Unit 1. Digital Representation

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1 1 Unit 1 Digital Representation

2 2 Information Representation All information is represented as sequences of 1's and 0's 5 kinds of information Numbers Text Instructions Sound Image/Video

3 3 Why 1 s and 0 s Transistors are electronic devices used to build computer hardware Like a switch (2 positions) => On (conducting) / Off (non-conducting) Output voltage of a transistor will either be high or low voltage 2 states of the transistor (on or off) lead to only 2 values in computer hardware 1 s and 0 s are arbitrary symbols representing high or low voltage A single 1 or 0 is known as a bit The voltage here determines if current can flow between drain and source Controlling Input (Gate ) Schematic Symbol of a Transistor Output (Drain ) Source Circuit Diagram of a Switch circuit is open (off) no current can flow circuit is closed (on) current can flow Functional View of a Transistor as a Switch on off Circuit Diagram of a Switch High Voltage +5V +12V or Low Voltage 0V -12V 1 0

4 4 Starting With Numbers A single bit can only represent 1 and 0 To represent more than just 2 values we need to use combinations/sequences of many bits A byte is defined as a group 8-bits A word varies in size but is usually 32- bits So how do we interpret those sequences of bits? Let's learn about number systems 0 or 1 A bit Two bit Combinations An example byte An example word

5 5 Interpreting Binary Strings Given a string of 1 s and 0 s, you need to know the representation system being used, before you can understand the value of those 1 s and 0 s. Information (value) = Bits + Context (System) Unsigned Binary system =? Instructions ASCII system A ASCII "add 1"

6 6 Unsigned Binary Number System Humans use the decimal number system Based on number digits: [0-9] Because computer hardware uses digital signals with 2 values, computers use the binary number system Based on number 2 2 binary digits (a.k.a bits): [0,1]

7 7 Number System Theory Let's understand how number systems work by examining decimal and then moving to binary The written digits have implied place values Place values are powers of the base (decimal = 10) Place value of digit to left of decimal point is 10 0 and ascend from there, negative powers of 10 to the right of the decimal point The value of the number is the sum of each digit times its implied place value base (852.7) 10 =

8 8 Number System Theory The written digits have implied place values Place values are powers of the base (decimal = 10) Place value of digit to left of decimal point is 10 0 and ascend from there, negative powers of 10 to the right of the decimal point The value of the number is the sum of each digit times its implied place value base Most Significant Digit (MSD) Least Significant Digit (LSD) (852.7) 10 = 8* * * *10-1 digits place values

9 9 Binary Number System Place values are powers of 2 The value of the number is the sum of each bit times its implied place value (power of 2) base (110.1) 2 =

10 10 Binary Number System Place values are powers of 2 The value of the number is the sum of each bit times its implied place value (power of 2) base Most Significant Bit (MSB) Least Significant Bit (LSB) (110.1) 2 = 1 * * * * 2-1 bits place values (110.1) 2 = 1*4 + 1*2 + 1*.5 = =

11 11 Interpreting Binary Strings Information (65) = Bits + Context (Unsigned Binary System) Unsigned Binary system =? Instructions ASCII system A ASCII "add 1"

12 12 Unique Combinations Given n digits of base r, how many unique numbers can be formed? 2-digit, decimal numbers 3-digit, decimal numbers combinations: - combinations: - 4-bit, binary numbers 6-bit, binary numbers combinations: - combinations: - Main Point: Given n digits of base r, unique numbers can be made with the range [ ]

13 13 Unique Numbers Computers represent binary numbers using a fixed number of bits Given a fixed number of bits, n, what is the range of numbers we can make? If n=1 bit: If n=2 bits: If n=3 bits: = 0 = = 0 = 1 = 2 = = 0 = 1 = 2 = 3 = 4 = 5 = 6 = 7 Given n bits, 2 n numbers can be made 4 2 1

14 14 Powers of 2 Might help to memorize the powers of = = = = = = = = = = =

15 15 Sign Is there any limitation if we only use the powers of some base as our weights? Can't make negative numbers What if we change things How do humans represent negative numbers? Can we do something similar?

