Hexadecimal

Verbal representation

Most European languages lack non-decimal-based words for some of the numerals eleven to fifteen. Some people read hex numbers digit by digit, like a phone number, or using the NATO phonetic alphabet, the Joint Army/Navy Phonetic Alphabet, or a similar ad hoc system. In the wake of the adoption of hex among IBM System/360 programmers, Magnuson (1968) suggested a pronunciation guide that gave short names to the letters of hex – for instance, "A" was pronounced "ann", B "bet", C "chris", etc. Another naming-system was published online by Rogers (2007) that tries to make the verbal representation distinguishable in any case, even when the actual number does not contain numbers A–F. Examples are listed in the tables below. Yet another naming system was elaborated by Babb (2015), based on a joke in Silicon Valley.

Others have proposed using the verbal Morse code conventions to express four-bit hex digits, with "dit" and "dah" representing zero and one, respectively, so that "0000" is voiced as "dit-dit-dit-dit" (....), "dah-dit-dit-dah" (-..-) voices the digit with a value of nine, and "dah-dah-dah-dah" (----) voices the hex digit for decimal 15.

Systems of counting on digits have been devised for both binary and hex. Arthur C. Clarke suggested using each finger as an on/off bit, allowing finger counting from zero to 1023₁₀ on ten fingers. Another system for counting up to FF₁₆ (255₁₀) is illustrated on the right.

Conversion

Binary conversion

Most computers manipulate binary data, but it is difficult for humans to work with a large number of digits for even a relatively small binary number. Although most humans are familiar with the base 10 system, it is much easier to map binary to hex than to decimal because each hex digit maps to a whole number of bits (4₁₀). This example converts 1111₂ to base ten. Since each position in a binary numeral can contain either a 1 or a 0, its value may be easily determined by its position from the right:

• 0001₂ = 1₁₀
• 0010₂ = 2₁₀
• 0100₂ = 4₁₀
• 1000₂ = 8₁₀

Therefore:

With little practice, mapping 1111₂ to F₁₆ in one step becomes easy. The advantage of using hex rather than decimal increases rapidly with the size of the number. When the number becomes large, conversion to decimal is very tedious. However, when mapping to hex, it is trivial to regard the binary string as 4-digit groups and map each to a single hex digit.

This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.

Compare this to the conversion to hex, where each group of four digits can be considered independently and converted directly:

The conversion from hex to binary is equally direct.

Other simple conversions

Although quaternary (base 4) is little used, it can easily be converted to and from hex or binary. Each hex digit corresponds to a pair of quaternary digits, and each quaternary digit corresponds to a pair of binary digits. In the above example 2 5 C₁₆ = 02 11 30₄.

The octal (base 8) system can also be converted with relative ease, although not quite as trivially as with bases 2 and 4. Each octal digit corresponds to three binary digits, rather than four. Therefore, we can convert between octal and hex via an intermediate conversion to binary followed by regrouping the binary digits in groups of either three or four.

Division-remainder in source base

As with all bases there is a simple algorithm for converting a representation of a number to hex by doing integer division and remainder operations in the source base. In theory, this is possible from any base, but for most humans, only decimal and for most computers, only binary (which can be converted by far more efficient methods) can be easily handled with this method.

Let d be the number to represent in hex, and the series h_ih_i−1...h₂h₁ be the hex digits representing the number.

1. i ← 1
2. h_i ← d mod 16
3. d ← (d − h_i) / 16
4. If d = 0 (return series h_i) else increment i and go to step 2

"16" may be replaced with any other base that may be desired.

The following is a JavaScript implementation of the above algorithm for converting any number to a hex in String representation. Its purpose is to illustrate the above algorithm. To work with data seriously, however, it is much more advisable to work with bitwise operators.

