information as bits what kinds of data do we need to
play

Information as Bits What kinds of data do we need to represent? - - PowerPoint PPT Presentation

Jul 10, 2023 •247 likes •782 views

CSC 2400: Computer Systems Information as Bits What kinds of data do we need to represent? - Numbers integers, floating point, - Text characters, strings, - Images pixels, colors, shapes, - Sound - Instructions -

  1. CSC 2400: Computer Systems Information as Bits

  2. What kinds of data do we need to represent? - Numbers – integers, floating point, … - Text – characters, strings, … - Images – pixels, colors, shapes, … - Sound - Instructions - …

  3. Integers

  4. Unsigned Integers q An n -bit unsigned integer represents 2 n values: from 0 to 2 n -1 q Example for n = 3: 2 2 2 1 2 0 0 0 0 0 0 0 1 1 0 1 0 2 0 1 1 3 1 0 0 4 1 0 1 5 1 1 0 6 7 = 2 3 - 1 1 1 1

  5. Signed Integers q How do computers differentiate between positive and negative integers? - Positive integers have the most significant bit (left bit) 0 - Negative integers have the most significant bit (left bit) 1 q Negative integer representations: 1. Sign-Magnitude 2. One’s Complement 3. Two’s Complement

  6. 1. Sign-Magnitude q Reserve the leftmost bit to represent the sign: 0 means positive 1 means negative q Examples 0 0 1 0 1 1 0 0 è 44 1 0 1 0 1 1 0 0 è -44 Magnitude Sign q Hard to do arithmetic this way, so it is rarely used - What is the result of 44 – 44?

  7. Exercise • Assume 8-bit sign-magnitude representation for integers • What is the decimal value of 11010110

  8. Exercise q Assume 8-bit sign-magnitude representation for integers q What is the smallest value you can represent in this system? q What is the largest value you can represent in this system?

  9. 1. Sign-Magnitude (contd.) 0 0 1 0 1 1 0 0 è 44 1 0 1 0 1 1 0 0 è -44 Sign Magnitude q For numbers represented on n bits: - Range of positive integers: from 0 to (2 n-1 – 1) - Range of negative integers: from –( 2 n-1 – 1) to – 1

  10. 2. One’s Complement q Leftmost bit is 0 for positive numbers 0 0 1 0 1 1 0 0 è 44 q To obtain the corresponding negative number (-44), flip every bit: 1 1 0 1 0 0 1 1 è -44

  11. 2. One’s Complement (contd.) q What is the result of 44 – 44? 0 0 1 0 1 1 0 0 ( 44) 1 1 0 1 0 0 1 1 (-44) _______________ q Issue: two different representations for zero

  12. 3. Two’s Complement q Leftmost bit is 0 for positive numbers 0 0 1 0 1 1 0 0 è 44 q To obtain the corresponding negative number - 44, add 1 to the one’s complement of 44: 1 1 0 1 0 0 1 1 è one’s complement + 0 0 0 0 0 0 0 1 _______________ 1 1 0 1 0 1 0 0 è two’s complement

  13. 3. Two’s Complement (contd.) q What is the result of 44 – 44? 0 0 1 0 1 1 0 0 ( 44) 1 1 0 1 0 1 0 0 (-44) _______________ q Used by most computer systems q For numbers represented on n bits: - Range of positive integers: from 0 to ________ - Range of negative integers: from – 2 n-1 to ________

  14. Exercise • Assume 8-bit two’s complement representation for integers • What is the decimal value of 11010110

  15. Exercises q Assuming 4-bit two's complement representation, what is the decimal value of 1011 (2) ? q Assuming 5-bit two's complement representation, what is the decimal value of 1011 (2) ? q What is -2 in 4-bit two’s complement representation? q What is -2 in 6-bit two’s complement representation?

  16. Exercise q Assume 8-bit 2’s complement representation for integers q What is the smallest value you can represent in this system? q What is the largest value you can represent in this system?

