Algebraic Rules

Here is a list of algebraic rules for hyperreal vectors. Suppose the scalars and vectors ε, δ, are infinitesimal, c, A, are finite but not infinitesimal, and H, K are infinite.

Negatives:

• -δ is infinitesimal.
• -A is finite but not infinitesimal.
• -K is infinite.

Sums:

• δ₁ + δ₂ is infinitesimal.
• A + δ is finite but not infinitesimal.
• A₁ + A₂ is finite (possibly infinitesimal).
• K + δ and K + A are infinite.

Scalar multiples:

• εδ, εδ, and εA are infinitesimal.
• cA is finite but not infinitesimal.
• cK, HA, and HK are infinite.

Inner products:

δ₁ · δ₂ and δ · A are infinitesimal.

A₁ · A₂ is finite (possibly infinitesimal).

Each of these rules can be proved using Theorem 1.For example εA is infinitesimal because each of its components εa₁, εa₂, and εa₃ is infinitesimal.

Other combinations, such as εK and Hδ, can be either infinitesimal, finite, or infinite.