R*m . The distance between two points a, b is the Euclidean distance, which is a non-negative value in R*. The relation d « e holds for two distances d, e, if d/e is infinitesimal.

The distance operator and the comparator are related by a number of axioms, specified below. The most interesting of these is called the om-triangle inequality. If ab and bc are both much smaller than xy , then ac is much smaller than xy . This combines the ordinary triangle inequality "The distance ac is less than or equal to distance ab plus distance bc" together with the rule from order-of-magnitude algebra, "If p ≪ r and q ≪ r then p+q ≪ r,"
It will simplify the exposition below if, rather than talking about distances, we talk about orders of magnitude. These are defined as follows. We say that two distances d and e have the same order of magnitude if neither d ≪ e nor e ≪ d . In R* this is the condition that d/e is finite: neither infinitesimal nor infinite. (Raiman, 1990 uses the notation " d Co e " for this relation.) By the rules of the order-of-magnitude calculus, this is an equivalence relation. Hence we can define an order of magnitude to be an equivalence class of distances under the relation "same order of magnitude". For two points a, b, we define the function od( a, b ) to be the order of magnitude of the distance from a to b. For two orders of magnitude p, q, we define p ≪ q if, for any representatives d ∈ p and e ∈ q, d ≪ e. By the rules of the orderof-magnitude calculus, if this holds for any representatives, it holds for all representatives. The advantage of using orders-of-magnitude and the function "od", rather than distances and the distance function, is that it allows us to deal with logical equality rather than the equivalence relation "same order of magnitude".

For example, in the non-standard real line, let δ be a positive infinitesimal value. Then values such as {1, 100, 2 - 50 δ + 100 δ 2 . . }, are all of the same order of magnitude, o1. The values { δ , 1.001δ, 3δ + e- 1/δ . . } are of a different order of magnitude o2 ≪ o1. The values
{1/6, 10/6 + 65 . . .} are of a third order of magnitude o3 » o1.

Definition 1: An order-of-magnitude space (om-space) Ω consists of:
- A set of points P : - A set of orders of magnitude D ;
- A distinguished value 0 E D ;
- A function "od(a, b)" mapping two points a, b E P to an order of magnitude;
- A relation "d ≪ e" over two orders of magnitude d, e E D
st isfying the following axioms:
A.1 For any orders of magnitude d, e E D, exactly one of the following holds: d « e, e « d, d = e.

A.2 For d, e, f E D, if d « e and e « f then d « f .

(Transitivity. Together with A.1, this means that ≪ is a total ordering on orders of magnitude.)
A.3 For any d E D, not d ≪ 0.

(0 is the minimal order of magnitude.)