# Chapter 2 Bits, Data Types, and Operations

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1 Chapter Bits, Data Types, and Operations How do we represent data in a computer? At the lowest level, a computer is an electronic machine. works by controlling the flow of electrons Easy to recognize two conditions:. presence of a voltage we ll call this state. absence of a voltage we ll call this state Could base state on value of voltage, but control and detection circuits more complex. compare turning on a light switch to measuring or regulating voltage -

2 Computer is a binary digital system. Digital system: finite number of symbols Binary (base two) system: has two states: and Basic unit of information is the binary digit, or bit. Values with more than two states require multiple bits. A collection of two bits has four possible states:,,, A collection of three bits has eight possible states:,,,,,,, A collection of n bits has n possible states. - What kinds of data do we need to represent? Numbers signed, unsigned, integers, floating point, complex, rational, irrational, Text characters, strings, Images pixels, colors, shapes, Sound Logical true, false Instructions Data type: representation and operations within the computer We ll start with numbers -

3 Unsigned Integers Non-positional notation could represent a number ( ) with a string of ones ( ) problems? Weighted positional notation like decimal numbers: 9 is worth, because of its position, while 9 is only worth 9 most significant 9 least significant x + x + 9x = 9 x + x + x = - Unsigned Integers (cont.) An n-bit unsigned integer represents n values: from to n -. -

4 Unsigned Binary Arithmetic Base- addition just like base-! add from right to left, propagating carry carry Subtraction, multiplication, division, - Signed Integers With n bits, we have n distinct values. assign about half to positive integers ( through n- ) and about half to negative (- n- through -) that leaves two values: one for, and one extra Positive integers just like unsigned zero in most significant (MS) bit = Negative integers sign-magnitude set MS bit to show negative, other bits are the same as unsigned = - one s complement flip every bit to represent negative = - in either case, MS bit indicates sign: =positive, =negative -

5 Two s Complement Problems with sign-magnitude and s complement two representations of zero (+ and ) arithmetic circuits are complex How to add two sign-magnitude numbers? e.g., try + (-) How to add to one s complement numbers? e.g., try + (-) Two s complement representation developed to make circuits easy for arithmetic. for each positive number (X), assign value to its negative (-X), such that X + (-X) = with normal addition, ignoring carry out () (9) + (-) + (-9) () () -9 Two s Complement Representation 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 () (9) ( s comp) ( s comp) + + (-) (-9) -

6 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 ( s comp) (flip) + (copy) - Two s Complement Signed Integers MS bit is sign bit it has weight n-. Range of an n-bit number: - n- through n-. The most negative number (- n- ) has no positive counterpart

7 Converting Binary ( s C) to Decimal. If leading bit is one, take two s complement to get a positive number.. Add powers of that have in the corresponding bit positions.. If original number was negative, add a minus sign. X = two = + + = ++ = ten n 9 n Assuming -bit s complement numbers. - More Examples X = two = = +++ = 9 ten X = two -X = = + + = ++ = ten X = - ten n 9 n Assuming -bit s complement numbers. -

8 Converting Decimal to Binary ( s C) First Method: Division. Find magnitude of decimal number. (Always positive.). Divide by two remainder is least significant bit.. Keep dividing by two until answer is zero, writing remainders from right to left.. Append a zero as the MS bit; if original number was negative, take two s complement. X = ten / = r bit / = r bit / = r bit / = r bit / = r bit / = r bit X = two / = r bit - Converting Decimal to Binary ( s C) Second Method: Subtract Powers of Two. Find magnitude of 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.. Append a zero as MS bit; if original was negative, take two s complement. n 9 n X = ten - = bit - = bit - = bit X = two -

