Consider the following expression:

thing = ((1 + I Sqrt[3]) (1/10 (9 + I Sqrt[2319]))^(1/3))/(
  2 3^(2/3)) + (1 - I Sqrt[3])/(3/10 (9 + I Sqrt[2319]))^(1/3) + 
  Sin[1]

It's actually a real constant, as you can get Mathematica to verify if you ask with enough "pretty please":

FullSimplify[Element[thing, Reals]]
(* True *)

However, it grows the common imaginary fungus when you try to get a numeric value:

thing // N
(* 0.941979 - 1.11022*10^-16 I *)

Root objects behave better as algebraic numbers than radical expressions do, and the troublesome part of thing is the algebraic part. So, one idea I tried yesterday was defining a function that turns radical expressions to Root objects:

mkAlgebraic[x_] := AlgebraicNumber[x, {0, 1}]

I was thinking of it as a TransformationFunction for Simplify, but then I found that AlgebraicNumber recognizes and gracefully excludes non-algebraic parts of expressions!

mkAlgebraic[thing]
(* Root[1 - 10 #1 + 5 #1^3 &, 2] + Sin[1] *)

This is better behaved in numerical evaluation:

mkAlgebraic[thing] // N
(* 0.941979 *)

I wonder if this is more reliable than // N // Chop.