Class Note for ENGIN 112 at UMass(32)
Class Note for ENGIN 112 at UMass(32)
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Date Created: 02/06/15
ENGIN 112 Intro to Electrical and Computer Engineering Lectu re 3 More Number Systems quot ELECTRICAL 9 quot COMPUTER ENGINEERING umvsnswv or MASSACHUSETTS AMHERST ENGIN11Z L3 Mum Number Systems SepIemhernznn3 Overview Hexadecimal numbers Related to binary and octal numbers Conversion between hexadecimal octal and binary Value ranges of numbers Representing positive and negative numbers Creating the complement of a number Make a positive number negative and vice versa Why binary ENGN112 L3 More Number Systems September 8 2003 Understanding Binary Numbers Binary numbers are made of binary digits bits 0 and 1 How many items does an binary number represent 10112 1x23 0x221x211x2 1110 What about fractions 110102 1x22 1x210x20 1x21 0x22 Groups of eight bits are called a byte 11oo1oo12 Groups of four bits are called a nibble 1101 2 ENGN112 L3 More Number Systems September 8 2003 Understanding Hexadecimal Numbers Hexadecimal numbers are made of digits 0123456789A B C D E How many items does an hex number represent 3A9F16 3x163 10x162 9x16115x16 1499910 What about fractions 203516 2x162 13x161 3x160 5x16391 72331251o Note that each hexadecimal digit can be represented with four bits 1110 2 E16 Groups of four bits are called a nibble 1110 2 ENGN112 L3 More Number Systems September 8 2003 Putting It All Together DCL i mu O 1 Z 5 ENGINHZ L3 More Number Systems Binary 0 1 10 11 100 101 110 111 1000 1001 1010 1011 1100 1101 1110 1111 OCHKI 16 Hexadecimal zj m pmxumw awmwo o Binar octal anq hexa ecrmal sImIIar Easy to build circuits to operate on these representations Possible to convert between the three formats September a zoos Converting Between Base 16 and Base 2 3A9F16 0011 10101001 11112 3 A 9 F Conversion is easy gt Determine 4bit value for each hex digit shite that there are 24 16 different values of four I s Easier to read and write in hexadecimal Representations are equivalent ENGN112 L3 More Number Systems September 8 2003 Converting Between Base 16 and Base 8 3A9F16 0011 10101001 11112 3 A 9 F i 352378 m 1m w w 1112 3 5 2 3 7 Convert from Base 16 to Base 2 Regroup bits into groups of three starting from right Ignore leading zeros PPNT Each group of three bits forms an octal digit ENGN112 L3 More Number Systems September 8 2003 How To Represent Signed Numbers Plus and minus sign used for decimal numbers 25 or 25 16 etc For computers desirable to represent everything as bits Three types of signed binary number representations signed magnitude 1 s complement 2 s complement In each case leftmost bit indicates sign positive 0 or negative 1 Consider signed magnitude 0000110021210 100011002 4210 Sign bit Magnitude Sign bit Magnitude ENGN112 L3 More Number Systems September 8 2003 One s Complement Representation The one s complement of a binary number involves inverting all bits 1 s comp of 00110011 is 11001100 1 s comp of 10101010 is 01010101 For an n bit number N the 1 s complement is Zn1 N Called diminished radix complement b Mano since 1 s complement for base radix To find negative of 1 s complement number take the 1 s complement OOOO110021210 111100112 1210 Sign bit Magnitude Sign bit Magnitude ENGN112 L3 More Number Systems September 8 2003 Two s Complement Representation The two s complement of a binary number involves inverting all bits and adding 1 2 s comp of 00110011 is 11001101 2 s comp of 10101010 is 01010110 For an n bit number N the 2 s complement is 2 4 N 1 Called radix complement by Mano since 2 s complement for base radix 2 To find negative of 