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In mathematics and computing, hexadecimal (also base 16, or hex) is a positional numeral system with a radix, or base, of 16. 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.
Hexadecimal numerals are widely used by computer system designers and programmers, as they provide a more human-friendly representation of binary-coded values. Each hexadecimal digit represents four binary digits, also known as a nibble, which is half a byte. For example, a single byte can have values ranging from 0000 0000 to 1111 1111 in binary form, which can be more conveniently represented as 00 to FF in hexadecimal.
In mathematics, a subscript is typically used to specify the radix. For example the decimal value would be expressed in hexadecimal as 2AF3_{16}. In programming, a number of notations are used to support hexadecimal representation, usually involving a prefix or suffix. The prefix 10,9950x
is used in C and related languages, which would denote this value by 0x2AF3.
Hexadecimal is used in the transfer encoding Base16, in which each byte of the plaintext is broken into two 4-bit values and represented by two hexadecimal digits.
0_{hex} | = | 0_{dec} | = | 0_{oct} | 0 | 0 | 0 | 0 | |
1_{hex} | = | 1_{dec} | = | 1_{oct} | 0 | 0 | 0 | 1 | |
2_{hex} | = | 2_{dec} | = | 2_{oct} | 0 | 0 | 1 | 0 | |
3_{hex} | = | 3_{dec} | = | 3_{oct} | 0 | 0 | 1 | 1 | |
4_{hex} | = | 4_{dec} | = | 4_{oct} | 0 | 1 | 0 | 0 | |
5_{hex} | = | 5_{dec} | = | 5_{oct} | 0 | 1 | 0 | 1 | |
6_{hex} | = | 6_{dec} | = | 6_{oct} | 0 | 1 | 1 | 0 | |
7_{hex} | = | 7_{dec} | = | 7_{oct} | 0 | 1 | 1 | 1 | |
8_{hex} | = | 8_{dec} | = | 10_{oct} | 1 | 0 | 0 | 0 | |
9_{hex} | = | 9_{dec} | = | 11_{oct} | 1 | 0 | 0 | 1 | |
A_{hex} | = | 10_{dec} | = | 12_{oct} | 1 | 0 | 1 | 0 | |
B_{hex} | = | 11_{dec} | = | 13_{oct} | 1 | 0 | 1 | 1 | |
C_{hex} | = | 12_{dec} | = | 14_{oct} | 1 | 1 | 0 | 0 | |
D_{hex} | = | 13_{dec} | = | 15_{oct} | 1 | 1 | 0 | 1 | |
E_{hex} | = | 14_{dec} | = | 16_{oct} | 1 | 1 | 1 | 0 | |
F_{hex} | = | 15_{dec} | = | 17_{oct} | 1 | 1 | 1 | 1 |
In contexts where the base is not clear, hexadecimal numbers can be ambiguous and confused with numbers expressed in other bases. There are several conventions for expressing values unambiguously. A numerical subscript (itself written in decimal) can give the base explicitly: 159_{10} is decimal 159; 159_{16} is hexadecimal 159, which is equal to 345_{10}. Some authors prefer a text subscript, such as 159_{decimal} and 159_{hex}, or 159_{d} and 159_{h}.
