Suppose f(x,y) is a binary form of degree d with coefficients in a field K ⊆ ℂ. The K-rank of f  is the smallest number of d-th powers of linear forms over K of which f is a K-linear combination. We prove that for d≥5, there always exists a form of degree d with at least three different ranks over various fields. We also study the relationship between the relative rank and the algebraic properties of the underlying field. In particular, we show that the K-rank of a form f (such as x^3y^2) may depend on whether −1 is a sum of two squares in K. We provide lower bounds for the ℂ-rank (Waring rank) and for the ℝ-rank (real Waring rank) of binary forms depending on their factorization. We also give the rank of quartic and quintic binary forms based on their factorization over ℂ. We investigate the structure of binary forms with unique ℂ-minimal representation.