Sorting non-numerical lists [duplicate]

Question

The Sort function will by default sort from lowest to highest.

In[1]:= Sort[{3, 1, 4}]
Out[1]= {1, 3, 4}

However, if the list contains non-numerical values, it will sort it incorrectly

In[2]:= zeroes = {2 Pi - 2 ArcTan[Sqrt[2 + Sqrt[5]]], 
   2 ArcTan[Sqrt[2 + Sqrt[5]]], 2 Pi + 2 ArcTan[Sqrt[2 + Sqrt[5]]]};

In[3]:= zeroes // N
Out[3]= {4.04615, 2.23704, 8.52022}

In[4]:= Sort[zeroes] // N
Out[4]= {4.04615, 2.23704, 8.52022}

One could use the Greater function as a parameter to Sort and then reverse the input

In[5]:= Reverse@Sort[zeroes, Greater] // N
Out[5]= {2.23704, 4.04615, 8.52022}

There is no Lesser function in Mathematica, but there must be a better way to do this, without converting the data to floats.

Answer 1

The function Less is what you can use.

list = {2 Pi - 2 ArcTan[Sqrt[2 + Sqrt[5]]], 
   2 ArcTan[Sqrt[2 + Sqrt[5]]], 2 Pi + 2 ArcTan[Sqrt[2 + Sqrt[5]]]};
Sort[list, Less]

Or,

Sort[list, #1 < #2 &]

Or you could use:

SortBy[list, N@# &]

Answer 2

Sort[zeros// N]
(* {2.23704,4.04615,8.52022} *)

Mathematica is doing exactly what it should. You were sorting the expressions...

Ah, ninja'd... As ubpdqn posted, SortBy if you wish to sort (and keep form) of expressions by numeric value.