{"id":34,"date":"2019-08-12T09:44:29","date_gmt":"2019-08-12T08:44:29","guid":{"rendered":"http:\/\/pb.ee.pw.edu.pl\/pb\/microproc\/?post_type=chapter&p=34"},"modified":"2019-12-31T10:02:27","modified_gmt":"2019-12-31T10:02:27","slug":"numbers-in-computing","status":"publish","type":"chapter","link":"http:\/\/pb.ee.pw.edu.pl\/pb\/microproc\/chapter\/numbers-in-computing\/","title":{"raw":"Numbers in Computing","rendered":"Numbers in Computing"},"content":{"raw":"
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Mathematics is the most beautiful and most<\/em>\r\npowerful creation of the human spirit.<\/em><\/p>\r\n

Stefan Banach<\/p>\r\n<\/blockquote>\r\nComputer processors might be perceived as extremely efficient \u201cnumber crunchers\u201d which means that they process enormous streams of information expressed only with numbers. It is worth for engineers to understand various ways by which electronic systems process these numbers.\r\n\r\nHumans in almost every corner of the earth are used to decimal system. This is so easy and natural to count with fingers hence positional system based on powers of tens is the most common nowadays. It is well known in practice but revising it might be helpful in understanding other systems discussed here. Look at the example:\r\n\r\n$latex 40375 = 4\\times10^4 + 0\\times10^3 + 3\\times10^2 + 7\\times10^1 + 5\\times10^0 = 40000 + 0 + 300 + 70 + 5 $<\/span>\r\n\r\nIn computer systems decimal numbers usually have no prefix or suffix but sometimes they are suffixed with \u201cd\u201d: $latex 40375d $<\/span>.\r\n

Binary System<\/h2>\r\nComputers do not use decimal system. They are working on bits<\/em> where bit is unit of information. It can have only one of two possible but different values such as:\r\n\r\n\r\n\r\n\r\n\r\n\r\n
True<\/th>\r\nFalse<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
1<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
Day<\/td>\r\nNight<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nBinary system has only two digits: 0 and 1. It is also positional as decimal system, but the base equals 2 (that is number of digits). Binary numers sometimes are suffixed with letter \u201cb\u201d or prefixed with \u201c0b\u201d. In the following example every multiplicand is expressed in decimal mode but it was omitted to simplify the equation and make it easier to follow the process:\r\n\r\n$latex 0b1011 = 1\\times2^3 + 0\\times2^2 + 1\\times2^1 + 1\\times2^0 = 1\\times8 + 0\\times4 + 1\\times2 + 1\\times1 = 8d + 2d + 1d = 11d $<\/span>\r\n\r\nAbove example also shows how binary number is converted to decimal mode. The series of multiplicands (starting from the rightmost position) in this conversion is easy to understand, reproduce or even remember: 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048, 4096, 8129, 16384, 32768, 65536, \u2026\r\n\r\nConversion in from decimal mode to binary is a bit more tricky and there are several ways to do it. First approach is repetitive division by 2 with remainder that indicates value of bit at specific position. Example:\r\n\r\n$latex \\begin{aligned}\r\n11d = 2\\times5d + 1\\\\\r\n5d = 2\\times2d + 1\\\\\r\n2d = 2\\times1d + 0\\\\\r\n1d = 2\\times0d + 1\\end{aligned} $<\/span>\r\n\r\nIf we write remainders starting from the last one we will get binary value: 1011<\/strong>b.\r\n\r\nThe above method works well for any value. But for small values simpler method might be employed which is based on subtraction. However, it requires to know (e.g. remember) weights at different digit positions. In every step a biggest possible weight is subtracted from the number that is being converted provided that result will remain positive number. At positions with corresponding weights \u201cones\u201d are set leaving all other positions with \u201czeros\u201d. Example:\r\n\r\n$latex \\begin{aligned}\r\n11d - 8d &= 3d \\qquad\\text{8d subtracted successfully so there should be ``1''}\\\\\r\n3d - 0d &= 3d \\qquad\\text{4d cannot be subtracted within positive integers so ``0''}\\\\\r\n3d - 2d &= 1d \\qquad\\text{2d subtracted successfully so there should be ``1''}\\\\\r\n1d - 1d &= 0d \\qquad\\text{1d left so there should be ``1'' in the last position}\\end{aligned} $<\/span>\r\n\r\nThis method may be much faster than previous one for small numbers. There is no threshold given how \u201csmall\u201d the number should be as it depends on experience of the person that is doing the conversion. Range of one nibble (4 bits) is easy to achieve while full byte (8 bits) is little harder, although even 12-bit values are possible to be converted by human being without neither calculator or pen and paper at hand.\r\n\r\nBinary system is foundation for boolean algebra. In typical computer implementations there are usually four basic operations available as presented in the table [1.1]. It is worth to remember that XOR and NOT operations are reversible. XOR is reversible because $latex \\forall x: x\\oplus x = 0 $<\/span> then $latex x\\oplus y \\oplus y = x $<\/span>. NOT operation is reversible because doubling negation cancels it so $latex \\neg \\neg x = x $<\/span>.\r\n\r\nSometimes these four operations are displayed on truth tables that let easily and quickly find out the result of operation assuming known values of two arguments: $latex x $<\/span> and $latex y $<\/span>. They are presented in tables [1.2], [1.3], [1.4], [1.5].\r\n\r\nIn computer logic operations usually are not done on single bits but whole bytes, words and so on. In such case bits are compared pairwise and the result has length equal to number of bits compared.\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Table 1.1: Four basic boolean operations.<\/caption>\r\n
Name<\/strong><\/th>\r\nOperation<\/strong><\/th>\r\nSymbol<\/strong><\/th>\r\n<\/th>\r\n<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Conjunction<\/td>\r\nAND<\/td>\r\n$latex \\wedge $<\/span><\/td>\r\n$latex x\\wedge y = 1 $<\/span> if $latex x = y = 1 $<\/span><\/td>\r\notherwise $latex x\\wedge y = 0 $<\/span><\/td>\r\n<\/tr>\r\n
Disjunction<\/td>\r\nOR<\/td>\r\n$latex \\vee $<\/span><\/td>\r\n$latex x\\vee y = 0 $<\/span> if $latex x = y = 0 $<\/span><\/td>\r\notherwise $latex x\\vee y = 1 $<\/span><\/td>\r\n<\/tr>\r\n
Exclusive or<\/td>\r\nXOR<\/td>\r\n$latex \\oplus $<\/span><\/td>\r\n$latex x\\oplus y = 1 $<\/span> if $latex x \\neq y $<\/span><\/td>\r\notherwise $latex x\\oplus y = 0 $<\/span><\/td>\r\n<\/tr>\r\n
Negation<\/td>\r\nNOT<\/td>\r\n$latex \\neg $<\/span><\/td>\r\n$latex \\neg x = 1 $<\/span> if $latex x = 0 $<\/span><\/td>\r\notherwise $latex \\neg x = 0 $<\/span><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n
Table 1.2: Conjunction truth table.<\/caption>\r\n
<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
<\/td>\r\nAND <\/strong>$latex x\\wedge y$<\/span><\/td>\r\n0<\/strong><\/td>\r\n1<\/strong><\/td>\r\n<\/tr>\r\n
<\/td>\r\n0<\/strong><\/td>\r\n0<\/td>\r\n0<\/td>\r\n<\/tr>\r\n
<\/td>\r\n1<\/strong><\/td>\r\n0<\/td>\r\n1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n
Table 1.3: Disjunction truth table.<\/caption>\r\n
<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
<\/td>\r\nOR $latex x\\vee y$<\/span><\/strong><\/td>\r\n0<\/strong><\/td>\r\n1<\/td>\r\n<\/tr>\r\n
<\/td>\r\n0<\/strong><\/td>\r\n0<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
<\/td>\r\n1<\/strong><\/td>\r\n1<\/td>\r\n1<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n\r\n
Table 1.4: Exclusive disjunction truth table.<\/caption>\r\n
<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
<\/td>\r\nXOR $latex x\\oplus y$<\/span><\/strong><\/td>\r\n0<\/strong><\/td>\r\n1<\/strong><\/td>\r\n<\/tr>\r\n
<\/td>\r\n0<\/strong><\/td>\r\n0<\/td>\r\n1<\/td>\r\n<\/tr>\r\n
<\/td>\r\n1<\/strong><\/td>\r\n1<\/td>\r\n0<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n\r\n\r\n\r\n
Table 1.5: Negation truth table.<\/caption>\r\n
<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
NEG<\/strong> $latex \\neg x$<\/span><\/td>\r\n0<\/strong><\/td>\r\n1<\/strong><\/td>\r\n<\/td>\r\n<\/tr>\r\n
<\/td>\r\n1<\/td>\r\n0<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n

