Bits, Data Types, and Operations

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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Agenda 1. Data Types 2. Unsigned & Signed Integers 3. Arithmetic Operations Chapter 2 4. Logical Operations 5. Shifting Bits, Data Types, 6. Hexadecimal & Octal Notation and Operations 7. Other Data Types

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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 1. Data Types Computer is a binary digital system. How do we represent data in a computer? Digital system: Binary (base two) system: At the lowest level, a computer is an electronic machine. • finite number of symbols • has two states: 0 and 1 • works by controlling the flow of electrons

Easy to recognize two conditions: 1. presence of a voltage – we’ll call this state “1” 2. absence of a voltage – we’ll call this state “0” 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: Could base state on value of voltage, 00, 01, 10, 11 but control and detection circuits more complex. • A collection of three bits has eight possible states: • compare turning on a light switch to 000, 001, 010, 011, 100, 101, 110, 111 measuring or regulating voltage • A collection of n bits has 2n possible states. COMPSCI210 S1C 2009 3 COMPSCI210 S1C 2009 4

1 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. What kinds of data do we need to represent? 2. Unsigned & Signed Integers

Decimal Numbers • Numbers – signed, unsigned, integers, floating point, complex, rational, irrational, … “decimal” means that we have ten digits to use in our • Text – characters, strings, … representation (the symbols 0 through 9) • Images – pixels, colors, shapes, … • Sound What is 3,546? • Logical – true, false • it is three thousands plus five hundreds plus four tens plus six ones. • Instructions • i.e. 3,546 = 3.103 + 5.102 + 4.101 + 6.100 • … How about negative numbers? • we use two more symbols to distinguish positive and negative: Data type: + and – • representation and operations within the computer Note that not everyone uses base 10 We’ll start with numbers… • Vigesimal (base 20)http://en.wikipedia.org/wiki/Vigesimal

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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Unsigned Integers Unsigned Binary Integers Non-positional notation Y = “abc” = a.22 + b.21 + c.20 • could represent a number (“5”) with a string of ones (“11111”) • problems? (where the digits a, b, c can each take on the values of 0 or 1 only)

N = number of bits 3-bits 5-bits 8-bits Weighted positional notation 0 000 00000 00000000 • like decimal numbers: “329” Range is: • “3” is worth 300, because of its position, while “9” is only worth 9 0  i < 2N - 1 1 001 00001 00000001 2 010 00010 00000010 most least significant significant 3 011 00011 00000011 329 101 Problem: 4 100 00100 00000100 102 101 100 22 21 20 • How do we represent 3x100 + 2x10 + 9x1 = 329 1x4 + 0x2 + 1x1 = 5 negative numbers?

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2 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Unsigned Binary Arithmetic Signed Integers Base-2 addition – just like base-10! With n bits, we have 2n distinct values. • add from right to left, propagating carry • assign about half to positive integers (1 through 2n-1) and about half to negative (- 2n-1 through -1) carry • that leaves two values: one for 0, and one extra Positive integers • just like unsigned – zero in most significant (MS) bit 00101 = 5 Negative integers • sign-magnitude – set MS bit to show negative, other bits are the same as unsigned 10101 = -5 -1 * (0×23 + 1×22 + 0×21 + 1×20) • one’s complement – flip every bit to represent negative 11010 = -5 (-00101) Subtraction, multiplication, division,… • in either case, MS bit indicates sign: 0=positive, 1=negative COMPSCI210 S1C 2009 9 COMPSCI210 S1C 2009 10

Y = “abc” = (-1)a (b.21 + c.20) -1 101 If msb (most significant bit) is 1 then the -1 110 -0 100 number is negative (same as signed -0 111 Range is: magnitude) -2N-1 + 1 <= i <= 2N-1 - 1 +0 000 +0 000 +1 001 Range is: +1 001 E.g. 3 bits: -3 <= i <= 3 +2 010 -2N-1 + 1 <= i <= 2N-1 - 1 +2 010 Problems: +3 011 +3 011 E.g. 3 bits: -3 <= i <= 3 • How do we do addition/subtraction? Problems: • We have two numbers for zero (+/-)! •Same as for signed magnitude