  17. Two’s Complement to Decimal (Method 1) 1. If leading bit is one, take two’s complement to get a positive number n 2 n 2. Convert to decimal: add powers of 2 that 0 1 have “1” in corresponding bit positions 1 2 2 4 3. If original number was negative, 3 8 add a minus sign 4 16 5 32 X = 01101000 (2) 6 64 7 128 = 2 6 +2 5 +2 3(10) = 64+32+8 (10) 8 256 = 104 (10) 9 512 10 1024 Assuming 8-bit two’s complement numbers.

  18. Another Example Assume 8-bit two’s complement numbers. X = 11100110 (2) 1. Leading bit is one, so take two’s complement to get a positive number -X = 00011001 + 00000001 (2) = 00011010 (2) 2. Convert to decimal -X = 2 4 +2 3 +2 1(10) = 16+8+2 (10) = 26 (10) 3. Add a minus sign X = -(-X) = -26 (10)

  19. More Examples n 2 n X = 00100111 two 0 1 = 2 5 +2 2 +2 1 +2 0 = 32+4+2+1 1 2 = 39 ten 2 4 3 8 4 16 X = 11100110 two 5 32 6 64 -X = 00011010 7 128 = 2 4 +2 3 +2 1 = 16+8+2 8 256 = 26 ten 9 512 10 1024 X = -26 ten Assuming 8-bit 2 ’ s complement numbers.

  20. Two’s Complement to Decimal (Method 2) q Most significant (left, sign) bit has weight –2 n-1 q All other bits have positive weights q Example: 4-bit two’s complement -2 3 2 2 2 1 2 0 -2 3 2 2 2 1 2 0 0 0 0 0 0 1 0 0 0 -8 0 0 0 1 1 1 0 0 1 ___ 0 0 1 0 2 1 0 1 0 ___ 0 0 1 1 3 1 0 1 1 ___ 0 1 0 0 4 1 1 0 0 ___ 0 1 0 1 5 1 1 0 1 ___ 0 1 1 0 6 1 1 1 0 ___ 0 1 1 1 7 1 1 1 1 ___

  21. Two’s Complement to Decimal (Method 2) q Most significant (left, sign) bit has weight –2 n-1 q All other bits have positive weights q Example: 4-bit two’s complement -2 3 2 2 2 1 2 0 -2 3 2 2 2 1 2 0 0 0 0 0 0 1 0 0 0 -8 0 0 0 1 1 1 0 0 1 -7 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

  22. Binary Number Representation Summary q Leftmost bit 0 indicates positive number q Leftmost bit 1 indicates negative number q To negate a binary value: - sign-magnitude: flip the sign bit - one’s complement: take the one’s complement - two’s complement: take the two’s complement q Binary to decimal (two’s complement): - normal conversion from binary to decimal, with the most significant bit having negative weight

  23. Fill in the Table Bit Value Value Value Pattern (Sign Magnitude) (One’s Complement) (Two’s Complement) 000 001 010 011 100 101 110 111

  24. Fill in the Table Bit Value Value Value Pattern (Sign Magnitude) (One’s Complement) (Two’s Complement) 000 001 010 011 100 -0 -3 -4 101 -1 -2 -3 110 -2 -1 -2 111 -3 -0 -1

  25. Sign Extension q Assuming 8-bit two’s complement numbers, write -5 in binary. q Method 1: - Represent 5 (10) = ________ (2) - Take the two’s complement: ________ (2) + 1 = ________ (2) q Method 2: - Represent -5 = -8+2+1 on a minimum number of bits: 1011 - Use sign extension to fill in all 8 bits (value does not change): 11111011

  26. Question q What decimal value does 10011001 (2) represent?

  27. Floating-Point Numbers IEEE Floating-Point Standard

  28. Floating-Point Numbers q Decimal System: 11.625 analyzed as 10 1 10 0 10 -1 10 -2 10 -3 1 1 . 6 2 5 11.625 = (1 x 10) + 1 + (6 x 10 -1 ) + (2 x 10 -2 ) + (5 x 10 -3 ) q Binary System:

  29. Floating-Point Numbers You try it: 10010.01001 (2) = ____________________________ (10)

  30. How to Store Floating-Point Numbers? q We have no way to store the point separating the whole part from the fractional part! q Standard committees (IEEE) came up with a way to store floating point numbers

