Chapter 2 Bits, Data Types, and Operations

Transcription

1 How do we represent data in a computer? t the lowest level, a computer has electronic plumbing Operates by controlling the flow of electrons Chapter Bits, Data Types, and Operations Easy to recognize two conditions:. Presence of a voltage we ll call this state. bsence of a voltage we ll call this state Based on slides McGraw-Hill dditional material // Lewis/Martin lternative: Base state on value of voltage On/Off light switch versus dimmer switch Problem: Control/detection circuits more complex CSE - Computer is a binary digital system What kinds of data do we need to represent? Digital system: Finite number of symbols (base two) system: Has two states: and Basic unit of information: the binary digit, or bit + state values require multiple bits collection of two bits has four possible states:,,, Numbers signed, unsigned, integers, real, floating point, complex, rational, irrational, Text characters, strings, Images pixels, colors, shapes, Sound Logical true, false Instructions collection of three bits has eight possible states:,,,,,,, collection of n bits has n possible states Data type: Representation and operations within the computer side: why binary? We ll start with numbers CSE - CSE -

2 Unsigned Integers Non-positional notation Could represent a number ( ) with a string of ones ( ) Problems? Weighted positional notation Like decimal numbers: is worth, because of its position, while is only worth base- (decimal) x + x + x = most significant least significant x + x + x = base- (binary) Unsigned Integers (cont.) n n-bit unsigned integer represents n values From to n - val CSE - CSE - N = Number Represented Unsigned Unsigned rithmetic From GTech s CS, Kishore Ramachandran Used with permission Base- addition just like base-! dd from right to left, propagating carry CSE carry () () + () + () + () () + Subtraction, multiplication, division, () () () () () () -

3 Signed Integers With n bits, we have n distinct values ssign half to positive integers ( through ~ n- ) and half to negative (~- n- through -) That leaves two values: one for, and one extra Positive integers Just like unsigned with zero in most significant bit = Negative integers Sign-magnitude: set high-order bit to show negative, other bits are the same as unsigned = - One s complement: flip every bit to represent negative = - In either case, most significant bit indicates sign:!=positive, =negative CSE - N = Number Represented Unsigned Signed Mag From GTech s CS, Kishore Ramachandran Used with permission N = Number Represented Unsigned Signed Mag 's Comp From GTech s CS, Kishore Ramachandran Used with permission Problem Signed-magnitude and s complement Two representations of zero (+ and ) rithmetic circuits are complex!how do we add two sign-magnitude numbers? CSE e.g., try + (-) () + (-) (-)?!How do we add to one s complement numbers? e.g., try + (-) ()? + (-) () -

4 Two s Complement Idea Find representation to make arithmetic simple and consistent Specifics For each positive number (X), assign value to its negative (-X), such that X + (-X) = with normal addition, ignoring carry out () () + (-) + (-) () () Two s Complement (cont.) If number is positive or zero Normal binary representation, zeroes in upper bit(s) If number is negative Start with positive number Flip every bit (i.e., take the one s complement) Then add one () () ( s comp) ( s comp) + + (-) (-) CSE - CSE - Two s Complement Shortcut To take the two s complement of a number: Copy bits from right to left until (and including) the first Flip remaining bits to the left Two s Complement Signed Integers MS bit is sign bit: it has weight n- Range of an n-bit number: - n- through n- Note: most negative number (- n- ) has no positive counterpart CSE ( s comp) (flip) + (copy) - CSE

5 N = Number Represented Unsigned Signed Mag 's Comp 's Comp Two s complement is used by all modern computers From GTech s CS, Kishore Ramachandran Used with permission Converting ( s C) to Decimal. If leading bit is one, take two s complement to get a positive number. dd powers of that have in the corresponding bit positions. If original number was negative, add a minus sign CSE X = two = + + = ++ = ten ssuming -bit s complement numbers. n n - More Examples Converting Decimal to ( s C) X = two = = +++ = ten X = two -X = = + + = ++ = ten X = - ten ssuming -bit s complement numbers. n n First Method: Division. Change to positive decimal number. Divide by two remainder is least significant bit. Keep dividing by two until answer is zero, recording remainders from right to left. ppend a zero as the MS bit; if original number negative, take two s complement X = ten / = r bit / = r bit / = r bit / = r bit / = r bit / = r bit / = r bit X= two CSE - CSE -

6 Converting Decimal to ( s C) Second Method: Subtract Powers of Two. Change to positive decimal number. Subtract largest power of two less than or equal to number. Put a one in the corresponding bit position. Keep subtracting until result is zero. ppend a zero as MS bit; if original was negative, take two s complement CSE X = ten - = bit - = bit - = bit X = two n n - Operations: rithmetic and Logical Recall data type includes representation and operations Operations for signed integers ddition Subtraction Sign Extension Logical operations are also useful ND OR NOT nd... Overflow conditions for addition CSE - ddition s comp. addition is just binary addition ssume all integers have the same number of bits Ignore carry out For now, assume that sum fits in n-bit s comp. representation Subtraction Negate nd operand and add ssume all integers have the same number of bits Ignore carry out For now, assume that difference fits in n-bit s comp. representation () (-) + (-) + (-) () (-) () (-) - () - (-) () (-) + (-) + () () (-) ssuming -bit s complement numbers. ssuming -bit s complement numbers. CSE - CSE -