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3 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Two’s Complement Two’s Complement Representation Problems with sign-magnitude and 1’s complement If number is positive or zero, • two representations of zero (+0 and –0) • normal binary representation, zeroes in upper bit(s) • arithmetic circuits are complex If number is negative, How to add two sign-magnitude numbers? – e.g., try 2 + (-3) • start with positive number How to add to one’s complement numbers? • flip every bit (i.e., take the one’s complement) – e.g., try 4 + (-3) • then add one Two’s complement representation developed to make circuits easy for arithmetic. (5) (9) • for each positive number (X), assign value to its negative (-X), such that X + (-X) = 0 with “normal” addition, ignoring carry out (1’s comp) (1’s comp)

(5) (9) (-5) (-9) (-5) (-9) (0) (0) COMPSCI210 S1C 2009 13 COMPSCI210 S1C 2009 14

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Two’s Complement Shortcut Two’s Complement Signed Integers MS bit is sign bit – it has weight –2n-1. To take the two’s complement of a number: n-1 n-1 • copy bits from right to left until (and including) the first “1” Range of an n-bit number: -2 through 2 – 1. • flip remaining bits to the left • The most negative number (-2n-1) has no positive counterpart. -23 22 21 20 -23 22 21 20 0 0 0 0 0 1 0 0 0 -8 0 0 0 1 1 1 0 0 1 -7 (1’s comp) (flip) (copy) 0 0 1 0 2 1 0 1 0 -6 0 0 1 1 3 1 0 1 1 -5 0 1 0 0 4 1 1 0 0 -4 0 1 0 1 5 1 1 0 1 -3 0 1 1 0 6 1 1 1 0 -2 0 1 1 1 7 1 1 1 1 -1

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4 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Applet: Bin2Dec.html Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Converting Binary (2’s C) to Decimal More Examples 1. If leading bit is one, take two’s complement to get a positive number. X = 00100111two 5 2 1 0 2. Add powers of 2 that have “1” in the n 2n = 2 +2 +2 +2 = 32+4+2+1 n 2n corresponding bit positions. 0 1 = 39 0 1 1 2 ten 1 2 3. If original number was negative, 2 4 2 4 add a minus sign. 3 8 X = 11100110 3 8 4 16 two 4 16 5 32 -X = 00011010 5 32 4 3 1 X = 01101000two 6 64 = 2 +2 +2 = 16+8+2 6 64 6 5 3 7 128 7 128 = 2 +2 +2 = 64+32+8 8 256 = 26ten 8 256 = 104ten 9 512 X = -26 9 512 10 1024 ten 10 1024

Assuming 8-bit 2’s complement numbers. Assuming 8-bit 2’s complement numbers. COMPSCI210 S1C 2009 17 COMPSCI210 S1C 2009 18

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Applet: Dec2Bin.html Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

Converting Decimal to Binary (2’s C) Converting Decimal to Binary (2’s C) n 2n First Method: Division Second Method: Subtract Powers of Two 0 1 1. Find magnitude of decimal number. (Always positive.) 1 2 1. Find magnitude of decimal number. 2 4 2. Divide by two – remainder is least significant bit. 2. Subtract largest power of two 3 8 3. Keep dividing by two until answer is zero, less than or equal to number. 4 16 writing remainders from right to left. 5 32 4. Append a zero as the MS bit; 3. Put a one in the corresponding bit position. 6 64 if original number was negative, take two’s complement. 4. Keep subtracting until result is zero. 7 128 8 256 5. Append a zero as MS bit; 9 512 X = 104ten 104/2 = 52 r0 bit 0 52/2 = 26 r0 bit 1 if original was negative, take two’s complement. 10 1024 26/2 = 13 r0 bit 2 13/2 = 6 r1 bit 3 X = 104ten 104 - 64 = 40 bit 6 6/2 = 3 r0 bit 4 40 - 32 = 8 bit 5 3/2 = 1 r1 bit 5 8 - 8 = 0 bit 3 X = 01101000 1/2 = 0 r1 bit 6 two X = 01101000two COMPSCI210 S1C 2009 19 COMPSCI210 S1C 2009 20

