The following terminology and notation is used here.

A line above a number is used to denote its negation. For example the digit minus one (-1) is usually written .

An arrow above a digit is used to denote an infinite stream of that digit. For example denotes an infinite stream of ones.

A double colon (::) is used to separate elements of a list or
stream. The last element printed represents the tail of the list or
stream unless stated otherwise. Hence is a stream whose
first two elements are the digits one and minus one, and the remainder
of which is a stream which would be referred to as *x*.

The term *numeral* is used to refer to the symbol or group of
symbols representing a *number*. A number is a mathematical
object, whereas a numeral is a syntactic object. When working with
numerals and numbers in equations where the potential for confusion
exists, we use semantic brackets to denote the number or
interval represented by the numeral *x*.

If *x* is one of the representations of reals defined in this chapter,
is simply the real number it represents. When working with
exact reals as infinite streams, however, we often know only a finite
portion of the stream. We define the *range* of a finite portion
of a stream representing a real number to be the set of all possible
reals that may be represented by a stream starting with the
elements known already, but with all possible suffixes.

The symbol is used to represent the average operation, . Average is a mathematical operation, but the symbol is also used to denote the result of applying an average algorithm to two numerals. In the latter case, the actual algorithm required will depend on whether the numerals are dyadic digits (see section A.2), a digit and a stream (section 3.6.2), or two streams (sections 4.2.1 and 4.3.1).

Multiplication will be represented using the symbol , or as a single dot (). In general we will use when both operands are numerals in stream representation, and the dot notation for the multiplication of two numbers, or when the operands are numerals and one or both are digits.