Base


Table of Contents

This article defines a word and clarifies its application and usefulness in scale studies. It is only the word's use in formal mathematics that applies to scale studies.

=Definition-
The word "base" in mathematics is used to refer to a particular mathematical object that is used as a building block. The most common uses are the related concepts of the number system whose digits are used to represent numbers and the number system in which logarithms are defined. It can also be used to refer to the bottom edge or surface of a geometric figure.

A real number can be represented using any integer number as a base (sometimes also called a radix or scale). The choice of a base yields to a representation of numbers known as a number system. In base , the digits 0, 1, ..., are used (where, by convention, for bases larger than 10, the symbols A, B, C, ... are generally used as symbols representing the decimal numbers 10, 11, 12, ...).


Radix

Base is sometimes also called radix, for root. It can be useful to refer to the base of the exponential numeric system as the root or radix, as opposed to the base of the logarithm expressed.

Definition

In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9. In any numeral system, the base is written as "10". In a radix-10 ten numeral system, "10" represents the number ten; in a radix-2 two system, "10" represents the number two. This follows Mayer Goldberg who wrote in 2004, “The term base has two different meanings, both of which are used in this work, and it is important to distinguish amongst them. The base of a number system has to do with the power series representation of a number… The base of a logarithm has to do with representing a number as the power of another: If logd x = y, then dy =x. These two distinct meanings of the term base are related in this work, so to prevent any ambiguity, we use the term radix-d to speak of the base-d representation of a number.”

Etymology


An older English term for root, late O.E. rædic, from L. radicem (nom. radix) "root," from PIE base *wrad- "twig, root" (cf. Gk. rhiza, Lesbian brisda "root;" Gk. hradamnos "branch;" Goth. waurts, O.E. wyrt; Welsh gwridd, O.Ir. fren "root").

Usage

In mathematical numeral systems, the base or radix for the simplest case is the number of unique digits, including zero, that a positional numeral system uses to represent numbers. For example, for the decimal system (the most common system in use today) the radix is ten, because it uses the ten digits from 0 through 9.

Radix may be used to avoid confusing the base of the numeral place-holding system with the base of the exponent.