In linear text systems, such as those used in most computer programming environments, a variety of methods have arisen:
%
: http://www.example.com/name%20with%20spaces
where %20
is the space (blank) character, ASCII code point 20 in hex, 32 in decimal.ode;
, where the x denotes that code is a hex code point (of 1- to 6-digits) assigned to the character in the Unicode standard. Thus ’
represents the right single quotation mark (’), Unicode code point number 2019 in hex, 8217 (thus ’
in decimal).^{[1]}U+
followed by the hex value, e.g. U+20AC
is the Euro sign (€).#
: white, for example, is represented #FFFFFF
.^{[2]} CSS allows 3-hexdigit abbreviations with one hexdigit per component: #FA3 abbreviates #FFAA33 (a golden orange: ).0x
for numeric constants represented in hex: 0x5A3
. Character and string constants may express character codes in hexadecimal with the prefix \x
followed by two hex digits: '\x1B'
represents the Esc control character; "\x1B[0m\x1B[25;1H"
is a string containing 11 characters (plus a trailing NUL to mark the end of the string) with two embedded Esc characters.^{[3]} To output an integer as hexadecimal with the printf function family, the format conversion code %X
or %x
is used.=
, as in Espa=F1a
to send "España" (Spain). (Hexadecimal F1, equal to decimal 241, is the code number for the lower case n with tilde in the ISO/IEC 8859-1 character set.)FFh
or 05A3H
. Some implementations require a leading zero when the first hexadecimal digit character is not a decimal digit, so one would write 0FFh
instead of FFh
$
as a prefix: $5A3
.H'ABCD'
(for ABCD_{16}). Similarly, Fortran 95 uses Z'ABCD'.16#5A3#
. For bit vector constants VHDL uses the notation x"5A3"
.^{[5]}8'hFF
, where 8 is the number of bits in the value and FF is the hexadecimal constant.16r
: 16r5A3
16#
: 16#5A3
. For PostScript, binary data (such as image pixels) can be expressed as unprefixed consecutive hexadecimal pairs: AA213FD51B3801043FBC
...#x
and #16r
. Setting the variables *read-base*^{[6]} and *print-base*^{[7]} to 16 can also be used to switch the reader and printer of a Common Lisp system to Hexadecimal number representation for reading and printing numbers. Thus Hexadecimal numbers can be represented without the #x or #16r prefix code, when the input or output base has been changed to 16.&H
: &H5A3
&
for hex.^{[9]}0h
prefix: 0h5A3
16r
to denote hexadecimal numbers: 16r5a3
. Binary, quaternary (base-4) and octal numbers can be specified similarly.X'5A3'
, and is used in Assembler, PL/I, COBOL, JCL, scripts, commands and other places. This format was common on other (and now obsolete) IBM systems as well. Occasionally quotation marks were used instead of apostrophes.:
). This, for example, is a valid IPv6 address: 2001:0db8:85a3:0000:0000:8a2e:0370:7334; this can be abbreviated as 2001:db8:85a3::8a2e:370:7334. By contrast, IPv4 addresses are usually written in decimal.{3F2504E0-4F89-41D3-9A0C-0305E82C3301}
.There is no universal convention to use lowercase or uppercase for the letter digits, and each is prevalent or preferred in particular environments by community standards or convention.
The use of the letters A through F to represent the digits above 9 was not universal in the early history of computers.
There are no traditional numerals to represent the quantities from ten to fifteen – letters are used as a substitute – and most European languages lack non-decimal names for the numerals above ten. Even though English has names for several non-decimal powers (pair for the first binary power, score for the first vigesimal power, dozen, gross and great gross for the first three duodecimal powers), no English name describes the hexadecimal powers (decimal 16, 256, 4096, 65536, ... ). Some people read hexadecimal 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.
Systems of counting on digits have been devised for both binary and hexadecimal. Arthur C. Clarke suggested using each finger as an on/off bit, allowing finger counting from zero to 1023_{10} on ten fingers.^{[citation needed]} Another system for counting up to FF_{16} (255_{10}) is illustrated on the right.
The hexadecimal system can express negative numbers the same way as in decimal: −2A to represent −42_{10} and so on.
Hexadecimal can also be used to express the exact bit patterns used in the processor, so a sequence of hexadecimal digits may represent a signed or even a floating point value. This way, the negative number −42_{10} can be written as FFFF FFD6 in a 32-bit CPU register (in two's-complement), as C228 0000 in a 32-bit FPU register or C045 0000 0000 0000 in a 64-bit FPU register (in the IEEE floating-point standard).
Just as decimal numbers can be represented in exponential notation, so too can hexadecimal numbers. By convention, the letter P (or p, for "power") represents times two raised to the power of, whereas E (or e) serves a similar purpose in decimal as part of the E notation. The number after the P is decimal and represents the binary exponent.
