COMS 161 Introduction to Computing Title: Digital Numbers Date: February 7, 2005 Lecture Number: 12

Announcements Test next (not this) Wednesday First draft of paper Due 2/11/05

Review Numbers Signed numbers Two’s complement numbers

Outline Numbers Numeric representation of letters Hexadecimal Binary to hexadecimal conversion Hexadecimal to binary conversion Binary Coded Decimal (BCD)

Digital Letters Digital system All entities are represented as numbers How do we represent the letters in the English language The letters form a discrete set (unique unambiguous, precise) No sampling is needed Simply need a mapping from each letter to a numerical representation A = 65 B = 66

Digital Letters Important that all converters use the same mapping Otherwise the inverse process (converting a number to a letter) would give incorrect results Computers in the US primarily use the American Standard Code for Information Interchange (ASCII) Unicode is an international standard Compatible but extends the ASCII standard

ASCII mapping How many bits will I need to encode the letters of the English alphabet? Upper case Lower case Decimal digits Punctuation Arithmetic symbols Printer control characters

ASCII mapping Letters in the English language A = 6510 = 0100 00012 B = 6610 = 0100 00102 … Z = 9010 = 0101 10102 a = 9710 = 0110 00012 z = 12210 = 0111 10102 Numbers are still left over for punctuation

ASCII Table

Binary number system Precision The number of bits used to represent an item Letter: precision of 8 bits Integer (whole number): precision of 32 or 64 bits Always finite Computers have finite precision Presents some limitations

Hexadecimal number system Sometimes called hex Positional, base-16 system Each digit is multiplied by a power of 16 Sixteen unique symbols (digits) 0, 1, 2, …, 15 Symbol a or A for 10 Symbol b or B for 11 Symbol e or E for 14 Symbol c or C for 12 Symbol f or F for 15 Symbol d or D for 13

Hexadecimal number system A hex number can represent 16 different items Equivalent to 4 bits Makes it easy to convert between binary and hex Group bits by 4’s from the right end Substitute the hex symbol 9010 = 0101 10102 = 5A16 Is the base 16 really needed? 6610 = 0100 00102 = 4216

Hexadecimal number system Use the backwards conversion to convert hex to binary One hex digit is equivalent to 4 bits Substitute the binary nibble Always start at the right end Add zeros to the left end as necessary to fill in 4 bits

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

Hexadecimal to decimal conversion Same procedure as converting a binary number to a decimal number The digits of the hex number are the coefficients of the corresponding positional weighting factor ABC16 = 0xABC = A * 162 + B * 161 + C * 160 = A * 256 + B * 16 + C * 1 = 10 * 256 + 11 * 16 + 12 * 1 = 2560 + 176 + 12 = 274810

Digitization The process of converting analog information into binary Discrete forms are unambiguous Text and numbers are discrete Conversion of discrete to digital Come up with a mapping As we did with the letters

Binary Coded Decimal Integers (whole numbers) One mapping is to use its binary equivalent Binary Coded Decimal (BCD) 010 = 00002 110 = 00012 … 910 = 10012 Need a minimum of 4 bits to represent 10 different values Some 4 bit quantities are wasted

Binary Coded Decimal String of decimal digits Each decimal digit is represented by 4 bits The number of bits needed to represent different numbers vary Performing arithmetic is complicated Why? 15910 = 00012 01012 10012