LECTURE 2: Number Systems

Transcription

1 LECTURE : Number Systems CIS Fall 7 Instructor: Dr. Song Xing Department of Information Systems California State University, Los Angeles Learning Objectives Review Boolean algebra. Describe numbering systems and their use in data representation. Compare and contrast various data representation methods. Introduce simple arithmetic operations. Automated Data is converted from native format into a form suitable for the processing device Computers represent data electrically and process it with electrical switches Laws of electricity can be stated as mathematical equations Electronic devices perform computational functions embedded in the equations 3

2 Processing Circuits Example Data and Signal A + B = C Data are entities that convey meaning (computer files, music on a CD, etc). There are two basic ways to store and manage data: Analog Digital Signals are the electric or electromagnetic encoding of data (telephone conversation, web page download). 5 Analog Signal Digital Signal Analog signal is a continuous waveform, with examples such as (naturally occurring) music and voice. a discrete or non-continuous waveform (called gate waveform as well) with examples such as music on a compact disc - the disc stores numbers representing specific voltage levels sampled at specific times 7

3 Digital Information Computers store all information digitally: numbers text graphics and images video audio program instructions In some way, all information is digitized - broken down into pieces and represented as numbers Representing Text Digitally For example, every character is stored as a number, including spaces, digits, and punctuation Corresponding upper and lower case letters are separate characters 9 Why Binary? Early computer design was decimal. Binary data processing was proposed in 95. Binary number Only one of two possible values ( or ) per digit Can be processed by two state electrical devices (relatively easy to design and fabricate). Correspond directly with values in Boolean logic. Can combine multiple binary On True Off False digits to form a single data value Yes No to represent large numbers.

4 Boolean Algebra Boolean Operators AND and OR Rules that govern constants and variables that can take on values True/false; on/off; yes/no; / Boolean logic Rules for handling Boolean constants and variables 3 fundamental operations: AND, OR and NOT AND Result C is TRUE () if and only if both input operands A and B are true OR Result C is TRUE () if any input operands A or B is true A A B B C C 3 Boolean Operator NOT Boolean Expressions in C/Java NOT Result C is TRUE () if input operand A is FALSE () A C 5

5 Counting and Arithmetic Base: the number of different symbols required to represent any given number Decimal or base has digits:,,,3,,5,,7,,9 Binary or base has digits: and Bit stands for binary digit. bits = byte Hexadecimal or base has digits:,,,3,,5,,7,,9,a,b,c,d,e,f Examples: A = ; B = Note: The subscript number indicates the base. 7 Decimal, Binary, and Hexadecimal Numbers Why Do We Need Hexadecimal? Modern computer operating systems and networks present variety of troubleshooting data in hex format Much easier to read and write than binary (base ) notation Hex F 7 Binary 9

6 Hex Example: Memory Address Windows /XP logical memory map Hex Example: MAC Address IP address is a logical address (possibly temporary), while the -bit MAC address on every NIC (Network Interface Card) is the physical, or permanent address. For example, a MAC address -33-A3-AA- BD-5 is as -bit binary Windows IPCONFIG.EXE Command Executed On A Workstation Hex Example: IPv Address The current version of the IP protocol of the Internet is IPv which is based on 3-bit-wide addresses. The Cal-State-LA web server ( address, for example, is divided into groups of bits written as one decimal digit. It is A new IP, IPv, uses bits for its address which is divided into groups of bits written as four hex digits. An example is FE::::A:DFF:FEA5:E9F5 3

7 Hex Example: Hexadecimal Values for Color in HTML 5 Conversion Between Binary and Hexadecimal bits binary represents a hexadecimal. The hexadecimal number F can be converted to binary: or. The binary number is equivalent to or. Its hexadecimal representation is 5B9A. 7

8 Recall Positional Notation for Decimal Binary Powers Table 57 = 5 x + x + 7 x s place s place s place Power Base Place 5 3 Value Evaluate Sum 5 x 5 x 7 x 7 i.e., =, =, =, 3 =, =, 5 = 3, =, 7 =, = 5, 9 = 5, =, 9 3 Conversion from Binary to Decimal From Binary to Decimal: ~ = x 7 +x +x 5 +x +x 3 +x +x +x = = Place Binary Number s ( 3 ) Equivalent s ( ) s ( ) x x s ( ) x x x x Decimal Number 3 Value 3 x x x 5 Evaluate x x x 3 x x x x x x x x x x 7 Sum for Base x 3 x 3 x 3 x x 9 3 3

9 Converting Binary IP Addresses to Dotted Decimal Notation Binary IP Address -Bit Segments Convert Segments to Decimal Dotted Decimal Notation Fractions: Decimal Fractions: Binary.59 = x +x - +5x - +x -3 +9x - = = x + x - + x - + x -3 + x - + x -5 + x - = =.775 Place Value / / / / Place Evaluate x / 5 x / x / 9 x/ Value / / / / /3 / Sum Evaluate x / x / x / x / x /3 x / Sum

10 Fractional Conversion from Binary to Decimal. = x 5 +x +x 3 +x + x + x + x - + x - + x -3 = = Recall the Division From Decimal to Binary = Power Base Quotient Integer /3 = / = / = / = / = / = Remainder 39

11 Range of Possible Numbers R = B n where R = range B = base n = number of digits Example #: Decimal, Base, digits R = = different numbers ( 99) Example #: Binary, Base, digits R = = 5,53 or K K = = I.e., -bit can represent 5,53 different number values Decimal Range for Binary Data Bit Widths Bit Permutations bit bits 3 bits bits Decimal Number

12 Bit Permutations (cont.) Each permutation can represent a particular item. There are N permutations. Therefore, N bits are needed to represent N unique items. For example, to represent unique items, 7 bits are required as < <, =, and 7 =. 5 Addition Binary Arithmetic Base Decimal Binary Problem +3 + Largest Single Digit 9 Addition + Base Decimal Binary Problem +5 + Carry Carry the Carry the Answer Multiplication x 7

13 Binary Addition Binary Addition (Cont.) carry bit Binary Multiplication Summary How data is represented and stored within computer hardware Compare and contrast various data representation methods. Understanding data representation is key to understanding hardware and software technology 5 5