16 16 C Integer Data Types Integer Types (signed by default unsigned with optional leading keyword) C Type Bytes Bits Signed Range Unsigned Range [unsigned] char 1 8 [unsigned] short 2 16 [unsigned] long [unsigned] int 4 32 [unsigned] long long to to to billion to +2 billion 0 to to 4 billion -8*10 18 to +8* to 16*10 18

17 17 Approximating Large Powers of 2 How did we know a 32-bit int could store up to 4 billion? Here's a quick way to estimate the decimal approximation of a large powers of 2 like 2 16, 2 32, etc. Use following approximations: 2 10 = 10 3 (1 thousand=1k) (1 million=1m) (1 billion=1g) For other powers of 2, decompose into product of 2 10 or 2 20 or 2 30 and a power of 2 that is less than 2 10 See examples 2 16 = 2 6 * * 10 3 = 64, = 2 4 * * 10 6 = 16,000, = 2 8 * * 10 6 = 256,000, = 2 2 * * 10 9 = 4,000,000,000

18 18 Text Text characters are usually represented with some kind of binary code (mapping of character to a binary number such as 'a' = bin = 97 dec) ASCII = Traditionally an 8-bit code How many combinations (i.e. characters)? English only UNICODE = 16-bit code How many combinations? Most languages w/ an alphabet In C/C++ a single printing/text character must appear between single-quotes (') Example: 'a', '!', 'Z'

19 19 Text Strings To represent words and sentences we can use a string of characters C++ uses double-quote (") to group the characters that are part of a string Example: "Hello\n" Each character is converted to ASCII equivalent 'H' = 72, 'e' = 101, '\n' = Represents a non-printing character ("Enter/Return" and moves the cursor to the start of the next line) '\n' = LF (0x0A) = line feed (moves cursor down a line)

20 INSTRUCTIONS 20

21 21 Instructions Tell the computer what operations to perform on the data Contains Operation to perform (OpCode) Source operands (input data) Destination location (where to put output) Z = X + 1 X = X - 2 High-level operations would be converted to "instructions" which have a direct translation to 1's and 0's Dst. Src1 Src2 These would be the binary representations of the instructions to perform: Z = X + 1 and X = X ADD Z X Dst. Src1 Src SUB X X 2

22 22 DATA STORAGE & COMPUTER MEMORY

23 23 Computer Components Processor Executes the program and performs all the operations Main Memory Stores data and program (instructions) RAM = read and write but volatile (lose values when power off) Let's look more at memory Processor (Reads instructions, operates on data) Instructions Data Combine 2c. Flour Mix in 3 eggs Processor Software Program Input Devices Data Program (Instructions) Data (Operands) Memory (RAM) Output Devices

24 24 Memory Set of cells that each store a group of bits (usually, 1 byte = 8 bits) Unique address assigned to each cell Used to reference the value in that location Analogy: Safe-deposit or mail boxes Each has an identifying number and a value stored inside The value can be an instruction, a number, a character, etc. (You the programmer must know what to expect and how to interpret it!) Address Data Memory Device

25 25 Memory Operations Memories perform 2 operations Read: retrieves data value in a particular location (specified using the address) You, the programmer, must know what type (integer, character, etc.) that data is. Write: changes data in a location to a new value To perform these operations a set of address, data, and control inputs/outputs are used Note: A group of wires/signals is referred to as a bus Thus, we say that memories have an address, data, and control bus Read Write Addr. Data Control A Read Operation Addr. Data Control A Write Operation

26 26 One At a Time Recall that while we see the image of a man, the computer "sees" a collection of numbers (aka pixels)? Now we can understand why Every number is stored as bits in memory Memory can only be accessed one data value at a time This limitation of accessing one value at a time leads to a fundamental issue of programming: How do we break abstract Data tasks into a sequence of "1 at a time" operations? Individual Pixels Image taken from the photo "Robin Jeffers at Ton House" (1927) by Edward Weston Addr

27 27 Exercise Show how "cs 102" would be stored in the memory below Use decimal to represent each byte How do we indicate the string is done ("terminated") With special NULL character (i.e. 0)

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