5 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Examples: (Decimal -> Binary) Examples: (Binary -> Decimal)

Convert –2310 to binary numbers in Convert 10101010 to Decimal if the number is • 8-bit, Signed Magnitude represented as 10010111 • 8-bit, Signed Magnitude 10101010 -> Negative numbers • 8-bit, One’s complement = -42 23 -> Binary: 00010111 One’s complement -> 11101000 • Signed 8-bit in two’s complement 10101010 -> Negative numbers => Complement it = 01010101 = 85 • 8-bit, Two’s complement: => Add 1 = 86 23 -> Binary: 00010111 = -86 One’s complement -> 11101000 Two’s complement -> 11101001 • Unsigned 8-bit 10101010 = 170

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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Exercises (Decimal -> Binary) Exercises (Binary -> Decimal) Convert Decimal –17 to 8-bit binary numbers in Convert 10110011 to Decimal if the number is represented • Signed Magnitude as • Signed Magnitude

• One’s complement: • Signed 8-bit two’s complement

• Two’s complement: • Unsigned 8-bit

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6 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Applet: NumOps.html 3. Arithmetic Operations Addition Recall: As we’ve discussed, 2’s comp. addition is just a data type includes representation and operations. binary addition. We now have a good representation for signed integers, • assume all integers have the same number of bits so let’s look at some arithmetic operations: • ignore carry out • Addition • for now, assume that sum fits in n-bit 2’s comp. representation • Subtraction • Sign Extension We’ll also look at overflow conditions for addition. (104) (-10) Multiplication, division, etc., can be built from these basic operations. (-16) (-9) (98) (-19)

Assuming 8-bit 2’s complement numbers. COMPSCI210 S1C 2009 25 COMPSCI210 S1C 2009 26

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Subtraction Exercise Negate subtrahend (2nd no.) and add. 01101011 – 00010110 • assume all integers have the same number of bits • ignore carry out • for now, assume that difference fits in n-bit 2’s comp. representation

(104) (-10) 00101100 – 01101001 (16 (-9) (104) (-10) (-16) (9) (88) (-1)

Assuming 8-bit 2’s complement numbers. COMPSCI210 S1C 2009 27 COMPSCI 105 28

7 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Sign Extension Sign extension: How it works To add two numbers, we must represent them The sign bit is replicated: with the same number of bits. 0x9876: 1001 1000 0111 0110: -2^15 + 2^12+…… If we just pad with zeroes on the left: 0xff9876: 1111 1111 1001 1000 0111 0110: 4-bit 8-bit (4) (still 4) -2^23+2^22+2^21+2^20+2^19+…+2^15+2^12+…. = (-4) (12, not -4) -2^22 +2^21+………………………………… = -2^21 +2^20+………………………….. =

Instead, replicate the MS bit -- the sign bit: 2^16+2^15+2^12+………. = -2^15+2^12+………. = 0x9876 4-bit 8-bit By the same way you can demonstrate that the quadword 0xff…ff9876 is (4) (still 4) equal to the word 0x9876. Using 2’s representation convention you can (-4) (still -4) also see that the quadword 0xff…ff9876 (which is a negative number) is equal to the word –0x678a

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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Overflow Examples (4-bit) If operands are too big, then sum cannot be represented Assuming 4-bit 2’s complement numbers as an n-bit 2’s comp number. • Range - 8 <= x <= 7 -6 -6 + -7 = -13 1010 (outside the range) -7 6 6 + 7 = 13 + 1001 (8) (-8) 0110 (outside the range) 7 10000 carries (9) (-9) + 0111 01100 carries 10011(4-bit)= 0011 (-15) (+15) 01101 (4-bit) => 1101 Answer = 3 Answer = -3 Invalid answer We have overflow if: Invalid answer 2 • signs of both operands are the same, and -2 -2 + 3 = 1 0010 2 + 3 = 5 1110 3 • sign of sum is different. 3 +0011 + 0011 Answer = 1 0100carries Another test -- easy for hardware: 11100 carries valid answer • carry into MS bit does not equal carry out 0101 Answer = 5 10001 (4-bit)=0001 Valid answer

