Hexadecimal system

Hexadecimal digits,
binary and decimal:

Hex.	Dual system				Dec


0	0	0	0	0	00
1	0	0	0	1	01
2	0	0	1	0	02
3	0	0	1	1	03



4th	0	1	0	0	04
5	0	1	0	1	05
6th	0	1	1	0	06
7th	0	1	1	1	07



8th	1	0	0	0	08
9	1	0	0	1	09
A.	1	0	1	0	10
B.	1	0	1	1	11



C.	1	1	0	0	12
D.	1	1	0	1	13
E.	1	1	1	0	14th
F.	1	1	1	1	15th

In the hexadecimal system , numbers are represented in a place value system based on 16. "Hexadecimal" (from Greek hexa "six" and Latin decem "ten") is a Latin-Greek mixed word; Another correct but seldom used designation is hexadecimal (from the Latin sedecim "sixteen"). Another alternative name is hexadecadic (Greek). On the other hand, the expression hexa ges imal, which is synonymous with sexa ges imal and denotes the number system based on base 60, is incorrect .

In the data processing the hexadecimal system is used very often, because it is ultimately up to a more comfortable administration of the binary system is. The data words exist in the computer science mostly of octets that can be held as an eight-digit binary numbers represented as two-digit hexadecimal numbers. In contrast to the decimal system, the hexadecimal system with its base as a fourth power of two (16 = 2 ⁴ ) is suitable for simpler notation of binary numbers, as a fixed number of characters is always required to reproduce the data word. Nibbles can be represented exactly with one hexadecimal digit and bytes with two hexadecimal digits.

In the 1960s and 1970s, the octal system was also often used in computer science with its base as a third power of two (8 = 2 ³ ), since it uses the usual digits from 0 to 7. However, it is rarely used today, for example to represent characters in the C programming language. There are also other number systems with different base values.

We are used to calculating in the decimal system . This means that our Indo-Arabic number system uses ten symbols for the notation of the digits ( 0 to 9 ). The hexadecimal system, on the other hand, contains sixteen digits. Since the mid-1950s, the letters A to F or a to f have been used as numerals to represent the six additional digits . This goes back to the practice of the IBM computer scientists at the time .

Representation of hexadecimal numbers

In order to be able to differentiate between hexadecimal and decimal numbers, there are several notations. Usually hexadecimal numbers are given an index or prefix .

Common spellings are: 72 ₁₆ , 72 _hex , 72h, 72H, 72 _H , 0x72, $ 72, "72 and X'72 ', whereby the prefix 0x and the suffix h are particularly used in programming and technical informatics h to the hex number is also known as the Intel convention. The notation with the dollar prefix is common in the assembly languages of certain processor families , especially Motorola , for example the Motorola 68xx and 68xxx , but also the MOS 65xx ; the notation X '72' is common in the world of IBM mainframes , as in REXX .

Separation points for clarity can be set at all four places in hexadecimal numbers, thus separating groups of sixteen bits each . The meaning of the 1,0000 ₁₆ = 65,536 ₁₀ among the hexadecimal numbers corresponds to that of the 1,000 ₁₀ among the decimal numbers.

For comparison, a full sixty-four-bit bus with and without breakpoints: FFFF.FFFF.FFFF.FFFF and FFFFFFFFFFFFFFFF

Decimal numbers are indexed where they are not the normal case to be expected: 114 ₁₀

Counting in the hexadecimal system

It is counted as follows:

0	1	2	3	4th	5	6th	7th	8th	9	A.	B.	C.	D.	E.	F.
10	11	12	13	14th	15th	16	17th	18th	19th	1A	1B	1C	1D	1E	1F
20th	21st	22nd	23	24	25th	26th	27	28	29	2A	2 B	2C	2D	2E	2F
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
F0	F1	F2	F3	F4	F5	F6	F7	F8	F9	FA	FB	FC	FD	FE	FF
100	101	102	103	104	105	106	107	108	109	10A	10B	10C	10D	10E	10F
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
FF0	FF1	FF2	FF3	FF4	FF5	FF6	FF7	FF8	FF9	FFA	FFB	FFC	FFD	FFE	FFF
1000	1001	1002	1003	1004	1005	1006	1007	1008	1009	100A	100B	100C	100D	100E	100F
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...
FFF0	FFF1	FFF2	FFF3	FFF4	FFF5	FFF6	FFF7	FFF8	FFF9	FFFA	FFFB	FFFC	FFFD	FFFE	FFFF
10,000	10001	10002	10003	10004	10005	10006	10007	10008	10009	1000A	1000B	1000C	1000D	1000E	1000F
...	...	...	...	...	...	...	...	...	...	...	...	...	...	...	...

