Number Systems

The best known number system among the infinite number of possible systems is without doubt the decimal number system, the reason probably being that people usually have ten fingers to count with.

The position of a digit in a decimal number denotes how often the corresponding decimal power is present in the whole number. You should keep in mind, that any number to the power of Zero always equals One (x⁰=1).

Example for decimal number 1234

1234 =	{	1∙103	+	2∙102	+	3∙101	+	4∙100
		1000	+	200	+	30	+	4
		1234

A computer does not have ten fingers to count with but has to fall back on some other intelligent mechanism for calculating numbers. The logic it uses defines two states, off and on, or 0 and 1. These two states can be easily expressed using binary numbers, which only use the digits 0 and 1. Here the position of the digit denotes how often the corresponding binary power is present in the whole number.

Example of converting binary 10011001 to decimal number 153

10011001 =

{

1∙27

+

0∙26

+

0∙25

+

1∙24

+

1∙23

+

0∙22

+

0∙21

+

1∙20

128

+

0

+

0

+

16

+

8

+

0

+

0

+

1

153

A single binary digit is known as 1bit in computer science. The above example, having eight digits can then be said to consist of 8bit. A unit of 4bit is usually called 1nybble (or nibble) and a unit of 8bit is called 1Byte.

Now binaries are something for geeks but the humble user is seldom confronted with them. Far more important and widespread throughout specifications are hexadecimal numbers, which are for example used for encoding colours and characters. A hexadecimal number may consist of the digits 0 to 9 and/or the letters a to f. This scope is sometimes called hexit in analogy to digit. In a hexadecimal number the position of a hexit denotes how often the corresponding hexadecimal power is present in the whole number. The letters a to f represent the decimal numbers 10 to 15 respectively and no distinction is made between uppercase or lowercase lettering.

Example of converting hexadecimal number 2BA3 to decimal number 11171

2BA3 =	{	2∙163	+	B∙162	+	A∙161	+	3∙160
		2∙163	+	11∙162	+	10∙161	+	3∙160
		8192	+	2816	+	160	+	3
		11171

Using hexadecimal numbers has one great advantage, which is, nybbles and bytes can be easily expressed as hexadecimal numbers:

Relationship between nybbles and hexadecimal numbers

nybble	Hex	nybble	Hex
0000	0	1000	8
0001	1	1001	9
0010	2	1010	A
0011	3	1011	B
0100	4	1100	C
0101	5	1101	D
0110	6	1110	E
0111	7	1111	F

As one byte is made up of two nybble, the only thing you have to do is take the corresponding nybble from the former table to get the corresponding hexadecimal number. Consider the byte with value 10010010 (decimal 128+16+2=146). It consits of nybble 1001 and 0010. Its corresponding hexadecimal value is thus 92.

This allows for easy handling of numbers, which are large and complicated in decimal and binary systems. Example:

1111111111111111bin = 65535dec = FFFFhex

For the sake of completeness octal numbers shall be mentioned as well, as one might encounter these from time to time. Octal numbers consist of the digits 0 to 7 and follow the same rules as the above mentioned systems, that is, the digits represent the factors of octal powers.

Example of converting octal number 63571 to decimal number 26489

63571 =	{	6∙84	+	3∙83	+	5∙82	+	7∙81	+	1∙80
		6∙4096	+	3∙512	+	5∙64	+	7∙8	+	1∙1
		24576	+	1536	+	320	+	56	+	1
		26489

Octal numbers can be used to simplify bitstrings in a way like hexadecimal numbers do. Instead of 4bit an octal number can represent 3bit as shown in the following table:

Relationship between 3bit binaries and octal numbers

3bit	Oct
000	0
001	1
010	2
011	3
100	4
101	5
110	6
111	7

Thus long binaries may be expressed by their corresponding octal values like the following example shows:

111111111111111bin = 32767dec = 77777oct

The subsequent chart lists decimal numbers 0 to 255 with their corresponding hexadecimal values. These are used very often as they represent all 256 possible byte values and thus correspond to the 256 possible values the basic colours red, green, and blue can have in the 24bit RGB colour-space and, moreover, they are used to encode the extended ASCII charset.