Hexadecimal Numbers<\/h2>\r\nHexadecimal mode is also positional mode so it works exactly the same way as decimal and binary modes. The base in hexadecimal mode equals to 16 so 16 digits are necessary. First five letters from latin alphabet (A-F) expand standard decimal digits 0-9 making altogether 16 digits. Capital and small letters might be used but should not be mixed in single number. In the table below they are presented with their decimal values:\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Hexadecimal number<\/th>\r\n<\/th>\r\n0<\/th>\r\n1<\/th>\r\n2<\/th>\r\n3<\/th>\r\n4<\/th>\r\n5<\/th>\r\n6<\/th>\r\n7<\/th>\r\n\u2026<\/th>\r\n<\/th>\r\n<\/tr>\r\n<\/thead>\r\n
Decimal value<\/td>\r\n<\/td>\r\n0<\/td>\r\n1<\/td>\r\n2<\/td>\r\n3<\/td>\r\n4<\/td>\r\n5<\/td>\r\n6<\/td>\r\n7<\/td>\r\n\u2026<\/td>\r\n<\/td>\r\n<\/tr>\r\n
Hexadecimal number<\/td>\r\n\u2026<\/td>\r\n8<\/td>\r\n9<\/td>\r\nA<\/td>\r\nB<\/td>\r\nC<\/td>\r\nD<\/td>\r\nE<\/td>\r\nF<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n
Decimal value<\/td>\r\n\u2026<\/td>\r\n8<\/td>\r\n9<\/td>\r\n10<\/td>\r\n11<\/td>\r\n12<\/td>\r\n13<\/td>\r\n14<\/td>\r\n15<\/td>\r\n<\/td>\r\n<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\nHexadecimal values are very useful and popular representation of integer numbers, memory addresses and other numbers in programming and computing machinery. One and only one of several suffixes and prefixes should be used to indicate that given number is hexadecimal value. Sometimes suffixes and prefixes are not present as it might be assumed or it is known that given number is hexadecimal. Examples:\r\n