Pronunciation of hexadecimal numbers

No separate names are used for the hexadecimal digits and numbers. Hexadecimal numbers are therefore usually read digit by digit.

Examples:

10 read: "one-zero" (rarely: "ten"),
1E say: "one-E",
F112 read: "F-one-one-two".

Hexadecimal multiplication table (multiplication table)

Example: 2 • 5 = A

Go vertically down from the column with the value 2 to the intersection of the line with the value 5. Result: A

*	1	2	3	4th	5	6th	7th	8th	9	A.	B.	C.	D.	E.	F.	10
1	1	2	3	4th	5	6th	7th	8th	9	A.	B.	C.	D.	E.	F.	10
2	2	4th	6th	8th	A.	C.	E.	10	12	14th	16	18th	1A	1C	1E	20th
3	3	6th	9	C.	F.	12	15th	18th	1B	1E	21st	24	27	2A	2D	30th
4th	4th	8th	C.	10	14th	18th	1C	20th	24	28	2C	30th	34	38	3C	40
5	5	A.	F.	14th	19th	1E	23	28	2D	32	37	3C	41	46	4B	50
6th	6th	C.	12	18th	1E	24	2A	30th	36	3C	42	48	4E	54	5A	60
7th	7th	E.	15th	1C	23	2A	31	38	3F	46	4D	54	5B	62	69	70
8th	8th	10	18th	20th	28	30th	38	40	48	50	58	60	68	70	78	80
9	9	12	1B	24	2D	36	3F	48	51	5A	63	6C	75	7E	87	90
A.	A.	14th	1E	28	32	3C	46	50	5A	64	6E	78	82	8C	96	A0
B.	B.	16	21st	2C	37	42	4D	58	63	6E	79	84	8F	9A	A5	B0
C.	C.	18th	24	30th	3C	48	54	60	6C	78	84	90	9C	A8	B4	C0
D.	D.	1A	27	34	41	4E	5B	68	75	82	8F	9C	A9	B6	C3	D0
E.	E.	1C	2A	38	46	54	62	70	7E	8C	9A	A8	B6	C4	D2	E0
F.	F.	1E	2D	3C	4B	5A	69	78	87	96	A5	B4	C3	D2	E1	F0
10	10	20th	30th	40	50	60	70	80	90	A0	B0	C0	D0	E0	F0	100

Hexadecimal fractions

Since the hexadecimal system is a place value system , the places after the comma (which is also sometimes written as a comma, sometimes as a point) have the place value , with the decimal base being 16 and the position of the respective decimal place. The first decimal place (n = 1) has the value , the second decimal place (n = 2) has the value , the third decimal place (n = 3) has the value and so on. ${\ displaystyle 1 \ over B ^ {n}}$ ${\ displaystyle B}$ ${\ displaystyle n}$ ${\ displaystyle {1 \ over 16 ^ {1}} = {1 \ over 16}}$ ${\ displaystyle {1 \ over 16 ^ {2}} = {1 \ over 256}}$ ${\ displaystyle {1 \ over 16 ^ {3}} = {1 \ over 4096}}$

Since the number 16 only has the single prime factor 2, all abbreviated fractions whose denominator is not a power of two have a periodic point representation in the hexadecimal system:

${\ displaystyle 1 \ over 1}$ = 1	${\ displaystyle 1 \ over 5}$ = 0, 3 ₁₆	${\ displaystyle 1 \ over 9}$ = 0.1C7 ₁₆	${\ displaystyle 1 \ over \ mathrm {D} _ {16}}$ = 0.13B ₁₆
${\ displaystyle 1 \ over 2}$ = 0.8 ₁₆	${\ displaystyle 1 \ over 6}$ = 0.2 A ₁₆	${\ displaystyle 1 \ over \ mathrm {A} _ {16}}$ = 0.1 9 ₁₆	${\ displaystyle 1 \ over \ mathrm {E} _ {16}}$ = 0.1 249 ₁₆
${\ displaystyle 1 \ over 3}$ = 0 5 ₁₆	${\ displaystyle 1 \ over 7}$ = 0.249 ₁₆	${\ displaystyle 1 \ over \ mathrm {B} _ {16}}$ = 0.1745D ₁₆	${\ displaystyle 1 \ over \ mathrm {F} _ {16}}$ = 0, 1 ₁₆
${\ displaystyle 1 \ over 4}$ = 0.4 ₁₆	${\ displaystyle 1 \ over 8}$ = 0.2 ₁₆	${\ displaystyle 1 \ over \ mathrm {C} _ {16}}$ = 0.1 5 ₁₆	${\ displaystyle 1 \ over 10_ {16}}$ = 0.1 ₁₆

