Computer numbers

Computer numbers are the basic units used and manipulated in a digital computer. The term binary is used to describe the raw data as it is stored in a computer. The smallest unit of the binary system is the "bit" which defines only a 1 or a 0. Multiple bits are used to define larger numbers.

Overview

Even if the computer data represents symbols or characters the computer hardware will still treat it as a binary number. It is up to the software program to interpret the information correctly.

The computer uses a binary numbering system as shown below on the left as compared to the decimal system we use today. Each binary digit is called a 'bit' in computer parlance. A bit can only have a value of 1 or 0 (on or off) so to count past 1 you need more than 1 bit.

table

decimal binary powers 0 0 1 1 20 2 10 21 3 11 4 100 22 5 101 6 110 7 111 8 1000 23 9 1001

hexadecimal binary decimal 0 0000 0 1 0001 1 2 0010 2 3 0011 3 4 0100 4 5 0101 5 6 0110 6 7 0111 7 8 1000 8 9 1001 9 A 1010 10 B 1011 11 C 1100 12 D 1101 13 E 1110 14 F 1111 15

Hexadecimal

Hexadecimal (also called base-16 or abbreviated to hex) numbers is a number system used to simplify the display and manipulation of numbers used in computers. As can be seen in the table it can take up to 4 binary digits to represent the symbols used in the decimal system. Working with binary digits can really be cumbersome so computer folks often work in hexadecimal. Hexadecimal is similar to decimal but adds 6 more characters to the number system (A, B, C, D, E, F) to fill out all the 16 combinations possible with 4 binary digits (bits). Two hexadecimal numbers can represent 8 bits, called a byte, that can count from 0 to 255 in decimal.

It is up to the program and computer to understand what a byte represents. It could be a binary number or a character or even interpreted as a logical representation of data such as true or false. However, it can sometimes be confusing as to what numbering system is in use. Hexadecimal numbers include a small x at the beginning to indicate that the number is being shown in hexadecimal thus x10 is equal to a decimal 16. In this notation the letters A-F can be in either upper or lower case. The x may be preceded with a 0.

Implications of binary

In decimal we use powers of ten thus we see progressions as 10, 100, 1000, 10000, etc. In binary we see powers of two thus the progression is 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, etc. This is why computer terminology for a thousand bytes is really 1024 instead of 1000.

It is easy to translate decimal integers into the equivalent binary but not so for fractions. There are many decimal fractions that have no exact binary equivalent. Binary fractions are actually division by 2 so fractions like 1/2, 1/4, 1/8 are easy to do but others are more difficult. Binary numbers also get long quickly and become more difficult to deal with. One choice is to group them in 4's and create bytes. For really big numbers or really small numbers in decimal we typically resort to scientific notation. For binary the answer is to convert them to a floating-point number which may also lose some accuracy.

Note that computer numbers are often used even to identify the binary value of characters or such as ASCII values or for any binary information since it is easier to identify values in Hexadecimal than to use pure binary. Sometimes binary can also be represented in groups of 3 bits using Octal notation which is simply the numbers from 0 to 7 as shown in the table.

Note that we have referred to this binary data as numbers but it may also be interpreted as printable characters or any other form of data. The computer understands it as numbers but the program or user can interpret it differently.

Byte

The term byte is used to collect a number of binary bits into a character. Generally the term Byte is used to represent 8 bits, two hexadecimal digits. Thus a single byte (2⁸) has 256 possible values ranging from 0 to 255. When abbreviated the term bit is in lower case while Byte is in upper case so a transfer rate of 300 b/s would indicate bits per second while 300 B/s would be 8 times faster. Memory and Disk capacity is normally specified in Bytes.

Larger Bytes

Multiple Bytes can have a prefix that specifies multiples of 1,000. These prefixes are K for Kilo, M for Mega, G for Gigi and T for Terra. For example 1000 bytes in decimal would be 1 KB (Note 1Kb would be 1000 bits). Similarly a 1000 KBs would be 1 MB (1 Megabyte). While 1,000 MB would be 1 GB (1 Gigabyte) that is 1,000,000,000 bytes.

Technically in binary or Hexadecimal it would be multiples of 1024 bytes to make 1 MB. Thus the hexadecimal number 400 is 1 KB. F FFFF would be just under 1 MB. This accounts for differences in how the capacity of a disk drive or memory module might be specified.

Word

To carry the analogy of one Byte representing a character a bit further multiple bytes can represent a word. A computer word is the generally the size of an instruction or meaningful memory location. Computers are often identified by the size of their word. A native 32 bit computer means the word size is 32 bits or 4 bytes. A 32 bit word can hold a number up to 4 Gig (2 Gig for a signed number). To represent a larger number you might use a double word or perhaps a 64 bit computer.

There are implications to the word size as it effects limits. For example a 32bit word limits the capacity of a computer to access storage directly using one word to 4 GB. Normally numbers in a computer are signed. This means that they could be positive or negative numbers and one bit is reserved for the sign, leaving 31 bits for the number itself. This effectively halves the size of the number that can be stored in a word to 2 Gigabytes. This, for example, is the cause of the need to change the format of an SD card when the capacity exceeded 2 GB. This also is the cause of a future problem where the date stored in computer files will overflow. Time in a computer is generally stored in total seconds in one word from with a finite date. In Unix this was Jan 1 1970 UT. Dates are computed using the number of seconds in the computer clock word and this will overflow after 2030. Negative time is assigned to dates prior to 1970.

Converting Hexadecimal to decimal

In the discussion to follow see the table above to translate the individual characters to their decimal equivalent or view below for the translation of full 2 Hex digits.

