  A real number is a rule that, when provided with a rational interval, answers
Yes or No depending on if the real number ought to be considered to be in the
given interval. Since the goal is to define the real numbers, this can only
motivate the definition of which rules should be considered a real number. The
rule must satisfy five properties and any rule that does so we call an oracle.
Three of the properties ensure that we do not have multiple oracles
representing the same real number. The other two properties ensure that the
oracle does narrow down to a single real number. The most important property is
the Separating property which ensures that if we divide a Yes interval into two
parts, then one part is a Yes interval while the other is a No interval; the
exception is if the division point, which is a rational number, happens to be
the desired real number in which case both intervals are Yes intervals. We
explore various examples and algorithms in using oracles in addition to
establishing that the oracles do, in fact, form the field of real numbers. The
concept of a Family of Overlapping, Notionally Shrinking Intervals is defined
and found to be an essential tool in working with oracle arithmetic. Mediant
approximations, which are related to continued fraction representations,
naturally arise from an oracle perspective. We also compare and contrast with
other common definitions of real numbers, such as Cauchy sequences and Dedekind
cuts, in which the conclusion is that the oracle perspective is somewhat of a
master map to the other definitions. We do an explicit example to contrast
oracle arithmetic with decimal arithmetic and continued fraction arithmetic.