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8 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Valid or invalid? Examples Overflow occurs if • Look at the carries into and out of the sign bit. If they • +ve number add +ve number => -ve number are not equal, there is an overflow. • -ve number add –ve number => +ve number 0110 1010 Carry out of Example: Carry into + 0111 + 1001 the sign bit the sign bit • 6 + 7 = -3 (actual answer = 13) 01100 carries 10000 carries • -6 + -7 = 3 (actual answer = -13) 01101 10011

A better way: 1110 0010 • Look at the carries into and out of the sign bit. If they are not Carry into & Carry out + 0011 + 0011 equal, there is an overflow. No carry 11100 carries 00100 carries • Not equal: -> overflow 10001 0101 • Equal = No Overflow

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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Applet: BinOps.html Exercises 4. Logical Operations 0111 Logical operations are also useful: 1001 • AND + 0101 + 1100 • OR • NOT • XOR

1100 1001 + 0111 + 0011

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9 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Logical Operations Examples of Logical Operations Operations on logical TRUE or FALSE AND • useful for clearing bits • two states -- takes one bit to represent: TRUE=1, FALSE=0 AND AND with zero = 0 A NOT A A B A OR B AND with one = no change 0 1 0 0 0 1 0 0 1 1 OR 1 0 1 A B A AND B A B A XOR B • useful for setting bits OR with zero = no change OR 0 0 0 1 1 1 0 0 0 OR with one = 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 1 1 1 0 View n-bit number as a collection of n logical values • operation applied to each bit independently COMPSCI210 S1C 2009 37 COMPSCI210 S1C 2009 38

XOR • Same = 0 • Different = 1 XOR

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10 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. -43 5. Shifting Sll << x 1 1 0 1 0 1 0 1 -86 Shift operators move bits to the left or to the right. Examples 0 • Used to shift the bit patterns left and right. • 1101 0101 << 1 1 0 1 0 1 0 1 0 • Shift corresponds to division/multiplication by powers of 2 (no • 0001 0101 << 2 21 overflow problem) • Three shift instructions: “ sll” (shift left logical), “ sra” (shift right • 0000 0101 << 3 0 0 0 1 0 1 0 1 arithmetic), and “srl” (shift right logical), corresponding to <<, >> 84 and >>>. Exercise 0 1 0 1 0 1 0 0 • The shift logical instructions fill the vacated bits with 0 • 0000 1100 << 3 • The shift right arithmetic instruction fills the vacated 5 bits with the sign bit. 0 0 0 0 0 1 0 1 • These instructions can be used to extract fields out of a 40 bit pattern, and interpret them as either unsigned or 0 0 1 0 1 0 0 0 signed numbers.

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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. -43 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. SRA (sign fill) >> x 1 1 0 1 0 1 0 1 SRL (zero fill) >>> x 1 1 0 1 0 1 0 1 Examples -22 Example 1 1 1 0 1 0 1 0 0 1 1 0 1 0 1 0 • 1101 0101 >> 1 • 11010101 >>> 1 • 0001 0101 >> 2 21 • 0001 0101 >>> 2 22 • 1001 0011 >> 3 x 0 0 0 1 0 1 0 1 • 1001 0011 >>> 2 x 0 0 0 1 0 1 0 1

5 5 Exercise 0 0 0 0 0 1 0 1 Exercise 0 0 0 0 0 1 0 1 • 1010 0001 >> 3 • 1010 0001 >>> 3 -109 -109 x 1 0 0 1 0 0 1 1 x 1 0 0 1 0 0 1 1 -14 36 1 1 1 1 0 0 1 0 0 0 1 0 0 1 0 0 -14 -13 5 6 5.25 -13.625 COMPSCI210 S1C 2009 43 COMPSCI210 S1C 2009 44