7 Sign Extension To add Must represent numbers with same number of bits What if we just pad with zeroes on the left? -bit -bit () (still ) (-) (, not -) No, let s replicate the MSB (the sign bit) -bit -bit () (still ) (-) (still -) Overflow What if operands are too big? Sum cannot be represented as n-bit s comp number () (-) + () + (-) (-) (+) We have overflow if Signs of both operands are the same, and Sign of sum is different nother test (easy for hardware) Carry into most significant bit does not equal carry out CSE - CSE - Logical Operations Operations on logical TRUE or FLSE Two states: TRUE=, FLSE= B ND B B OR B NOT Examples of Logical Operations ND Useful for clearing bits!nd with zero =!ND with one = no change OR Useful for setting bits!or with zero = no change!or with one = ND OR View n-bit number as a collection of n logical values Operation applied to each bit independently NOT Unary operation -- one argument Flips every bit NOT CSE - CSE -

8 Hexadecimal Notation It is often convenient to write binary (base-) numbers as hexadecimal (base-) numbers instead Fewer digits: four bits per hex digit Less error prone: easy to misread long string of s and s Hex Decimal Hex B C D E F Decimal Converting from to Hexadecimal Every group of four bits is a hex digit Start grouping from right-hand side This is not a new machine representation, just a convenient way to write the number. F D CSE - CSE - Fractions: Fixed-Point Very Large and Very Small: Floating-Point How can we represent fractions? Use a binary point to separate positive from negative powers of two (just like decimal point ) s comp addition and subtraction still work!if binary points are aligned - =. CSE - =. - =.. (.) +. (-.). (.) No new operations -- same as integer arithmetic - Problem Large values:. x -- requires bits Small values:. x - -- requires > bits Use equivalent of scientific notation : F x E Need to represent F (fraction), E (exponent), and sign IEEE Floating-Point Standard (-bits): N = " N = " CSE S S b b b S Exponent!.fraction!!.fraction! " Fraction exponent", # exponent #, exponent = -

9 Floating Point Example Single-precision IEEE floating point number - bits Sign is : number is negative Exponent field is = (decimal) Fraction is. = / =. (decimal) Value = -. x (-) = -. x - = -. Double-precision IEEE floating point - bits -bit exponent field, -bit fraction field CSE sign exponent fraction - Floating Point Specials S N = " S N = " b b b S Exponent!.fraction!!.fraction! " Fraction exponent", exponent = If exponent bits are, denormalized numbers Gradual underflow (also used for representing zero) Other specials Two zeros (-, ) Two Infinities (-infinity, infinity) Not a number (negative and positive)! When does this occur? Lots of corner cases (difficult to implement correctly) Example: rounding modes CSE, # exponent # - Floating-Point Operations Text: SCII Characters Will regular s complement arithmetic work for Floating Point numbers? (Hint: In decimal, how do we compute. x +. x?) CSE - SCII: Maps characters to -bit code. Both printable and non-printable (ESC, DEL, ) characters CSE nul dle sp soh dc! stx dc " etx dc # eot dc $ enq nak % ack syn & bel etb ' bs can ( ht em ) a nl a sub a * b vt b esc b + c np c fs c, d cr d gs d - e so e rs e. f si f us f / a b c d e f : ; < = >? a b c d e B C D E F G H I J K L M N O a b c d e f P Q R S T U V W X Y Z [ \ ] ^ _ a b c d e f ` a b c d e f g h i j k l m n o p q r s t u v w x y a z b { c d } e ~ f del -

10 Interesting Properties of SCII Code What is relationship between a decimal digit ('', '', ) and its SCII code? What is the difference between an upper-case letter ('', 'B', ) and its lower-case equivalent ('a', 'b', )? Given two SCII characters, how do we tell which comes first in alphabetical order? re characters enough? ( CSE No new operations -- integer arithmetic and logic. - Other Data Types Text strings Sequence of characters, terminated with NULL () Typically, no hardware support Image rray of pixels!monochrome: one bit (/ = black/white)!color: red, green, blue (RGB) components (e.g., bits each)!other properties: transparency Hardware support!typically none, in general-purpose processors!mmx: multiple -bit operations on -bit word Sound Sequence of fixed-point numbers CSE - LC- Data Types Some data types are supported directly by the instruction set architecture For LC-, there is only one supported data type -bit s complement signed integer Operations: DD, ND, NOT (and sometimes MUL) Other data types? Supported by interpreting -bit values as logical, text, fixedpoint, etc., in the software that we write Next Time Lecture Digital logic structures: transistors and gates Reading Chapter -. Quiz Online Upcoming HW due this Friday! CSE - CSE -