digital number system

1 Digital System

Most written number systems are built upon the concept of a digital system (or positional system) for their functioning and conveying of quantitative meaning. In these systems, meaning is derived from two things: symbols and positions. A particular symbol in a specific place is called a digit.

The representation of a value in a digital system follows the schema:

$$\mathrm{\dots }{s}_2{s}_1{s}_0.s_{-1}{s}_{-2}{s}_{-3}\mathrm{\dots }$$

Where each $s_i$ is some symbol that has a quantitative value (a digit). Places to the left of the point ($.$) are worth whole units, and places to the right are worth fractional units. It is the base that tells us how much of a fraction or how many whole units. Once a base $b$ is chosen, the value of a number $s_2{s}_1{s}_0.s_{-1}{s}_{-2}{s}_{-3}$ would be calculated like:

$$s_2{s}_1{s}_0.s_{-1}{s}_{-2}{s}_{-3}=s_2\cdot b^2+s_1\cdot b^1+s_0\cdot b^0+s_{-1}\cdot b^{-1}+s_{-2}\cdot b^{-2}+s_{-3}\cdot b^{-3}$$

In our now-standard, Arabic-derived decimal system, the base $b$ is equal to 10. Other very common (and useful) systems are binary, hexadecimal, and octal, having $b=2$, $b=16$, and $b=8$ respectively .

Each $s_i$ is a member of an alphabet of symbols which must have $b$ members. Intuitively this makes sense: when we try to represent the number which follows “9” in the decimal system, we know it must be “10”, since there is no symbol after “9.” Hence, position as well as symbol conveys the meaning, and base tells us how much a unit in each position is worth.

2 Remark

Curiously, though one would think that the choice of base leads to merely a different way of rendering the same information, there are instances where things are variously provable or proven in some bases, but not others. For instance, there exists a non-recursive formula for the $n$th binary digit of $\pi$, but not for decimal– one still must calculate all of the $n-1$ preceding decimal digits of $\pi$ to get the $n$th (see http://www.nersc.gov/ dhbailey/dhbpapers/digits.pdfthis paper).

Title	digital number system
Canonical name	DigitalNumberSystem
Date of creation	2013-03-22 12:57:12
Last modified on	2013-03-22 12:57:12
Owner	akrowne (2)
Last modified by	akrowne (2)
Numerical id	9
Author	akrowne (2)
Entry type	Definition
Classification	msc 11-01
Related topic	DecimalExpansion
Defines	base
Defines	numerical base
Defines	digital base
Defines	positional systems
Defines	positional number systems
Defines	place systems
Defines	digit