<span id="mwBAY"></span><span class="kd" id="mwBAc">function</span><span class="w" id="mwBAg"> </span><span class="nx" id="mwBAk">toHex</span><span class="p" id="mwBAo">(</span><span class="nx" id="mwBAs">d</span><span class="p" id="mwBAw">)</span><span class="w" id="mwBA0"> </span><span class="p" id="mwBA4">{</span> <span class="w" id="mwBA8"> </span><span class="kd" id="mwBBA">var</span><span class="w" id="mwBBE"> </span><span class="nx" id="mwBBI">r</span><span class="w" id="mwBBM"> </span><span class="o" id="mwBBQ">=</span><span class="w" id="mwBBU"> </span><span class="nx" id="mwBBY">d</span><span class="w" id="mwBBc"> </span><span class="o" id="mwBBg">%</span><span class="w" id="mwBBk"> </span><span class="mf" id="mwBBo">16</span><span class="p" id="mwBBs">;</span> <span class="w" id="mwBBw"> </span><span class="k" id="mwBB0">if</span><span class="w" id="mwBB4"> </span><span class="p" id="mwBB8">(</span><span class="nx" id="mwBCA">d</span><span class="w" id="mwBCE"> </span><span class="o" id="mwBCI">-</span><span class="w" id="mwBCM"> </span><span class="nx" id="mwBCQ">r</span><span class="w" id="mwBCU"> </span><span class="o" id="mwBCY">==</span><span class="w" id="mwBCc"> </span><span class="mf" id="mwBCg">0</span><span class="p" id="mwBCk">)</span><span class="w" id="mwBCo"> </span><span class="p" id="mwBCs">{</span> <span class="w" id="mwBCw"> </span><span class="k" id="mwBC0">return</span><span class="w" id="mwBC4"> </span><span class="nx" id="mwBC8">toChar</span><span class="p" id="mwBDA">(</span><span class="nx" id="mwBDE">r</span><span class="p" id="mwBDI">);</span> <span class="w" id="mwBDM"> </span><span class="p" id="mwBDQ">}</span> <span class="w" id="mwBDU"> </span><span class="k" id="mwBDY">return</span><span class="w" id="mwBDc"> </span><span class="nx" id="mwBDg">toHex</span><span class="p" id="mwBDk">((</span><span class="nx" id="mwBDo">d</span><span class="w" id="mwBDs"> </span><span class="o" id="mwBDw">-</span><span class="w" id="mwBD0"> </span><span class="nx" id="mwBD4">r</span><span class="p" id="mwBD8">)</span><span class="w" id="mwBEA"> </span><span class="o" id="mwBEE">/</span><span class="w" id="mwBEI"> </span><span class="mf" id="mwBEM">16</span><span class="p" id="mwBEQ">)</span><span class="w" id="mwBEU"> </span><span class="o" id="mwBEY">+</span><span class="w" id="mwBEc"> </span><span class="nx" id="mwBEg">toChar</span><span class="p" id="mwBEk">(</span><span class="nx" id="mwBEo">r</span><span class="p" id="mwBEs">);</span> <span class="p" id="mwBEw">}</span> <span class="kd" id="mwBE0">function</span><span class="w" id="mwBE4"> </span><span class="nx" id="mwBE8">toChar</span><span class="p" id="mwBFA">(</span><span class="nx" id="mwBFE">n</span><span class="p" id="mwBFI">)</span><span class="w" id="mwBFM"> </span><span class="p" id="mwBFQ">{</span> <span class="w" id="mwBFU"> </span><span class="kd" id="mwBFY">const</span><span class="w" id="mwBFc"> </span><span class="nx" id="mwBFg">alpha</span><span class="w" id="mwBFk"> </span><span class="o" id="mwBFo">=</span><span class="w" id="mwBFs"> </span><span class="s2" id="mwBFw">"0123456789ABCDEF"</span><span class="p" id="mwBF0">;</span> <span class="w" id="mwBF4"> </span><span class="k" id="mwBF8">return</span><span class="w" id="mwBGA"> </span><span class="nx" id="mwBGE">alpha</span><span class="p" id="mwBGI">.</span><span class="nx" id="mwBGM">charAt</span><span class="p" id="mwBGQ">(</span><span class="nx" id="mwBGU">n</span><span class="p" id="mwBGY">);</span> <span class="p" id="mwBGc">}</span>

