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The TCP/IP Guide  9  Networking Fundamentals       9  Backgrounder: Data Representation and the Mathematics of Computing

Backgrounder: Data Representation and the Mathematics of Computing 1 2 3 Decimal, Binary, Octal and Hexadecimal Numbers

The essence of computing is information. Computer hardware and software products are designed to allow the input, storage, transfer and expression of various types of information. A primary way in which types of information are differentiated is based on whether they are analog or digital. In highly simplified terms, analog information is continuous, while digital is discrete. The distinction between them can be seen by considering, for example, a light switch and a dimmer. A light switch allows a light to be turned on or off only—no “in-between” states. That's digital information—on or off. In contrast, a dimmer allows you to fine-tune the light output from fully on to fully off, with an infinite number of intermediate states in between: this is analog.

Modern digital computers store all of their information in digital form, because of how they work at a fundamental level. Much the way a light bulb has only an “on” or “off” value, so too do the components that store and manipulate information within computers. Millions of transistors comprise computer processors and other circuits, and are, in highly-simplified form, digital switches. Thus, all information in computers is manipulated as collections of information pieces that can be only “on” or “off”, like a switch.

Since there are only two possible states, “on” or “off”, this is called binary information (the prefix “bi” meaning “two”, of course.) There are several advantages to using binary representation for information. It is a simple way to represent many types of information, such as the state of a light bulb, or whether a file has been successfully copied, or whether a temperature should be expressed in Celsius or Fahrenheit. It is also possible to collect multiple binary values together to represent more complex information.

Perhaps most importantly, binary information is unambiguous. “On” is “on” and “off” is “off”, and this is important as it allows devices to detect clearly the value of a particular piece of information. Computers like black and white; they are not particularly good at dealing with “shades of gray”. This becomes especially important in the field of networking when transmission of data can cause signals to become polluted by noise.

The “on” or “off” condition of a binary value can be expressed in a number of different ways. In logical expressions, we may consider the value to be “true” or “false”. For representing mathematical values, the most common representation is “one” (“on”) or “zero” (“off”).

Backgrounder: Data Representation and the Mathematics of Computing 1 2 3 Decimal, Binary, Octal and Hexadecimal Numbers