  31. Floating-Point Normalization q Every floating-point binary number ( except for zero ) can be normalized by choosing the exponent so that the radix point falls to the right of the leftmost 1 bit 37.25 (10) = 100101.01 (2) = 1.0010101 x 2 5 7.625 (10) = 111.101 (2) = 1.11101 x 2 2 0.3125 (10) = 0.0101 (2) = 1. 01 x 2 -2 exponent fraction mantissa significand

  32. IEEE Floating-Point Standard (Single Precision, 32 bits) q Sign-Magnitude: sign bit S , exponent E and fraction F q The binary exponent is not stored directly. Instead, E is the sum of the true exponent and 127. This biased exponent is always non-negative (seen as magnitude only). q The fraction part assumes a normalized significand in the form 1.F (so we get the extra leading bit for free)

  33. IEEE Floating-Point Standard (Single Precision, 32 bits) q Sign-Magnitude: sign bit S , exponent E and fraction F S × 1.fraction × 2 exponent − 127 , 1 ≤ exponent ≤ 254 N = − 1 q Special values: - E = 0, F = 0 represents 0.0 - Exponent with all bits 1 (value 255) is reserved to represent ±infinity (if F = 0) and NaN (Not a Number, if F != 0)

  34. How would 15213.0 be stored? q First, 15213 (10) = 11101101101101 (2) q Normalize to 1.1101101101101 (2) x 2 13 - The true exponent is 13, so the biased E is E = 13 + 127 (Bias) = 140 (10) = 10001100 (2) - The fraction is F = 11011011011010000000000 (2) Floating Point Representation: 4 6 6 D B 4 0 0 Hex: 0100 0110 0110 1101 1011 0100 0000 0000 Binary:

  35. How would 15213.5 be stored? q First, 15213.5 (10) = 11101101101101.1 (2) q Normalize to 1.11011011011011 (2) x 2 13 - The true exponent is 13, so the biased E is E = 13 + 127 (Bias) = 140 (10) = 10001100 (2) - The fraction is F = 11011011011011000000000 (2) Floating Point Representation: 4 6 6 D B 5 0 0 Hex: 0100 0110 0110 1101 1011 0110 0000 0000 Binary:

  36. How would 23.75 be stored? q First, 23.75 (10) = 10111.11 (2) q Normalize to 1.011111 (2) x 2 4 • The true exponent is 4, so the biased E is E = 4 + 127 (Bias) = 131 (10) = 10000011 (2) q The fraction is F = 01111100000000000000000 (2) Floating-Point Representation: 4 6 6 D B 4 0 0 Hex: 0100 0001 1011 1110 0000 0000 0000 0000 Binary:

  37. How would -23.75 be stored? q Just change the sign bit: 4 6 6 D B 4 0 0 Hex: 1100 0001 1011 1110 0000 0000 0000 0000 Binary: q Do not take the two’s complement!

  38. Exercise 1 q Find the IEEE representation of 40.0

Download Presentation
Download Policy: The content available on the website is offered to you 'AS IS' for your personal information and use only. It cannot be commercialized, licensed, or distributed on other websites without prior consent from the author. To download a presentation, simply click this link. If you encounter any difficulties during the download process, it's possible that the publisher has removed the file from their server.

Recommend

Bits and Bytes Topics Topics Why bits? Representing information as bits

Bits and Bytes Topics Topics Why bits? Representing information as bits

Systems I Bits and Bytes Topics Topics Why bits? Representing information as bits Binary/Hexadecimal Byte representations numbers characters and strings Instructions Bit-level manipulations Boolean algebra

614 views • 33 slides
2 4 0 = 2 x 10 2 + 4 x 10 1 + 0 x 10 0 100 10 1 weight Representing Data with Bits 10 2 10 1

2 4 0 = 2 x 10 2 + 4 x 10 1 + 0 x 10 0 100 10 1 weight Representing Data with Bits 10 2 10 1

Wellesley CS 240 - Data as Bits 8/31/16 positional number representation 2 4 0 = 2 x 10 2 + 4 x 10 1 + 0 x 10 0 100 10 1 weight Representing Data with Bits 10 2 10 1 10 0 position 2 1 0 Base determines: bits, bytes, numbers, and