Conversion through addition and multiplication

It is also possible to make the conversion by assigning each place in the source base the hex representation of its place value –before carrying out multiplication and addition to get the final representation. For example, to convert the number B3AD to decimal, one can split the hex number into its digits: B (11₁₀), 3 (3₁₀), A (10₁₀) and D (13₁₀), and then get the final result by multiplying each decimal representation by 16^p (p being the corresponding hex digit position, counting from right to left, beginning with 0). In this case, we have that:

B3AD = (11 × 16³) + (3 × 16²) + (10 × 16¹) + (13 × 16⁰)

which is 45997 in base 10.

Tools for conversion

Many computer systems provide a calculator utility capable of performing conversions between the various radices frequently including hex.

In Microsoft Windows, the Calculator, on its Programmer mode, allows conversions between hex and other common programming bases.

Elementary arithmetic

Elementary operations such as division can be carried out indirectly through conversion to an alternate numeral system, such as the commonly used decimal system or the binary system where each hex digit corresponds to four binary digits.

Alternatively, one can also perform elementary operations directly within the hex system itself –by relying on its addition/multiplication tables and its corresponding standard algorithms such as long division and the traditional subtraction algorithm.

Real numbers

Rational numbers

As with other numeral systems, the hex system can be used to represent rational numbers, although repeating expansions are common since sixteen (10₁₆) has only a single prime factor: two.

For any base, 0.1 (or "1/10") is always equivalent to one divided by the representation of that base value in its own number system. Thus, whether dividing one by two for binary or dividing one by sixteen for hex, both of these fractions are written as 0.1. Because the radix 16 is a perfect square (4²), fractions expressed in hex have an odd period much more often than decimal ones, and there are no cyclic numbers (other than trivial single digits). Recurring digits are exhibited when the denominator in lowest terms has a prime factor not found in the radix; thus, when using hex notation, all fractions with denominators that are not a power of two result in an infinite string of recurring digits (such as thirds and fifths). This makes hex (and binary) less convenient than decimal for representing rational numbers since a larger proportion lies outside its range of finite representation.

All rational numbers finitely representable in hex are also finitely representable in decimal, duodecimal and sexagesimal: that is, any hex number with a finite number of digits also has a finite number of digits when expressed in those other bases. Conversely, only a fraction of those finitely representable in the latter bases are finitely representable in hex. For example, decimal 0.1 corresponds to the infinite recurring representation 0.19 in hex. However, hex is more efficient than duodecimal and sexagesimal for representing fractions with powers of two in the denominator. For example, 0.0625₁₀ (one-sixteenth) is equivalent to 0.1₁₆, 0.09₁₂, and 0;3,45₆₀.

Irrational numbers

The table below gives the expansions of some common irrational numbers in decimal and hex.

Powers

The first 16 powers of 2 are below as hex to show relative simplicity compared to decimal representation.

Cultural history

The traditional Chinese units of measurement were base-16. For example, one jīn (斤) in the old system equals sixteen taels. The suanpan (Chinese abacus) can be used to perform hex calculations such as additions and subtractions.

As with the duodecimal system, there have been occasional attempts to promote hex as the preferred numeral system. These attempts often propose specific pronunciation and symbols for the individual numerals. Some proposals unify standard measures so that they are multiples of 16. An early such proposal was put forward by John W. Nystrom in Project of a New System of Arithmetic, Weight, Measure and Coins: Proposed to be called the Tonal System, with Sixteen to the Base, published in 1862. Nystrom among other things suggested hexadecimal time, which subdivides a day by 16, so that there are 16 "hours" (or "10 tims", pronounced tontim) in a day.