Usually the number is normalised so that the leading hexadecimal digit is 1 (unless the value is exactly 0).
Example: 1.3DEp42 represents 1.3DE_{16} × 2^{42}.
Hexadecimal exponential notation is required by the IEEE 754-2008 binary floating-point standard. This notation can be used for floating-point literals in the C99 edition of the C programming language.^{[18]} Using the %a or %A conversion specifiers, this notation can be produced by implementations of the printf family of functions following the C99 specification^{[19]} and Single Unix Specification (IEEE Std 1003.1) POSIX standard.^{[20]}
Most computers manipulate binary data, but it is difficult for humans to work with the 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 hexadecimal than to decimal because each hexadecimal digit maps to a whole number of bits (4_{10}). This example converts 1111_{2} 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:
Therefore:
1111_{2} | = 8_{10} + 4_{10} + 2_{10} + 1_{10} |
= 15_{10} |
With little practice, mapping 1111_{2} to F_{16} in one step becomes easy: see table in Written representation. The advantage of using hexadecimal 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 hexadecimal, it is trivial to regard the binary string as 4-digit groups and map each to a single hexadecimal digit.
This example shows the conversion of a binary number to decimal, mapping each digit to the decimal value, and adding the results.
(01011110101101010010)_{2} | = 262144_{10} + 65536_{10} + 32768_{10} + 16384_{10} + 8192_{10} + 2048_{10} + 512_{10} + 256_{10} + 64_{10} + 16_{10} + 2_{10} |
= 387922_{10} |
Compare this to the conversion to hexadecimal, where each group of four digits can be considered independently, and converted directly:
(01011110101101010010)_{2} | = | 0101_{ } | 1110_{ } | 1011_{ } | 0101_{ } | 0010_{2} |
= | 5 | E | B | 5 | 2_{16} | |
= | 5EB52_{16} |
The conversion from hexadecimal to binary is equally direct.
Although quaternary (base 4) is little used, it can easily be converted to and from hexadecimal or binary. Each hexadecimal digit corresponds to a pair of quaternary digits and each quaternary digit corresponds to a pair of binary digits. In the above example 5 E B 5 2_{16} = 11 32 23 11 02_{4}.
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 hexadecimal via an intermediate conversion to binary followed by regrouping the binary digits in groups of either three or four.
As with all bases there is a simple algorithm for converting a representation of a number to hexadecimal 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 hexadecimal, and the series h_{i}h_{i−1}...h_{2}h_{1} be the hexadecimal digits representing the number.
"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 hexadecimal 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.
function toHex(d) {
var r = d % 16;
if (d - r == 0) {
return toChar(r);
}
return toHex( (d - r)/16 ) + toChar(r);
}
function toChar(n) {
const alpha = "0123456789ABCDEF";
return alpha.charAt(n);
}
It is also possible to make the conversion by assigning each place in the source base the hexadecimal representation of its place value and then performing multiplication and addition to get the final representation. That is, to convert the number B3AD to decimal one can split the hexadecimal number into its digits: B (11_{10}), 3 (3_{10}), A (10_{10}) and D (13_{10}), and then get the final result by multiplying each decimal representation by 16^{p}, where p is the corresponding hex digit position, counting from right to left, beginning with 0. In this case we have B3AD = (11 × 16^{3}) + (3 × 16^{2}) + (10 × 16^{1}) + (13 × 16^{0}), which is 45997 base 10.
Most modern computer systems with graphical user interfaces provide a built-in calculator utility, capable of performing conversions between various radices, in general including hexadecimal.
In Microsoft Windows, the Calculator utility can be set to Scientific mode (called Programmer mode in some versions), which allows conversions between radix 16 (hexadecimal), 10 (decimal), 8 (octal) and 2 (binary), the bases most commonly used by programmers. In Scientific Mode, the on-screen numeric keypad includes the hexadecimal digits A through F, which are active when "Hex" is selected. In hex mode, however, the Windows Calculator supports only integers.
