1. A root of a polynomial f (x ) has multiplicity n if (x-a )ⁿ is a factor of f (x ) and n is the largest possible integer for which this is true.

2. The geometric multiplicity of an eigenvalue λ of a linear transformation T is the dimension of the null space of the transformation T - λI, where I denotes the identity transformation.

3. The algebraic multiplicity of an eigenvalue λ of a linear transformation T on a finite-dimensional vector space is the multiplicity of λ as a root of the characteristic polynomial of T.

In a system having Russell-Saunders coupling, the quantity 2S+1, where S is the total spin quantum number.