9 Operations: Arithmetic and Logical Recall: a data type includes representation and operations. We now have a good representation for signed integers, so let s look at some arithmetic operations: Addition Subtraction Sign Extension We ll also look at overflow conditions for addition. Multiplication, division, etc., can be built from these basic operations. Logical operations are also useful: AND OR NOT - Addition As we ve discussed, s comp. addition is just binary addition. assume all integers have the same number of bits ignore carry out for now, assume that sum fits in n-bit s comp. representation () (-) + (-) + (-9) (9) (-9) Assuming -bit s complement numbers. - 9

10 Subtraction Negate subtrahend (nd no.) and add. assume all integers have the same number of bits ignore carry out for now, assume that difference fits in n-bit s comp. representation () (-) - () - (-9) () (-) + (-) + (9) () (-) Assuming -bit s complement numbers. -9 Sign Extension To add two numbers, we must represent them with the same number of bits. If we just pad with zeroes on the left: -bit -bit () (still ) (-) (, not -) Instead, replicate the MS bit -- the sign bit: -bit -bit () (still ) (-) (still -) -

11 Overflow If operands are too big, then sum cannot be represented as an n-bit s comp number. () (-) + (9) + (-9) (-) (+) We have overflow if: signs of both operands are the same, and sign of sum is different. Another test -- easy for hardware: carry into MS bit does not equal carry out - Logical Operations Operations on logical TRUE or FALSE two states -- takes one bit to represent: TRUE=, FALSE= A B A AND B A B A OR B A NOT A View n-bit number as a collection of n logical values operation applied to each bit independently -

12 Examples of Logical Operations AND useful for clearing bits AND with zero = AND with one = no change AND OR useful for setting bits OR with zero = no change OR with one = OR NOT unary operation -- one argument flips every bit NOT - 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 corrupt long string of s and s Binary Hex Decimal Binary Hex Decimal 9 9 A B C D E F -

13 Converting from Binary to Hexadecimal Every four bits is a hex digit. start grouping from right-hand side A F D This is not a new machine representation, just a convenient way to write the number. - Fractions: Fixed-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 - =. - =. - =.. (.) +. (-.). (9.) No new operations -- same as integer arithmetic. -

14 Very Large and Very Small: Floating-Point Large values:. x -- requires 9 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): b b b S Exponent Fraction N = (" ) N = (" ) S S!.fraction!!.fraction! exponent" ", # exponent #, exponent = - Floating Point Example Single-precision IEEE floating point number: sign exponent fraction Sign is number is negative. Exponent field is = (decimal). Fraction is. =. (decimal). Value = -. x (-) = -. x - = -.. -

15 Floating-Point Operations Will regular s complement arithmetic work for Floating Point numbers? (Hint: In decimal, how do we compute. x + 9. x?) -9 Text: ASCII Characters ASCII: Maps characters to -bit code. both printable and non-printable (ESC, DEL, ) characters nul dle soh dc stx dc etx dc eot dc enq nak ack syn bel etb bs can 9 ht 9 em 9 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 sp! " # \$ % & ' ( ) * +, -. / 9 a b c d e f 9 : ; < = >? 9 a b c d e A B C D E F G H I J K L M N O 9 a b c d e f P Q R S T U V W X Y Z [ \ ] ^ _ 9 a b c d e f ` a b c d e f g h i j k l m n o 9 a b c d e f p q r s t u v w x y z { } ~ del -

16 Interesting Properties of ASCII Code What is relationship between a decimal digit ('', '', ) and its ASCII code? What is the difference between an upper-case letter ('A', 'B', ) and its lower-case equivalent ('a', 'b', )? Given two ASCII characters, how do we tell which comes first in alphabetical order? Are characters enough? ( No new operations -- integer arithmetic and logic. - Other Data Types Text strings sequence of characters, terminated with NULL () typically, no hardware support Image array 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 -

17 LC- Data Types Some data types are supported directly by the instruction set architecture. For LC-, there is only one hardware-supported data type: -bit s complement signed integer Operations: ADD, AND, NOT Other data types are supported by interpreting -bit values as logical, text, fixed-point, etc., in the software that we write. -

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