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1、Chapter 2 Number systems and codes Unsigned and Signed numbersBinary numbers with certain width;The numbers always be set as 0.xxxxxxxx;We can take these numbers as integers for operations!MSB/LSB Unsigned numberOperations for unsigned numbers0+0=0 1+0=1 1+1=10AdditionC:carry S:sum 0A=0 1A=A Multipl
2、ication Shift-add:use adders and shift-registers;The product width will be 2n!Operations for unsigned numbersNew rule differ from addMaybe get negative number subtraction Should use signed number to express positive and negativeOperations for unsigned numbersOnce the signed numbers be used,subtracti
3、on can be made by add!Operations for unsigned numbersSubtraction rule is not necessary!Signed-magnitude code(S-M code):Signed numberSigned-magnitude code(S-M code):Add may get wrong result!Signed numberTwos complement code:Signed numberTwos complement code:Change the sign means change every bit,and
4、add 1 at LSBSigned numberConverting rules for signed numbersFrom unsigned to signed:Its MSB is changed to sign bit.The number must be a positive!Beware the overflow!Key point:A positive signed number with any code must be the same!Its MSB is“0”!From one code to another Step 1:Check the MSB(sign-bit)
5、;For positive number(MSB=0):no changes!Step 2:For negative number(MSB=1):first change its sign by the former code way;then change its sign by the next code way!Examples:From one code to another 1two numbers must be the same width,and keep the width in addition!2subtract a number means add the negati
6、ve number;Examples:1110+1010=1000 0110 0011=0110+1101=0011 Addition for signed numbersOverflow:the result exceeds the number ranges;condition:two numbers has the same sign;checking:the sign is wrong!solving:increase the data width!Examples:0101+0101=1010 1101+1010=0111Addition for signed numbersMultiplication for signed numbersOnly S-Ms code can be easy done!The sign-bit of the result is the XOR of the sign-bits of the inputs!The value-bits of the result is decided by the multiplication rules for the unsigned numbers!