Numbers

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2.3 Numbering Systems

2.3.1 Binary and BOOL

The Binary numbering system is a base 2 system where each digit may only have a zero or one state. Computers use this system internally because of the gate or switch logic nature of computing systems. The ones and zeros can be grouped in such a way as to easily be converted into other numbering systems.

Though not directly related to the Binary numbering system, logical operations on strings of binary characters are called Boolean or bitwise operations. For this reason the Binary system is sometimes referred to as a Boolean system, though not entirely correctly.

2.3.2 Decimal

The system most people are used to is the Decimal or base 10 system. This system has a radix of 10 and allows fractional numbers to be represented conveniently using a Radix point for Base 10, or Decimal point. Decimal numbers are not as easily converted to and from binary numbers because its radix of 10 is not a power of 2 like the Hexadecimal and Octal numbering systems.

2.3.3 Hexadecimal and Octal

The Hexadecimal system is a base 16 system. Its primary use is as an easily convertible representation of groups of Binary digits. It uses 16 symbols, most often 0-9 for the first ten digits and A-F for the values 10-15. Since each digit represents four binary digits it serves as a convenient shorthand for values in base two. A Hexadecimal number is sometimes referred to as a nibble.

The Octal numbering system is a base 8 system which uses the numerical values of 0-7. Like the Hexadecimal system, Octal is easily converted from Binary since it is grouped as three binary digits (its radix is an even power of 2). The Octal and Hexadecimal numbering systems are sometimes used as an I/O numbering base.

2.3.4 Floating Point and REAL

Non-integer numbers are represented as Floating Point or Real numbers. These are usually represented using 32 bits, also known as single precision. 64 bit Floating Point numbers are known as double precision.

Floating Point numbers allow the Radix Point (Decimal Point in Base 10) to be variable depending on whether a very large or very small number is being represented. The term floating point refers to the fact that the radix point (decimal point, or, more commonly in computers, binary point) can “float”; that is, it can be placed anywhere relative to the significant digits of the number. This position is indicated separately in the internal representation, and floating-point representation can thus be thought of as a computer realization of scientific notation.
The advantage of floating-point representation over fixed-point (and integer) representation is that it can support a much wider range of values. For example, a fixed-point representation that has seven decimal digits, with the decimal point assumed to be positioned after the fifth digit, can represent the numbers 12345.67, 8765.43, 123.00, and so on, whereas a floating-point representation with seven decimal digits could in addition represent 1.234567, 123456.7, 0.00001234567, 1234567000000000, and so on. The floating-point format needs slightly more storage (to encode the position of the radix point), so when stored in the same space, floating-point numbers achieve their greater range at the expense of slightly less precision.

2.3.5 Bytes and Words

Bits can be grouped for convenience into 8 bit “Bytes” and 16 bit “Words”. These numerical structures are convenient for passing packets of information that include numbers and text characters. Bytes can be further subdivided into four bit “Nibbles” which can then be used to represent Hexadecimal values. 32 bit double words are also commonly used as a grouping technique.

2.3.6 ASCII

ASCII, or the American Standard Code for Information Interchange, is a coding standard that can be used for representing words and text characters. It is implemented as a character encoding scheme primarily in computers and communications equipment. ASCII includes definitions for 128 characters. 33 are non-printing, mostly obsolete control characters that affect how text is processed. 94 are printable characters, and the space is considered an invisible graphic.
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