In our decimal system each digit is multiplied by powers of 10 to achieve larger numbers. For hexadecimal you would need to multiply by powers of 16 instead. Beginning right to left, take the decimal equivalent digits and add them together after multiplying by the appropriate value. For the right digit multiply by 1. For the second digit multiply by 16. For example hex 2A would be 10 x 1 + 2 x 16 = 42. For each successive digit multiply by 256, 4096, 65536, 1048576, etc. using successive powers of 16. Two digits as shown below can count to 255 from 0. Three digits can go to 4095 while four can reach 65535.

Hexadecimal translation tables

Hex	Dec		Hex	Dec		Hex	Dec		Hex	Dec		Hex	Dec		Hex	Dec		Hex	Dec		Hex	Dec
00	00		10	16		20	32		30	48		40	64		50	80		60	96		70	112
01	01		11	17		21	33		31	49		41	65		51	81		61	97		71	113
02	01		12	18		22	34		32	50		42	66		52	82		62	98		72	114
03	03		13	19		23	35		33	51		43	67		53	83		63	99		73	115
04	04		14	20		24	36		34	52		44	68		54	84		64	100		74	116
05	05		15	21		25	37		35	53		45	69		55	85		65	101		75	117
06	06		16	22		26	38		36	54		46	70		56	86		66	102		76	118
07	07		17	23		27	39		37	55		47	71		57	87		67	103		77	119
08	08		18	24		28	40		38	56		48	72		58	88		68	104		78	120
09	09		19	25		29	41		39	57		49	73		59	89		69	105		79	121
0A	10		1A	26		2A	42		3A	58		4A	74		5A	90		6A	106		7A	122
0B	11		1B	27		2B	43		3B	59		4B	75		5B	91		6B	107		7B	123
0C	12		1C	28		2C	44		3C	60		4C	76		5C	92		6C	108		7C	124
0D	13		1D	29		2D	45		3D	61		4D	77		5D	93		6D	109		7D	125
0E	14		1E	30		2E	46		3E	62		4E	78		5E	94		6E	110		7E	126
0F	15		1F	31		2F	47		3F	63		4F	79		5F	95		6F	111		7F	127

Hex	Dec		Hex	Dec		Hex	Dec		Hex	Dec		Hex	Dec		Hex	Dec		Hex	Dec		Hex	Dec
80	128		90	144		A0	160		B0	176		C0	192		D0	208		E0	224		F0	240
81	129		91	145		A1	161		B1	177		C1	193		D1	209		E1	225		F1	241
82	130		92	146		A2	162		B2	178		C2	194		D2	210		E2	226		F2	242
83	131		93	147		A3	163		B3	179		C3	195		D3	211		E3	227		F3	243
84	132		94	148		A4	164		B4	180		C4	196		D4	212		E4	228		F4	244
85	133		95	149		A5	165		B5	181		C5	197		D5	213		E5	229		F5	245
86	134		96	150		A6	166		B6	182		C6	198		D6	214		E6	230		F6	246
87	135		97	151		A7	167		B7	183		C7	199		D7	215		E7	231		F7	247
88	136		98	152		A8	168		B8	184		C8	200		D8	216		E8	232		F8	248
89	137		99	153		A9	169		B9	185		C9	201		D9	217		E9	233		F9	249
8A	138		9A	154		AA	170		BA	186		CA	202		DA	218		EA	234		FA	250
8B	139		9B	155		AB	171		BB	187		CB	203		DB	219		EB	235		FB	251
8C	140		9C	156		AC	172		BC	188		CC	204		DC	220		EC	236		FC	252
8D	141		9D	157		AD	173		BD	189		CD	205		DD	221		ED	237		FD	253
8E	142		9E	158		AE	174		BE	190		CE	206		DE	222		EE	238		FE	254
8F	143		9F	159		AF	175		BF	191		CF	207		DF	223		EF	239		FF	255

floating-point

Floating-point is the computer method of handling very large and very small numbers. It can deal with both positive and negative values. There is often hardware built into the computer specifically to deal with these numbers. To see how floating-point works consider the example 1024=2¹⁰. In Hex this would be 400 or in binary it would be 100 0000 0000, a 1 with 10 zeros. If we stored the 1 and the 10 separately then we would need only 5 bits. One for the 1 and four for the 10. A floating-point number is stored exactly this way. If it were stored in a word (32 bits) we could devote 24 bits to the value and 8 bits for the exponent. We will actually use 7 bits for the exponent (0-127) and one bit for a sign. (a negative exponent is for very small numbers) Thus we can represent numbers clear up to 3.4 times 10³⁸ with about 7 digits of decimal accuracy using 24 bits. You might ask, what about leaving a bit for negative numbers? Since we don't need leading zeros in floating point we can assume the first digit is always a 1. Thus we don't need to store it and we can use that bit for the sign. For double precision floating-point we could use 64 bits.

Some of the image formats used today can represent each color with a floating-point number for HDR use but even single precision is overkill for this use. A half precision number has been developed using 16 bits (saving half the space). There are 11 bits reserved for the value and 4 digits are reserved for the exponent with 1 bit used for the exponent sign. Using 11 digits for precision is still more than the 10 bits per color (30 total bits) now be touted for high definition use. In addition the range, using the exponent, of over 65,000 distinct values provides brightness recording as good as the eye can see. Image formats supporting floating-point can also use half precision floating-point.

See https://en.wikipedia.org/wiki/Floating-point_arithmetic#Floating-point_numbers in wikipedia for more information.