COMS 161 Introduction to Computing Title: Digital Numbers Date: February 4, 2005 Lecture Number: 11

Announcements

Review Numbers Decimal to Binary Conversion Adding binary numbers Subtraction Method Adding binary numbers

Outline Numbers Signed numbers Hexadecimal Binary Coded Decimal (BCD) Binary to hexadecimal conversion Binary Coded Decimal (BCD)

Binary Number System How about representing negative numbers? Let the left most bit represent the sign (+, -) of the number Called signed magnitude representation [s][mag]

Signed Magnitude One less bit to represent the magnitude

Signed Magnitude Problems Two values of 0 Incorrect arithmetic More difficult to detect than one value of 0 Incorrect arithmetic 2 – 1 = 2 + (-1) = 1

Two’s Complement Representation Sign bit in a sense Positive numbers The leading bit (left most) is zero The same as signed magnitude Negative numbers The leading bit is one Defined so that when added to their corresponding positive number the answer is zero

Two’s Complement Representation Bit Pattern Value 0000 1000 -8 0001 1 1001 -7 0010 2 1010 -6 0011 3 1011 -5 0100 4 1100 -4 0101 5 1101 -3 0110 6 1110 -2 0111 7 1111 -1

Two’s Complement Representation Both problems with the signed magnitude representation are solved with two’s complement representation There is only value of zero Arithmetic is correct Solution is in two’s complement form 2 – 1 = 2 + (-1) = 1

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 Combatable 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