1. Introduction to Computer Engineering ECE/CS 252, Fall 2010 Prof. Mikko Lipasti Department of Electrical and Computer Engineering University of Wisconsin – Madison

2. Chapter 2 Bits, Data Types, and Operations - Part 1 Slides based on set prepared by Gregory T. Byrd, North Carolina State University

3. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-3 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: 1.presence of a voltage – we’ll call this state “1” 2.absence of a voltage – we’ll call this state “0” 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 We’ll see examples of these circuits in the next chapter.

4. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-4 Computer is a binary digital system. 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: 00, 01, 10, 11 A collection of three bits has eight possible states: 000, 001, 010, 011, 100, 101, 110, 111 A collection of n bits has 2 n possible states. Binary (base two) system: has two states: 0 and 1 Digital system: finite number of symbols

5. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-5 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…

6. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-6 Unsigned Integers Non-positional notation could represent a number (“5”) with a string of ones (“11111”) problems? Weighted positional notation like decimal numbers: “329” “3” is worth 300, because of its position, while “9” is only worth 9 329 10 2 10 1 10 0 1012 2121 2020 3x100 + 2x10 + 9x1 = 3291x4 + 0x2 + 1x1 = 5 most significant least significant

7. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-7 Unsigned Integers (cont.) An n-bit unsigned integer represents 2 n values: from 0 to 2 n -1. 2 2121 2020 0000 0011 0102 0113 1004 1015 1106 1117

8. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Adding Binary Numbers Just like decimal arithmetic Arithmetic tables: 1+1 = 2 1+2 = 3 … 9+9 = 18 If sum > 9 we have to carry Binary table is much smaller If sum > 1 we have to carry 2-8 ABCA+B+C (carry|sum) 00000 01001 100 11010 00101 01110 101 11111

9. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-9 Unsigned Binary Arithmetic Base-2 addition – just like base-10! add from right to left, propagating carry 10010100101111 +1001 +1011+1 110111110110000 10111 +111 11110 carry Subtraction, multiplication, division,…

10. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-10 Converting Binary to Decimal Add powers of 2 that have “1” in the corresponding bit positions. n2n2n 01 12 24 38 416 532 664 7128 8256 9512 101024 X = 01101000 two =2 6 +2 5 +2 3 = 64+32+8 =104 ten

11. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-11 Converting Decimal to Binary First Method: Subtract Powers of Two 1.Subtract largest power of two less than or equal to number. 2.Put a one in the corresponding bit position. 3.Keep subtracting until result is zero. X = 104 ten 104 - 64=40bit 6 40 - 32=8bit 5 8 - 8=0bit 3 X=01101000 two n2n2n 01 12 24 38 416 532 664 7128 8256 9512 101024

12. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-12 Converting Decimal to Binary Second Method: Division 1.Divide by two – remainder is least significant bit. 2.Keep dividing by two until answer is zero, writing remainders from right to left. X = 104 ten 104/2=52 r0bit 0 52/2=26 r0bit 1 26/2=13 r0bit 2 13/2=6 r1bit 3 6/2=3 r0bit 4 3/2=1 r1bit 5 X=01101000 two 1/2=0 r1bit 6

13. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-13 Operations: Arithmetic and Logical Recall: a data type includes representation and operations. We have a good representation for unsigned integers, and one operation: Addition Will look at other operations later Logical operations are also useful: AND OR NOT

14. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-14 Logical Operations Operations on logical TRUE or FALSE two states -- takes one bit to represent: TRUE=1, FALSE=0 View n-bit number as a collection of n logical values operation applied to each bit independently AB A AND B 000 010 100 111 AB A OR B 000 011 101 111 A NOT A 01 10

15. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-15 Examples of Logical Operations AND useful for clearing bits  AND with zero = 0  AND with one = no change OR useful for setting bits  OR with zero = no change  OR with one = 1 NOT unary operation -- one argument flips every bit 11000101 AND 00001111 00000101 11000101 OR 00001111 11001111 NOT 11000101 00111010

16. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-16 Hexadecimal Notation It is often convenient to write binary (base-2) numbers as hexadecimal (base-16) numbers instead. fewer digits -- four bits per hex digit less error prone -- easy to corrupt long string of 1’s and 0’s BinaryHexDecimal 000000 000111 001022 001133 010044 010155 011066 011177 BinaryHexDecimal 100088 100199 1010A10 1011B11 1100C12 1101D13 1110E14 1111F15

17. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-17 Converting from Binary to Hexadecimal Every four bits is a hex digit. start grouping from right-hand side 011101010001111010011010111 7D4F8A3 This is not a new machine representation, just a convenient way to write the number.

18. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-18 Text: ASCII Characters ASCII: Maps 128 characters to 7-bit code. both printable and non-printable (BEL, DEL, …) characters 00nul10dle20sp30040@50P60`70p 01soh11dc121!31141A51Q61a71q 02stx12dc222"32242B52R62b72r 03etx13dc323#33343C53S63c73s 04eot14dc424$34444D54T64d74t 05enq15nak25%35545E55U65e75u 06ack16syn26&36646F56V66f76v 07bel17etb27'37747G57W67g77w 08bs18can28(38848H58X68h78x 09ht19em29)39949I59Y69i79y 0anl1asub2a*3a:4aJ5aZ6aj7az 0bvt1besc2b+3b;4bK5b[6bk7b{ 0cnp1cfs2c,3c<4cL5c\6cl7c| 0dcr1dgs2d-3d=4dM5d]6dm7d} 0eso1ers2e.3e>4eN5e^6en7e~ 0fsi1fus2f/3f?4fO5f_6fo7fdel

19. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-19 Interesting Properties of ASCII Code What is relationship between a decimal digit ('0', '1', …) 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 128 characters enough? (http://www.unicode.org/) No new operations -- integer arithmetic and logic.

20. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 2-20 Other Data Types Text strings sequence of characters, terminated with NULL (0) typically, no hardware support Image array of pixels  monochrome: one bit (1/0 = black/white)  color: red, green, blue (RGB) components (e.g., 8 bits each)  other properties: transparency hardware support:  typically none, in general-purpose processors  MMX -- multiple 8-bit operations on 32-bit word Sound sequence of fixed-point numbers

21. Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display. Summary Binary digital system Data type: representation and operations Unsigned integers Weighted positional notation Addition Conversion from binary to decimal Converstion from decimal to binary (2 methods) Logical operations: AND/OR/NOT Hexadecimal notation ASCII representation for characters/text Other data types 2-21