It is split into classes of equivalence of mutual approximations, each of which contains exactly one finite number.
These numbers can be defined using Leibniz's notation through the use of clouds, shadows, and monads. The transfer principle is used to define this set by requiring all first-order statements that hold for this field's more familiar subfield to hold for it. This field does not form a metric space, but does support an order topology. It is split into classes of equivalence of mutual approximations, each of which contains exactly one finite number. The map from this field to its more familiar subfield is called the standard part function. The standard parts of these numbers are arbitrarily close to a corresponding real number; they are thus used extensively in non-standard analysis. For 10 points, name this extension of the real numbers that include both infinite and infinitesimal numbers.
These numbers can be defined using Leibniz's notation through the use of clouds, shadows, and monads.
hyperreal numbers [accept *R; prompt on R*; accept surreal numbers, of which the hyperreals as defined are a subfield; prompt on differentials until "clouds"]