Negative numbers

Negative numbers can also be displayed. In most cases, the two's complement representation is used for this. Due to their design, no changes need to be made to the mechanisms for calculations in the basic arithmetic operations.

application

Computer science

The hexadecimal system is very suitable for representing sequences of bits (used in digital technology ). Four digits of a bit sequence (a nibble , also called a tetrad) are interpreted like a binary number and thus correspond to a digit in the hexadecimal system, since 16 is the fourth power of 2. The hexadecimal representation of the bit sequences is easier to read and faster to write:

                                  Binär      Hexadezimal           Dezimal
                                   1111  =             F  =             15
                                 1.1111  =            1F  =             31
                      11.0111.1100.0101  =          37C5  =         14.277
                    1010.1100.1101.1100  =          ACDC  =         44.252
                  1.0000.0000.0000.0000  =        1.0000  =         65.536
1010.1111.1111.1110.0000.1000.0001.0101  =     AFFE.0815  =  2.952.661.013

In this illustration, the point is only used to group numbers .

Software therefore often represents machine language in this way.

mathematics

Since the Bailey-Borwein-Plouffe formula for calculating π was developed in 1995, the hexadecimal system has also been important beyond computer science. This sum formula can calculate any hexadecimal digit of π without needing the preceding digits.

Conversion to other number systems

Many pocket calculators , but also the similarly named auxiliary programs on personal computers , offer conversions for changing the number base . In particular, the Windows and macOS programs “Calculator” calculate binary, hexadecimal and octal numbers into decimals and back if you select the menu item “Programmer” under “View” (Windows) or “Display” (macOS). Many Linux distributions come pre-installed with a calculator utility that includes such a “programmer option”, or you can use the printf command on the command line (as a built-in bash command or a separate utility).

Conversion of decimal numbers into hexadecimal numbers

One way of converting a number in the decimal system into a number in the hexadecimal system is to consider the division remainders that arise when the number is divided by the base 16; the method is therefore also called the division method or the residual value method.

In the example of the 1278 _{10 it} would look like this:

1278 : 16 = 79 Rest: 14 (= E) (Nr:1278-(79*16)=14)
  79 : 16 =  4 Rest: 15 (= F) (Nr:79-(4*16)=15)
   4 : 16 =  0 Rest:  4       (Nr:4-(0*16)=4)

The hexadecimal number is read from bottom to top and thus gives 4.FE .

Conversion of hexadecimal numbers into decimal numbers

To convert a hexadecimal number into a decimal number , you have to multiply the individual digits by the respective power of the base . The exponent of the base corresponds to the place of the digit, whereby a zero is assigned to the number before the decimal point. To do this, however, you have to convert the digits A, B, C, D, E, F into the corresponding decimal numbers 10, 11, 12, 13, 14, 15.

Example for 4FE ₁₆ :

${\ displaystyle [4FE] _ {16} = 4 \ cdot 16 ^ {2} +15 \ cdot 16 ^ {1} +14 \ cdot 16 ^ {0} = [1278] _ {10}}$

Conversion from hexadecimal to octal

In order to convert numbers between the octal system, which used to be common in computer science, and the hexadecimal system used today, it is advisable to take the intermediate step using the binary system. This is quite easy, since both base 8 and base 16 are powers of two.

The hexadecimal number is converted into a sequence of binary digits according to the table above.
Convert the groups of four into groups of three.
The binary sequence is then translated into an octal sequence.

Example for 8D53 ₁₆ :

${\ displaystyle [8D53] _ {16} = 1000.1101.0101.0011_ {2} = 1'000'110'101'010'011_ {2} = 106523_ {8}}$

Conversion from octal to hexadecimal

The conversion from octal to hexadecimal is just as easy, only here is the way

Octal sequence → binary sequence in groups of three → binary sequence in groups of four → hexadecimal sequence

is gone.