11 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 6. Hexadecimal & Octal Notation 6. Hexadecimal & Octal Notation Hexadecimal Notation Octal Notation It is often convenient to write binary (base-2) numbers It is often convenient to write binary (base-2) numbers as hexadecimal (base-16) numbers instead. as octal (base-8) numbers instead. • fewer digits -- four bits per hex digit • fewer digits – three bits per octal digit • less error prone -- easy to corrupt long string of 1’s and 0’s Binary Octal Decimal 000 0 0 Binary Hex Decimal Binary Hex Decimal 001 1 1 0000 0 0 1000 8 8 010 2 2 0001 1 1 1001 9 9 011 3 3 0010 2 2 1010 A 10 100 4 4 0011 3 3 1011 B 11 101 5 5 0100 4 4 1100 C 12 110 6 6 0101 5 5 1101 D 13 111 7 7 0110 6 6 1110 E 14 0111 7 7 1111 F 15 COMPSCI210 S1C 2009 45 COMPSCI210 S1C 2009 46

Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Converting from Decimal to Octal/Hexadecimal Converting from Octal/Hexadecimal to Decimal Decimal -> Octal Octal to Decimal • 127 • 122 8 = 1 x 82 + 2 x 81 + 7 x 80 = 64 + 16 + 7 = 87

• 220 • 2358 = 2 x 82 + 3 x 81 + 5 x 80 = 128 + 24 + 5 = 157 Decimal -> Hexadecimal • 1940 Hexadecimal to Decimal

• 1A716 = 1 x 162 + 10 x 161 + 7 x 160 • 175 = 256 + 160 + 7 = 423

• 23E16 = 2 x 162 + 3 x 161 + 14 x 160 = 512 + 48 + 14 = 574

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12 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Exercise Binary -> Octal/Hexadecimal Decimal -> Octal Binary -> Octal • 1234 • Every three bits is a octal digit • Start grouping from right-hand side Decimal -> Hexadecimal • 0101 1111 • 689 • 1101 0001 • Exercise: 101101001 Octal to Decimal • 151 Binary -> Hexadecimal • Every four bits is a hex digit Hexadecimal to Decimal • Start grouping from right-hand side • 0101 1111 • 15B • 1101 0001 • Exercise: 101001101

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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Octal -> Binary/Hexadecimal Hexadecimal -> Binary/Octal Octal -> Binary Hexadecimal -> Binary • 363 • EA3 • 247 • 2A7 • Exercise: 123 • Exercise: 1F3

Octal -> Hexadecimal Hexadecimal -> Octal • EA3 • 363 • 2A7 • 247 • Exercise: 1F3 • Exercise: 123

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13 Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Text: ASCII Characters Interesting Properties of ASCII Code •What is relationship between a decimal digit ('0', '1', …) ASCII: Maps 128 characters to 7-bit code. and its ASCII code? • both printable and non-printable (ESC, DEL, …) characters •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?

Minimal memorizing effort allows you to retain more than 50 character ASCII code!!

Are 128 characters enough? (http://www.unicode.org/) No new operations -- integer arithmetic and logic.

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Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Other Data Types LC-3 Data Types Text strings Some data types are supported directly by the • sequence of characters, terminated with NULL (0) instruction set architecture. • typically, no hardware support Image For LC-3, there is only one hardware-supported data type: • array of pixels • 16-bit 2’s complement signed integer monochrome: one bit (1/0 = black/white) • Operations: ADD, AND, NOT color: red, green, blue (RGB) components (e.g., 8 bits each) other properties: transparency Other data types are supported by interpreting • hardware support: 16-bit values as logical, text, fixed-point, etc., typically none, in general-purpose processors in the software that we write. MMX -- multiple 8-bit operations on 32-bit word Sound • sequence of fixed-point numbers

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