand for Cos[symbol]q[/symbol] + I Sin[symbol]q[/symbol]] with the transformation R = SQRT(X[sup]2[/sup] + Y[sup]2[/sup]) and tan([symbol]q[/symbol]) = (Y/X). (-7+24I) = 25cis(73.74). In this form, finding the nth root of Rcis[symbol]q[/symbol] is easy: [R[sup]1/n[/sup] cis([symbol]q[/symbol]/n)][sup]n[/sup] = Rcis[symbol]q[/symbol], so the fourth root of 25cis(73.74) is sqrt(5)cis(18.43). Remembering that 360 can be added to the angle [symbol]q[/symbol] without altering the fundamental location of the point gives the other three roots: sqrt(5)cis(108.43), sqrt(5)cis(198.43), and sqrt(5)cis(288.43), conveniently 90 degrees from each other. Converting these back to cartesian form and checking my work is an exercise left to the reader. ---
**Author:** ![Jabba](https://avatars.discourse-cdn.com/v4/letter/j/b5e925/32.png) [@Jabba](https://boards.straightdope.com/u/Jabba)\ **Post date:** [June 17, 2002, 1:51pm UTC](https://boards.straightdope.com/t/help-me-find-the-complex-roots-of/114490/3 "2002-06-17T13:51:39Z")
Alternatively, solve it in two steps. If z is a root of z[sup]4[/sup] = -7+24i write z[sup]2[/sup] = w = x + iy. Then w[sup]2[/sup] = x[sup]2[/sup] - y[sup]2[/sup] + 2xyi, and so, equating real and imaginary parts x[sup]2[/sup] - y[sup]2[/sup] = -7 xy = 12 Eliminating y or solving by inspection gives (x,y) = (3,4) or (-3,-4). So the square roots of -7+24i are 3+4i and -3-4i. To finish the question you now need to repeat this procedure to find the square roots of both of these ( you should find four solutions in all). ---
**Author:** ![Phlip](https://avatars.discourse-cdn.com/v4/letter/p/ed8c4c/32.png) [@Phlip](https://boards.straightdope.com/u/Phlip)\ **Post date:** [June 17, 2002, 2:08pm UTC](https://boards.straightdope.com/t/help-me-find-the-complex-roots-of/114490/4 "2002-06-17T14:08:13Z")
> [@](#): > > \*Originally posted by zut \* > \*\* > > Switch the cartesian representation (X + YI) into a polar form (Rcis[symbol]q[/symbol]) [“cis” = shorthand for Cos[symbol]q[/symbol] + I Sin[symbol]q[/symbol]] with the transformation R = SQRT(X[sup]2[/sup] + Y[sup]2[/sup]) and tan([symbol]q[/symbol]) = (Y/X). (-7+24I) = 25cis(73.74). > > \*\* Ah HAH! THAT is what CIS means. I was looking online and saw that notation and wasnt sure what it meant. Both of your methods work and I did come up with the correct answer to the problem. Any of you ever had to deal with Krysiegs Advanced Engineering Math book? Its a hard read, Ill tell you that much. Hard to teach yourself anything out of it. Thanks for the help though. You were both right.