Mathematical representation of the hexadecimal system

Formulated in the decimal system:

${\ displaystyle h_ {m} h_ {m-1} \ cdots h_ {0}, h _ {- 1} h _ {- 2} \ cdots h _ {- n} = \ sum _ {i = -n} ^ {m } h_ {i} \ cdot {(16_ {10})} ^ {i} \ qquad m, n \ in \ mathbb {N} \ quad h_ {i} \ in \ {0; 1; \ cdots; 15 \ }}$

Formulated in the hexadecimal system:

${\ displaystyle h_ {m} h_ {m-1} \ cdots h_ {0}, h _ {- 1} h _ {- 2} \ cdots h _ {- n} = \ sum _ {i = -n} ^ {m } h_ {i} \ cdot {(10_ {16})} ^ {i} \ qquad m, n \ in \ mathbb {N} \ quad h_ {i} \ in \ {0; 1; \ cdots; 9; A; \ cdots; F \}}$

One- and two-handed counting with the fingertips and joints

Like the old Babylonian sexagesimal system, the hexadecimal system can also be counted with the fingers. The following technique is used to represent a byte with both hands together . Each hand represents a nibble . Its upper crumb shows on the used finger, while its lower crumb shows on the indicated joint or the fingertip. A bit flip can be brought about by mirroring the thumb position at the center of the finger surface.

One-handed counting from zero to F ₁₆

OK sign

If, like the ancient Babylonians, you use the thumb as a pointer and place it on the tip of the index finger, as with the OK sign of divers, and define this sign as the zero. Then you can define one on the upper joint of the index finger, followed by two on the middle and finally three on the lower joint. Likewise continued over the four at the tip of the middle finger, the eight at the tip of the ring finger and the twelve on the little one. You can then count to 15 = F ₁₆ when the thumb has reached the lower joint of the little finger, where it has grown.

The two crumbs of the nibble are displayed orthogonally on the hand. So that the lower two bits can be read at the level of the thumb on the respective finger and the two upper bits on the finger in question. This means that both a thumb on the fingertip and on the index finger stand for 00 ₂ in the respective crumb. The upper joint and the middle finger stand for 01 ₂ , the middle joint and ring finger stand for 10 ₂ and the lower joint and the little finger stand for 11 ₂ . Only four combinations have to be memorized in order to convert manually between the hexadecimal and binary system instead of 16.

A bit flip is easy to achieve by mirroring the position of the thumb at the intersection of the imaginary axes between the ring and middle fingers and the upper and middle row of joints. An example is given at the end of the table below.

Example of converting between hex and binary and bit flips by hand
Whole nibble	Upper crumb	finger	Lower crumb	Position of the thumb on the finger
0 ₁₆ = (00 00) ₂	00 ₂	index finger	00 ₂	Great, ok sign
1 ₁₆ = (00 01) ₂	00 ₂	index finger	01 ₂	Upper joint
2 ₁₆ = (00 10) ₂	00 ₂	index finger	10 ₂	Middle joint
3 ₁₆ = (00 11) ₂	00 ₂	index finger	11 ₂	Lower joint
4 ₁₆ = (01 00) ₂	01 ₂	Middle finger	00 ₂	top
8 ₁₆ = (10 00) ₂	10 ₂	Ring finger	00 ₂	top
C ₁₆ = (11 00) ₂	11 ₂	pinkie finger	00 ₂	top
Bit flip of 2 ₁₆ through point mirroring at the intersection of the aforementioned axes
D ₁₆ = (11 01) ₂	11 ₂	pinkie finger	01 ₂	Upper joint

Two-handed counting from zero to FF ₁₆

If you now count on the left hand with the procedure described above how often you have counted up to F ₁₆ on the right hand , a byte can be represented with two hands. Since four elements are counted on each finger, it follows that multiples of four appear on the fingertips. This means that when the thumbs of the respective hands start to count at zero on the respective index finger tip, the value on the tips of the fingers of the right hand increases by four, whereas when counting over the fingertips of the left hand it increases by 40 ₁₆ or 64 increased. If the left hand is only moved forward or backward by a phalanx, the displayed value changes by 10 ₁₆ or 16.

Web links

Wiktionary: Hexadecimal system - explanations of meanings, word origins, synonyms, translations

Online converter for different number systems
Number systems in comparison
Historical mechanical calculator for the hexadecimal system on one day

Individual evidence

↑ Conversion with different base values , accessed on October 30, 2018

[1] Conversion with different base values , accessed on October 30, 2018