As with other numeral systems, the hexadecimal system can be used to represent rational numbers, although repeating expansions are common since sixteen (10_{hex}) has only a single prime factor (two):
1/2 | = | 0.8 |
1/3 | = | 0.5 |
1/4 | = | 0.4 |
1/5 | = | 0.3 |
1/6 | = | 0.2A |
1/7 | = | 0.249 |
1/8 | = | 0.2 |
1/9 | = | 0.1C7 |
1/A | = | 0.19 |
1/B | = | 0.1745D |
1/C | = | 0.15 |
1/D | = | 0.13B |
1/E | = | 0.1249 |
1/F | = | 0.1 |
1/10 | = | 0.1 |
1/11 | = | 0.0F |
where an overline denotes a recurring pattern.
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 hexadecimal, both of these fractions are written as 0.1
. Because the radix 16 is a perfect square (4^{2}), fractions expressed in hexadecimal 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 hexadecimal 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 hexadecimal (and binary) less convenient than decimal for representing rational numbers since a larger proportion lie outside its range of finite representation.
All rational numbers finitely representable in hexadecimal are also finitely representable in decimal, duodecimal and sexagesimal: that is, any hexadecimal number with a finite number of digits 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 hexadecimal. For example, decimal 0.1 corresponds to the infinite recurring representation 0.199999999999... in hexadecimal. However, hexadecimal is more efficient than bases 12 and 60 for representing fractions with powers of two in the denominator (e.g., decimal one sixteenth is 0.1 in hexadecimal, 0.09 in duodecimal, 0;3,45 in sexagesimal and 0.0625 in decimal).
n | Decimal Prime factors of base, b = 10: 2, 5; b − 1 = 9: 3; b + 1 = 11: 11 |
Hexadecimal Prime factors of base, b = 16_{10} = 10: 2; b − 1 = 15_{10} = F: 3, 5; b + 1 = 17_{10} = 11: 11 | ||||
---|---|---|---|---|---|---|
Fraction | Prime factors | Positional representation | Positional representation | Prime factors | Fraction(1/n) | |
2 | 1/2 | 2 | 0.5 | 0.8 | 2 | 1/2 |
3 | 1/3 | 3 | 0.3333... = 0.3 | 0.5555... = 0.5 | 3 | 1/3 |
4 | 1/4 | 2 | 0.25 | 0.4 | 2 | 1/4 |
5 | 1/5 | 5 | 0.2 | 0.3 | 5 | 1/5 |
6 | 1/6 | 2, 3 | 0.16 | 0.2A | 2, 3 | 1/6 |
7 | 1/7 | 7 | 0.142857 | 0.249 | 7 | 1/7 |
8 | 1/8 | 2 | 0.125 | 0.2 | 2 | 1/8 |
9 | 1/9 | 3 | 0.1 | 0.1C7 | 3 | 1/9 |
10 | 1/10 | 2, 5 | 0.1 | 0.19 | 2, 5 | 1/A |
11 | 1/11 | 11 | 0.09 | 0.1745D | B | 1/B |
12 | 1/12 | 2, 3 | 0.083 | 0.15 | 2, 3 | 1/C |