The word hexadecimal is first recorded in 1952. It is macaronic in the sense that it combines Greek ἕξ (hex) "six" with Latinate -decimal. The all-Latin alternative sexadecimal (compare the word sexagesimal for base 60) is older, and sees at least occasional use from the late 19th century. It is still in use in the 1950s in Bendix documentation. Schwartzman (1994) argues that use of sexadecimal may have been avoided because of its suggestive abbreviation to sex. Many western languages since the 1960s have adopted terms equivalent in formation to hexadecimal (e.g. French hexadécimal, Italian esadecimale, Romanian hexazecimal, Serbian хексадецимални, etc.) but others have introduced terms which substitute native words for "sixteen" (e.g. Greek δεκαεξαδικός, Icelandic sextándakerfi, Russian шестнадцатеричной etc.)

Terminology and notation did not become settled until the end of the 1960s. In 1969, Donald Knuth argued that the etymologically correct term would be senidenary, or possibly sedenary, a Latinate term intended to convey "grouped by 16" modelled on binary, ternary, quaternary, etc. According to Knuth's argument, the correct terms for decimal and octal arithmetic would be denary and octonary, respectively. Alfred B. Taylor used senidenary in his mid-1800s work on alternative number bases, although he rejected base 16 because of its "incommodious number of digits".

The now-current notation using the letters A to F establishes itself as the de facto standard beginning in 1966, in the wake of the publication of the Fortran IV manual for IBM System/360, which (unlike earlier variants of Fortran) recognizes a standard for entering hexadecimal constants. As noted above, alternative notations were used by NEC (1960) and The Pacific Data Systems 1020 (1964). The standard adopted by IBM seems to have become widely adopted by 1968, when Bruce Alan Martin in his letter to the editor of the CACM complains that

With the ridiculous choice of letters A, B, C, D, E, F as hexadecimal number symbols adding to already troublesome problems of distinguishing octal (or hex) numbers from decimal numbers (or variable names), the time is overripe for reconsideration of our number symbols. This should have been done before poor choices gelled into a de facto standard!

Martin's argument was that use of numerals 0 to 9 in nondecimal numbers "imply to us a base-ten place-value scheme": "Why not use entirely new symbols (and names) for the seven or fifteen nonzero digits needed in octal or hex. Even use of the letters A through P would be an improvement, but entirely new symbols could reflect the binary nature of the system". He also argued that "re-using alphabetic letters for numerical digits represents a gigantic backward step from the invention of distinct, non-alphabetic glyphs for numerals sixteen centuries ago" (as Brahmi numerals, and later in a Hindu–Arabic numeral system), and that the recent ASCII standards (ASA X3.4-1963 and USAS X3.4-1968) "should have preserved six code table positions following the ten decimal digits -- rather than needlessly filling these with punctuation characters" (":;<=>?") that might have been placed elsewhere among the 128 available positions.

Base16

Base16 is a binary to text encoding in the family that also contains Base32, Base58, and Base64. Data is broken into 4-bit sequences, and each value (0-15) is encoded as a character. Although any 16 characters could be used, in practice, the ASCII digits "0"–"9" and letters "A"–"F" (or "a"–"f") are used to align with the typical notation for hex numbers.

Support for Base16 encoding is ubiquitous in modern computing. It is the basis for the W3C standard for URL percent encoding, where a character is replaced with a percent sign "%" and its Base16-encoded form. Most modern programming languages directly include support for formatting and parsing Base16-encoded numbers.

Advantages of Base16 encoding include:

• Most programming languages have facilities to parse ASCII-encoded hex
• Being exactly half a byte, 4-bits is easier to process than the 5 or 6 bits of Base32 and Base64, respectively
• The notation is well-known and easily understood without needing a symbol lookup table
• Many CPU architectures have dedicated instructions that allow access to a half-byte (aka nibble), making it more efficient in hardware than Base32 and Base64

Disadvantages include:

• Space efficiency is only 50%, since each 4-bit value from the original data will be encoded as an 8-bit byte; in contrast, Base32 and Base64 encodings have a space efficiency of 63% and 75% respectively
• Complexity of accepting both upper and lower case letters