2 s complement number take the 2 s complement 0000110021210 111101002 4210 Sign bit Magnitude Sign bit Magnitude ENGN112 L3 More Number Systems September 8 2003 Two s Complement Shortcuts Algorithm 1 Simply complement each bit and then add 1 to the result Finding the 2 s complement of 011001012 and of its 2 s complement N 01100101 N 10011011 10011010 01100100 1 1 10011011 01100101 Algorithm 2 Starting with the least significant bit copy all of the bits up to and including the first 1 bIt and then complementing the remaining bIts N o11oo1o1 N 1oo11o11 ENGN112 L3 More Number Systems September 8 2003 Finite Number Representation Machines that use 2 s complement arithmetic can represent integers in the range 2n1 lt N lt 2n11 where n is the number of bits available for representin N Note that 2n11 011112 and 2quot1 00002 oFor 2 s complement more negative numbers than posmve oFor 1 s complement two representations for zero oFor an n bit number in base radix 2 there are zn different unsigned values 0 1 1 ENGN112 L3 More Number Systems September 8 2003 1 s Complement Addition Using 1 s complement numbers adding numbers is easy For example suppose we wish to add 11002 and 00012 Let s compute 1210 110 121o 11002 011002in 1 s comp 110 ooo12 000012in1 s comp Step 1 Add binary numbers Step 2 Add carry to loworder bit ENGN112 L3 More Number Systems Add Final Result 0 l l O 1 September 8 2003 1 s Complement Subtraction Usin 1 scompement numbers subtracting num ers IS also easy For exam le suppose we wish to subtract 00012 rom 1 002 o O l l O O Lets compute 121o 110 0 0 0 0 1 1210 11002 011002 in 1 s comp 110 00012 111102in 1 s comp 1 s comp O l l O 0 Step 1 Take 1 s complement of 2nd operand Add l l l l 0 Step 2 Add binary numbers Step 3 Add carry to low order bit ENGN112 L3 More Number Systems September 8 2003 2 s Complement Addition is easy and 00012 Let s compute 1210 110 121o 11002 011002in 2 s 11o ooo12 000012in 2 s Step 1 Add binary numbers Step 2 Ignore carry bit ENGN112 L3 More Number Systems Using 2 s complement numbers adding numbers For example suppose we wish to add 11002 comp COmP 0 1 1 0 0 Add O O O O 1 Final 0 0 1 1 0 1 Result Ignore September 8 2003 2 s Complement Subtraction Using 2 s complement numbers follow steps for subtraction For exam le suppose we wish to subtract 00012 rom 1 002 Let s compute 1210110 0 0 O O l 1210 11002 011002 in 2 s comp 11o 00012 111112in 2 s comp 2 s comp O l l O 0 Step 1 Take 2 s complement of 2nd operand Add l l l l 1 Step 2 Add binary numbers Step 3 Ignore carry bit Final Result Ignore Carry ENGN112 L3 More Number Systems September 8 2003 2 s Complement Subtraction Example 2 Let s compute 1310 510 131o 11o12 o11o12 51o o1o12 11o112 Adding these two 5bit codes 1 carry l l O l O l O O 0 Discarding the carry bit the si n bit is seen to be zero indicating a correct resul Indeed 010002 10002 810 ENGN112 L3 More Number Systems September 8 2003 2 s Complement Subtraction Example 3 Let s compute 510 12 1210 11002 1o1002 510 o1o12 oo1o12 Adding these two 5bit codes 0 O 1 O 1 11001 Here there is no carry bit and the sign bit is 1 This indicates a negative result which is what we expect 110012 710 ENGN112 L3 More Number Systems September 8 2003 Summary Bina numbers can also be represented in octal and hexa ecumal Easy to convert between binary octal and hexadecimal Signed numbers represented in signed magnitude 1 s complement and 2 s complement 2 s com lement most important only 1 representation for zero Important to understand treatment of sign bit for 1 s and 2 s complement 34 g ENGN112 L3 More Number Systems September 8 2003
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