13 | 1/13 | 13 | 0.076923 | 0.13B | D | 1/D |
14 | 1/14 | 2, 7 | 0.0714285 | 0.1249 | 2, 7 | 1/E |
15 | 1/15 | 3, 5 | 0.06 | 0.1 | 3, 5 | 1/F |
16 | 1/16 | 2 | 0.0625 | 0.1 | 2 | 1/10 |
17 | 1/17 | 17 | 0.0588235294117647 | 0.0F | 11 | 1/11 |
18 | 1/18 | 2, 3 | 0.05 | 0.0E38 | 2, 3 | 1/12 |
19 | 1/19 | 19 | 0.052631578947368421 | 0.0D79435E5 | 13 | 1/13 |
20 | 1/20 | 2, 5 | 0.05 | 0.0C | 2, 5 | 1/14 |
21 | 1/21 | 3, 7 | 0.047619 | 0.0C3 | 3, 7 | 1/15 |
22 | 1/22 | 2, 11 | 0.045 | 0.0BA2E8 | 2, B | 1/16 |
23 | 1/23 | 23 | 0.0434782608695652173913 | 0.0B21642C859 | 17 | 1/17 |
24 | 1/24 | 2, 3 | 0.0416 | 0.0A | 2, 3 | 1/18 |
25 | 1/25 | 5 | 0.04 | 0.0A3D7 | 5 | 1/19 |
26 | 1/26 | 2, 13 | 0.0384615 | 0.09D8 | 2, D | 1/1A |
27 | 1/27 | 3 | 0.037 | 0.097B425ED | 3 | 1/1B |
28 | 1/28 | 2, 7 | 0.03571428 | 0.0924 | 2, 7 | 1/1C |
29 | 1/29 | 29 | 0.0344827586206896551724137931 | 0.08D3DCB | 1D | 1/1D |
30 | 1/30 | 2, 3, 5 | 0.03 | 0.08 | 2, 3, 5 | 1/1E |
31 | 1/31 | 31 | 0.032258064516129 | 0.08421 | 1F | 1/1F |
32 | 1/32 | 2 | 0.03125 | 0.08 | 2 | 1/20 |
33 | 1/33 | 3, 11 | 0.03 | 0.07C1F | 3, B | 1/21 |
34 | 1/34 | 2, 17 | 0.02941176470588235 | 0.078 | 2, 11 | 1/22 |
35 | 1/35 | 5, 7 | 0.0285714 | 0.075 | 5, 7 | 1/23 |
36 | 1/36 | 2, 3 | 0.027 | 0.071C | 2, 3 | 1/24 |
The table below gives the expansions of some common irrational numbers in decimal and hexadecimal.
Number | Positional representation | |
---|---|---|
Decimal | Hexadecimal | |
√2 (the length of the diagonal of a unit square) | 213562373095048... 1.414 | 1.6A09E667F3BCD... |
√3 (the length of the diagonal of a unit cube) | 050807568877293... 1.732 | 1.BB67AE8584CAA... |
√5 (the length of the diagonal of a 1×2 rectangle) | 067977499789696... 2.236 | 2.3C6EF372FE95... |
φ (phi, the golden ratio = (1+√5)/2) | 033988749894848... 1.618 | 1.9E3779B97F4A... |
π (pi, the ratio of circumference to diameter of a circle) | 592653589793238462643 3.141 279502884197169399375105... 383 |
3.243F6A8885A308D313198A2E0 3707344A4093822299F31D008... |
e (the base of the natural logarithm) | 281828459045235... 2.718 | 2.B7E151628AED2A6B... |
τ (the Thue–Morse constant) | 454033640107597... 0.412 | 0.6996 9669 9669 6996... |
γ (the limiting difference between the harmonic series and the natural logarithm) |
215664901532860... 0.577 | 0.93C467E37DB0C7A4D1B... |
Powers of two have very simple expansions in hexadecimal. The first sixteen powers of two are shown below.