Decimal Numbers 0-255 ↔ Hexadecimal Numbers 00-FF

DEC	HEX	DEC	HEX	DEC	HEX	DEC	HEX	DEC	HEX	DEC	HEX	DEC	HEX	DEC	HEX
DEC	HEX	DEC	HEX	DEC	HEX	DEC	HEX	DEC	HEX	DEC	HEX	DEC	HEX	DEC	HEX
000	00	001	01	002	02	003	03	004	04	005	05	006	06	007	07
008	08	009	09	010	0A	011	0B	012	0C	013	0D	014	0E	015	0F
016	10	017	11	018	12	019	13	020	14	021	15	022	16	023	17
024	18	025	19	026	1A	027	1B	028	1C	029	1D	030	1E	031	1F
032	20	033	21	034	22	035	23	036	24	037	25	038	26	039	27
040	28	041	29	042	2A	043	2B	044	2C	045	2D	046	2E	047	2F
048	30	049	31	050	32	051	33	052	34	053	35	054	36	055	37
056	38	057	39	058	3A	059	3B	060	3C	061	3D	062	3E	063	3F
064	40	065	41	066	42	067	43	068	44	069	45	070	46	071	47
072	48	073	49	074	4A	075	4B	076	4C	077	4D	078	4E	079	4F
080	50	081	51	082	52	083	53	084	54	085	55	086	56	087	57
088	58	089	59	090	5A	091	5B	092	5C	093	5D	094	5E	095	5F
096	60	097	61	098	62	099	63	100	64	101	65	102	66	103	67
104	68	105	69	106	6A	107	6B	108	6C	109	6D	110	6E	111	6F
112	70	113	71	114	72	115	73	116	74	117	75	118	76	119	77
120	78	121	79	122	7A	123	7B	124	7C	125	7D	126	7E	127	7F
128	80	129	81	130	82	131	83	132	84	133	85	134	86	135	87
136	88	137	89	138	8A	139	8B	140	8C	141	8D	142	8E	143	8F
144	90	145	91	146	92	147	93	148	94	149	95	150	96	151	97
152	98	153	99	154	9A	155	9B	156	9C	157	9D	158	9E	159	9F
160	A0	161	A1	162	A2	163	A3	164	A4	165	A5	166	A6	167	A7
168	A8	169	A9	170	AA	171	AB	172	AC	173	AD	174	AE	175	AF
176	B0	177	B1	178	B2	179	B3	180	B4	181	B5	182	B6	183	B7
184	B8	185	B9	186	BA	187	BB	188	BC	189	BD	190	BE	191	BF
192	C0	193	C1	194	C2	195	C3	196	C4	197	C5	198	C6	199	C7
200	C8	201	C9	202	CA	203	CB	204	CC	205	CD	206	CE	207	CF
208	D0	209	D1	210	D2	211	D3	212	D4	213	D5	214	D6	215	D7
216	D8	217	D9	218	DA	219	DB	220	DC	221	DD	222	DE	223	DF
224	E0	225	E1	226	E2	227	E3	228	E4	229	E5	230	E6	231	E7
232	E8	233	E9	234	EA	235	EB	236	EC	237	ED	238	EE	239	EF
240	F0	241	F1	242	F2	243	F3	244	F4	245	F5	246	F6	247	F7
248	F8	249	F9	250	FA	251	FB	252	FC	253	FD	254	FE	255	FF

The converter translates decimal, octal, hexadecimal and binary numbers into each other. The decimal input value is limited and not allowed to be greater than 2³¹-1 = 2147483647. There is also a popup-version available for independent use outside this page's context.

DOHB-Converter (limit: 2³¹-1)

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