# How are the complex roots ordered in the Root\[\] function ?

**URL:** <https://community.wolfram.com/t/how-are-the-complex-roots-ordered-in-the-root-function/8086>\
**Category:** Questions\
**Tags:** symbolic-computations\
**Created:** [March 10, 2016, 7:32pm UTC](https://community.wolfram.com/t/how-are-the-complex-roots-ordered-in-the-root-function/8086 "2016-03-10T19:32:11Z")\
**Posts on this page:** 3\
**Page:** 1

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**Author:** ![bobager](https://avatars.discourse-cdn.com/v4/letter/b/43a26b/32.png) [@bobager](https://community.wolfram.com/u/bobager)\
**Post date:** [March 10, 2016, 7:32pm UTC](https://community.wolfram.com/t/how-are-the-complex-roots-ordered-in-the-root-function/8086/1 "2016-03-10T19:32:11Z")

</div>

I looked at the documentation on the Root[] function, but could not find the definition on how the complex roots are ordered.  
Say I have a polynomial with complex coefficients and its root are: 3+i, 2-i and -3-i. Which one of these is 1, 2, 3 as returned  
by the Root[] function ?

---

<div class="post-metadata">

**Author:** ![murray](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/murray/32/39879_2.png) [@murray](https://community.wolfram.com/u/murray)\
**Post date:** [March 12, 2016, 8:22pm UTC](https://community.wolfram.com/t/how-are-the-complex-roots-ordered-in-the-root-function/8086/2 "2016-03-12T20:22:57Z")

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Can you provide a better example? For the cubic

```
     p = (z - (3 + I)) (z - (2 - I)) (z - (-3 - I))

```

_Mathematica_ gives:

```
    Solve[p == 0, z]
{{z -> -3 - I}, {z -> 2 - I}, {z -> 3 + I}}

```

so no `Root` objects at all.

---

<div class="post-metadata">

**Author:** ![murray](https://yyz1.discourse-cdn.com/wolfram/user_avatar/community.wolfram.com/murray/32/39879_2.png) [@murray](https://community.wolfram.com/u/murray)\
**Post date:** [March 12, 2016, 8:32pm UTC](https://community.wolfram.com/t/how-are-the-complex-roots-ordered-in-the-root-function/8086/3 "2016-03-12T20:32:56Z")

</div>

Does the following sort of experiment suggest an answer?

```
    p = -5 + 2 z + 3 z^2 - 9 z^3 - z^4 + 20 z^5 - 9 z^6;
    soln = Solve[p == 0, z]
(* output is list of rules with Root objects *)
    N/@soln
{{z -> 0.953232}, {z -> 1.9405}, 
  {z -> -0.668983 - 0.331246 I}, {z -> -0.668983 + 0.331246 I}, {z -> 0.333231 - 0.654154 I}, {z -> 0.333231 + 0.654154 I}}

```