Neutrices are convex subgroups of the nonstandard real number system, most of them are external sets. They may also be viewed as modules over the external set of all limited numbers, as such non-noetherian. Because of the convexity and the invariance under some translations and multiplications, the external neutrices are appropriate models of orders of magnitude of numbers. Using their strong algebraic structure a calculus of
has been developped. which includes solving of equations, and even an analysis, for the structure of external numbers has a property of completeness. This paper contains a further step, towards linear algebra and geometry. We show that in ℝ
every neutrix is the direct sum of two neutrices of ℝ. The components may be chosen orthogonal.