2^{x} | Value | Value (Decimal) |
---|---|---|
2^{0} | 1 | 1 |
2^{1} | 2 | 2 |
2^{2} | 4 | 4 |
2^{3} | 8 | 8 |
2^{4} | 10_{hex} | 16_{dec} |
2^{5} | 20_{hex} | 32_{dec} |
2^{6} | 40_{hex} | 64_{dec} |
2^{7} | 80_{hex} | 128_{dec} |
2^{8} | 100_{hex} | 256_{dec} |
2^{9} | 200_{hex} | 512_{dec} |
2^{A} (2^{10dec}) | 400_{hex} | 1024_{dec} |
2^{B} (2^{11dec}) | 800_{hex} | 2048_{dec} |
2^{C} (2^{12dec}) | 1000_{hex} | 4096_{dec} |
2^{D} (2^{13dec}) | 2000_{hex} | 8192_{dec} |
2^{E} (2^{14dec}) | 4000_{hex} | 16,384_{dec} |
2^{F} (2^{15dec}) | 8000_{hex} | 32,768_{dec} |
2^{10} (2^{16dec}) | 10000_{hex} | 65,536_{dec} |
The word hexadecimal is composed of hexa-, derived from the Greek ἕξ (hex) for six, and -decimal, derived from the Latin for tenth. Webster's Third New International online derives hexadecimal as an alteration of the all-Latin sexadecimal (which appears in the earlier Bendix documentation). The earliest date attested for hexadecimal in Merriam-Webster Collegiate online is 1954, placing it safely in the category of international scientific vocabulary (ISV). It is common in ISV to mix Greek and Latin combining forms freely. The word sexagesimal (for base 60) retains the Latin prefix. Donald Knuth has pointed out that the etymologically correct term is senidenary (or possibly, sedenary), from the Latin term for grouped by 16. (The terms binary, ternary and quaternary are from the same Latin construction, and the etymologically correct terms for decimal and octal arithmetic are denary and octonary, respectively.)^{[21]} 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".^{[22]}^{[23]} Schwartzman notes that the expected form from usual Latin phrasing would be sexadecimal, but computer hackers would be tempted to shorten that word to sex.^{[24]} The etymologically proper Greek term would be hexadecadic / ἑξαδεκαδικός / hexadekadikós (although in Modern Greek, decahexadic / δεκαεξαδικός / dekaexadikos is more commonly used).
The traditional Chinese units of weight were base-16. For example, one jīn (斤) in the old system equals sixteen taels. The suanpan (Chinese abacus) can be used to perform hexadecimal calculations.
As with the duodecimal system, there have been occasional attempts to promote hexadecimal as the preferred numeral system. These attempts often propose specific pronunciation and symbols for the individual numerals.^{[25]} Some proposals unify standard measures so that they are multiples of 16.^{[26]}^{[27]}^{[28]}
An example of unified standard measures is hexadecimal time, which subdivides a day by 16 so that there are 16 "hexhours" in a day.^{[28]}
Base16 (as a proper name without a space) can also refer to a binary to text encoding belonging to the same family as Base32, Base58, and Base64.
In this case, data is broken into 4-bit sequences, and each value (between 0 and 15 inclusively) is encoded using 16 symbols from the ASCII character set. Although any 16 symbols from the ASCII character set can be used, in practice the ASCII digits '0'-'9' and the letters 'A'-'F' (or the lowercase 'a'-'f') are always chosen in order to align with standard written notation for hexadecimal numbers.
There are several advantages of Base16 encoding:
The main disadvantages of Base16 encoding are:
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.
"\x1B[0m\x1B[25;1H"
specifies the character sequence Esc [ 0 m Esc [ 2 5 ; 1 H Nul. These are the escape sequences used on an ANSI terminal that reset the character set and color, and then move the cursor to line 25.
&
to prefix octal values. (Microsoft BASIC primarily uses &O
to prefix octal, and it uses &H
to prefix hexadecimal, but the ampersand alone yields a default interpretation as an octal prefix.
This base is used because a group of four bits can represent any one of sixteen different numbers (zero to fifteen). By assigning a symbol to each of these combinations we arrive at a notation called sexadecimal (usually hex in conversation because nobody wants to abbreviate sex). The symbols in the sexadecimal language are the ten decimal digits and, on the G-15 typewriter, the letters u, v, w, x, y and z. These are arbitrary markings; other computers may use different alphabet characters for these last six digits.
[…] They can be used interchangeable in present or future designs to offer designers a choice between two indicator fonts. The '46A, '47A, 'LS47, and 'LS48 compose the 6 and the 9 without tails and the '246, '247, 'LS247, and 'LS248 compose the 6 and the 0 with tails. Composition of all other characters, including display patterns for BCD inputs above nine, is identical. […] Display patterns for BCD input counts above 9 are unique symbols to authenticate input conditions. […]