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Click toToday’s edit Master title style Topics • Click to edit Master text styles • How information is encoded – Second level – signed •magnitude Third level – 1s complement – Fourth level » Fifth level – 2s complement • Base conversion • Arithmetic and logical operations 1 Click to edit Master title style Symbolic Representation of Voltage • Click edit text styles • Devices that to react to Master presence/absence of voltage–(within tolerance) are easier to Secondalevel build than•devices that react to a specific Third level voltage – Fourth level » Fifth level • We use 1 and 0 for presence/absence • bit = binary digit • 8 bits = 1 byte 2 Click toPositional edit Master title style Notation • Click to edit Master text styles 1 • 1 position: 0 or 1 (2 ) – Second level • Third level 2 • 2 positions:–00, 01, Fourth level10, 11 (2 ) » Fifth level • 3 positions: 000, 001, 010, 011, 100, 101, 110, 111 (23) A single wire can represent only one thing, a collection of k them can represent (2k) different things 3 Click to edit Master title style Data Types • Click to edit Master text styles • A data type is a set of values and – Second level operations over those values • Third level – Fourth level » Fifth level – ASCII codes for keyboard characters – 2s complement integers – others will be illustrated 4 Click Binary to edit Master title style or Base-2 Integers • Click to edit Master text styles • Decimal representation – Second level • 107 = 1x100+0x10+7 • Third level 1+ level Fourth • d2·102+ d1–·10 d0·100 » Fifth level • Binary representation • 107 = 01101011 • 0·27+ 1·26+ 1·25+ 0·24+ 1·23+ 0·22+ 1·21+ 1·20 • a7·27+ a6·26+ a5·25+ a4·24+ a3·23+ a2·22+ a1·21+ a0·20 5 Click toInteger edit Master title style Data Types • Click to edit Master text styles • Unsigned integers: non-negative values – Second level k from 0 to• 2Third -1 for k bits level – Fourthuse level half of the 2k • Signed integers: » Fifth level values for positive and half for negative – signed magnitude: leading bit represents the sign (0=positive) and remaining bits are the value – 1s complement: flip bits for negative value – 2s complement: 1s complement + 1 6 Click to edit Master title style 1s Complement • Click to edit Master text styles • Flip bits to represent negative values: – Second level • Third level – Fourth 4 islevel 0100 » Fifth level -4 is 1011 • Basic logic circuits for signed magnitude and 1s complement are more complex than 2s complement 7 Click to edit Master title style Motivation for 2s Complement • Click to edit Master text styles • From a device standpoint, it would be nice if levelthe representation for 0 adding–xSecond to -x yields level circuits that don’t care … then we• Third can build – Fourth level if the operands are positive or negative » Fifth level • Binary addition: 00011 00110 01001 8 ClickComputing to edit Master title style 2s Complement Click value to editfor Master textintegers styles • Use• binary positive – Second level • Compute representation for negative level integers: • Third – Fourth level – take 1s complement » Fifth level – add 1 • Example: what is 9 in binary? – what is its 1s complement value? – what is 1 + 1s complement value? – What do you get when you sum 9 and -9 in binary? 9 Click to edit Master title style Binary to Decimal Conversion • Click to edit Master text styles – Second level • Represent values from -2k-1 to 2k-1-1 with k • Third level bits – Fourth level » Fifth level • Algorithm: – if the leading bit is 1, apply 2s complement to obtain the magnitude of the negative number – sum the powers of 2 with coefficients of 1 10 Click to edit Master title style Decimal to Binary Conversion • For•positive do the computation Click numbers, to edit Master text styles below –and prefixlevel with a leading 0 Second • For negative numbers, • Third level use the absolute value in the computation below, add a leading 0, – Fourth level and take the 2s complement » Fifth level • Algorithm: – divide 2 into the dividend, write the remainder on the right – the quotient becomes the new dividend; repeat until the quotient is 0 – read the binary number from the bottom up 11 Click to edit Master title style Decimal to Binary Conversion • Example • Click to edit Master text styles Lets convert –124 to 8-bit 2’s complement binary representation – Second level 1) |-124|•=Third 124 level 2) Compute –binary of 124 Fourthrepresentation level 3) Add leading 0» and 2’s complement Fifthcompute level 2) 1242=62 remainder 0 3) 01111100 10000011 10000100 62 2=31 remainder 0 31 2=15 remainder 1 15 2=7 remainder 1 7 2=3 remainder 1 3 2=1 remainder 1 1111100 1 2=0 remainder 1 12 Click to edit Master title style Operations on Bits: Arithmetic • Click to edit Master text styles – Second level • Third level already … so we can • We can do addition – Fourth level do subtraction,» too Fifth level a - b = a + -b • Compute the 2s complement of b, add it to a 13 Click to edit Master title style Sign Extension • Click to edit Master text styles • To add –two numbers Second levelof different sizes, it is necessary to makelevel them the same length • Third – Fourth level » Fifth level • For positive numbers, pad with leading 0s • For negative numbers, pad with leading 1s • Does not change the values! 14 Click to editOverflow Master title style • Happens when of twotext positive • Click to the editsum Master styles numbers is largerlevel than the largest possible – Second positive number • Third level • Or alternately,– Fourth the sum of two negative level numbers is smaller than the smallest possible » Fifth level negative number • We check to see if we get a negative value when adding two positive numbers, or vice versa • Why is there never an overflow with the sum of a positive and negative value? 15 Click to edit Master title style Operations on Bits: Logical Ops • Click edit Master textOR, styles • Basic logic to functions: AND, NOT, XOR – Second level • Third level • Result from–treatment Fourth level of 0 as FALSE and 1 as TRUE » Fifth level • Truth tables: consider all possible combinations of input values and shows output values in last column • For n inputs, there are n+1 columns and 2n rows 16 Click to edit Master title style Truth table for the AND operator • Click to edit Master text styles a level – Second • Third 0 level b 0 – Fourth level 0 » Fifth level1 a AND b 0 0 1 0 0 1 1 1 17 Click toLogical edit Master title style Operations • • • • • Click to edit Master text styles AND: true when both inputs are 1, else false – Second level OR: true when either • Third level input is 1, else false Fourth level are the same, else true XOR: false if –both inputs » Fifth level NOT: invert/complement input • All ops can be applied to an entire bitstring (NOT) or pair of bitstrings (AND, OR, XOR) by corresponding positions 18 Click to edit Master title style Logic Functions • Click to edit Master text styles – Second level • Used for comparisons, e.g., if two integers • Thirdthe level are identical, output of XOR is all 0s – Fourth level » Fifth level • Basis for implementing all of the computer’s functionality! 19 Click toOther editRepresentations Master title style • Click to edit Master text styles – Second • Floating point: level real numbers using • Third level normalized scientific notation – Fourth level • ASCII: 256 possible symbols (keyboard » Fifth level input) • Hexadecimal: base 16 – each group of 4 bits is a hex digit – convenience for humans 20