Hexadecimal
Introduction
The hexadecimal system, also known as base-16, is a numeral system most commonly used in computing and digital systems. It uses sixteen distinct symbols, most often the symbols "0"–"9" to represent values zero to nine, and "A"–"F" (or alternatively "a"–"f") to represent values ten to fifteen.
History
The use of the hexadecimal system in computing dates back to the earliest days of computers. In the 1950s and 1960s, computer systems such as the IBM 1620 and the IBM 1401 used hexadecimal notation for their operations. The system was adopted due to its ease of conversion to and from binary, as well as its compactness compared to binary.
Mathematical Properties
In the hexadecimal system, each digit represents four binary digits, also known as a nibble, which makes it a useful tool for representing binary code. This is because binary code, the language of computers, is cumbersome to use in human-to-human communication. Hexadecimal, on the other hand, is more compact and easier to understand.
Usage in Computing
Hexadecimal is used in computing for a variety of purposes. It is often used in programming, especially in low-level programming languages such as assembly and machine code, where it is used to represent memory addresses and raw data. It is also used in debugging, where it can be used to represent character strings and pointers.
Hexadecimal and Binary
One of the main reasons hexadecimal is used in computing is because of its relationship with binary. Each hexadecimal digit represents exactly four binary digits, which means that a byte, which is eight binary digits, can be represented by two hexadecimal digits. This makes it much easier to read and write binary data, as it reduces the number of digits that need to be written.
Hexadecimal and Decimal
While hexadecimal is closely related to binary, it is also related to the decimal system, which is the system most commonly used in everyday life. Each hexadecimal digit can represent a decimal value from 0 to 15, which means that it can represent larger numbers in a more compact form than decimal.
Converting Between Hexadecimal and Other Systems
Converting between hexadecimal and other numeral systems is a straightforward process. To convert from hexadecimal to binary, each hexadecimal digit is replaced with its four-digit binary equivalent. To convert from binary to hexadecimal, the binary number is split into groups of four digits, starting from the right, and each group is replaced with its hexadecimal equivalent.
Converting between hexadecimal and decimal is slightly more complex. To convert from hexadecimal to decimal, each hexadecimal digit is multiplied by 16 raised to the power of its position, starting from 0 on the right, and the results are added together. To convert from decimal to hexadecimal, the decimal number is divided by 16, and the remainder is the rightmost hexadecimal digit. The process is repeated with the quotient until it is 0, and the hexadecimal digits are the remainders, read from right to left.
Hexadecimal in Modern Computing
In modern computing, hexadecimal is used in a variety of ways. It is used in web design, where it is used to specify colors in the CSS. It is also used in cryptography, where it is used to represent keys and hashes. In addition, it is used in file formats, where it is used to represent data.