Number SystemsBits and bytesNumber systemsEverything is a number?CountingCounting in three systemsComputers use binary numbersUsing octal and hex numbersWriting octal and hex integersASCIIExtended ASCIIUnicodeThe EndJan 14, 2019Number Systems2Bits and bytesA bit is a single two-valued quantity: yes or no, true or false, on or off, high or low, good or badOne bit can distinguish between two cases: T, FTwo bits can distinguish between four cases:TT, TF, FT, FFThree bits can distinguish between eight cases:TTT, TTF, TFT, TFF, FTT, FTF, FFT, FFFIn general, n bits can distinguish between 2n casesA byte is 8 bits, therefore 28 = 256 cases3Number systemsThe binary (base 2) number system uses two “binary digits, ” (abbreviation: bits) -- 0 and 1The octal (base 8) number system uses eight digits:0, 1, 2, 3, 4, 5, 6, 7The decimal (base 10) number system uses ten digits:0, 1, 2, 3, 4, 5, 6, 7, 8, 9The hexadecimal, or “hex” (base 16) number system uses sixteen digits:0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F4Everything is a number?Everything in the computer is stored as a pattern of bitsBinary distinctions are easy for hardware to work withNumbers are stored as a pattern of bitsComputers use the binary number systemCharacters are stored as a pattern of bitsOne byte (8 bits) can represent one of 256 charactersSo, is everything in the computer stored as a number?No it isn’t, it’s stored as a bit patternThere are many ways to interpret a bit pattern5CountingTo count up in any number system,add 1 to the rightmost digitif the result is higher than the largest digit,set that digit to zero and carry to the next placerepeat addition of 1 and carrying as many times as necessaryExample: In hex, F is the largest digitHence, 4A6FF + 1 = 4A7006Counting in three systemsDec Bin Hex 0 0 0 1 1 1 2 10 2 3 11 3 4 100 4 5 101 5 6 110 6 7 111 7 8 1000 8 9 1001 910 1010 A Dec Bin Hex 11 1011 B 12 1100 C 13 1101 D 14 1110 E 15 1111 F 16 10000 10 17 10001 11 18 10010 12 19 10011 13 20 10100 147Computers use binary numbersPeople like to use decimal numbersComputers use binary numbersJava translates decimal numbers into binaryThe computer does all its arithmetic in binaryJava translates binary results back into decimalYou occasionally have to use numbers in other number systemsIn Java, you can write numbers as octal, decimal, or hexadecimal (but not binary)Colors are usually specified in hexadecimal notation:#FF0000, #669966,8Using octal and hex numbersComputers use binary, but the numbers are too long and confusing for people--it’s easy to lose your placeOctal or hex is better for peopleTranslation between binary and octal or hex is easyOne octal digit equals three binary digits 101101011100101000001011 5 5 3 4 5 0 1 3One hexadecimal digit equals four binary digits 101101011100101000001011 B 5 C A 0 B9Writing octal and hex integersIntegers are usually written in decimal notation:7, 532, -28To write a number in octal, just start with a zero:02, 0657, -077...but don’t use the digits 8 or 9 !To write a number in hexadecimal, start with 0x or 0X: 0xA, 0X43AB5, -0xFFFFThe “digits” A through F can be upper or lower caseUppercase is usually preferredLowercase is more readable for long numbers10ASCIIUp until recently, all programming languages used ASCIISource: Wikipedia11Extended ASCIIIn ASCII, the first bit is always zeroEverybody got the bright idea of using a one for the first bit, to give an additional 128 charactersEverybody added different characters, in different ordersLots of people who don't know any better put Extended ASCII characters (curly quotes, letters with accents marks, etc.) into their documentsConsequently, other people see these characters as random garbageMoral: Don’t use Extended ASCII12UnicodeEventually, ASCII (American Standard Code for Information Interchange) wasn’t enoughUnicode is the new standard; it has more than 100,000 characters definedFor ease of transition, the first 128 characters are those defined in the ASCII standard, and require one byte each; other characters may require up to four bytesThis is the UTF-8 encodingA second Unicode standard uses either two or four bytes for each character (again, the first 128 are the same as ASCII)This is the UTF-16 encodingJava and all other modern languages now use Unicode13The End“There are 10 kinds of people in the world:those who understand binary, and those who don’t.” --Anon. “Real Programmers always confuse Christmas and Halloween because Oct31 == Dec25” --Andrew