556 views • 6 slides
Bits and Bytes Aug. 29, 2002 Topics Topics n Why bits? n Representing information as bits l

Bits and Bytes Aug. 29, 2002 Topics Topics n Why bits? n Representing information as bits l

15-213 The Class That Gives CMU Its Zip! Bits and Bytes Aug. 29, 2002 Topics Topics n Why bits? n Representing information as bits l Binary/Hexadecimal l Byte representations numbers characters and strings Instructions n

690 views • 34 slides
MIPS Instruction Formats 6 bits 5 bits 5 bits 5 bits 5 bits 6 bits for instance,

MIPS Instruction Formats 6 bits 5 bits 5 bits 5 bits 5 bits 6 bits for instance,

MIPS Instruction Formats 6 bits 5 bits 5 bits 5 bits 5 bits 6 bits for instance, add r1, r2, r3 has - 000000 00010 00011 00001 00000 100000 opcode (OP) tells the machine which format 1 VAX Instruction Formats (x86

431 views • 20 slides
? representing data with bits data SW- Processor Memory bits, bytes,

? representing data with bits data SW- Processor Memory bits, bytes,

9/4/15 1930s 1940s 1950s 1960s 1970s 1980s 1990s 2000s 2010s Modern Digital Computer von Neumann model HW- controlled instructions ? representing data with bits data SW- Processor

325 views • 4 slides
Bits and bytes Week 2: data representation Data is stored in the computer as bits. (20 cr

Bits and bytes Week 2: data representation Data is stored in the computer as bits. (20 cr

School of Computer Science, University of Birmingham. Java Lecture notes. M. D. Ryan. September 2000. Bits and bytes Week 2: data representation Data is stored in the computer as bits. (20 cr only) Bit stands for binary digit ; it has

76 views • 3 slides
1 Transducers Inherently Discrete values devices that convert from one representation to

1 Transducers Inherently Discrete values devices that convert from one representation to

Interpretation of bits depends on context meaning of a group of bits depends on how they are interpreted 1 byte could be Bits, Bytes, 1 bit in use, 7 wasted bits (e.g., M/F in a database) 8 bits storing a number between 0 and 255

543 views • 3 slides
1 0 1 1 = 1 x 2 3 + 0 x 2 2 + 1 x 2 1 + 1 x 2 0 8 4 2 1 weight Representing Data with Bits

1 0 1 1 = 1 x 2 3 + 0 x 2 2 + 1 x 2 1 + 1 x 2 0 8 4 2 1 weight Representing Data with Bits

binary = base 2 1 0 1 1 = 1 x 2 3 + 0 x 2 2 + 1 x 2 1 + 1 x 2 0 8 4 2 1 weight Representing Data with Bits 2 3 2 2 2 1 2 0 position 3 2 1 0 bits, bytes, numbers, and notation When ambiguous, subscript with base: 101 10 Dalmatians

279 views • 4 slides
Representing Data with Bits bits, bytes, numbers, and notation bit =

Representing Data with Bits bits, bytes, numbers, and notation bit =

Representing Data with Bits bits, bytes, numbers, and notation bit = binary digit = 0 or 1 Electronically: high voltage vs. low voltage 0 1 0 3.3V 2.8V 0.5V 0.0V Basis

389 views • 23 slides
Sidan # Address The Instruction space To work with a space we must be able to address

Sidan # Address The Instruction space To work with a space we must be able to address

Data representation String of bits! How do we represent data in a digital system? Everything we work with is represented as a string of bits . Digital data is stored as bits : Example: a bit is a binary number {0, 1} We don't work

365 views • 4 slides
Bits and Bytes At the smallest scale in the computer, information is stored as bits and bytes. In

Bits and Bytes At the smallest scale in the computer, information is stored as bits and bytes. In

Bits and Bytes At the smallest scale in the computer, information is stored as bits and bytes. In this section, we'll look at how that works. Bit Bit, like an atom, the smallest unit of storage A bit stores just a 0 or 1 "In the

302 views • 9 slides
Lecture Overview Lecture Topics How computers encode information Bits and Bytes How

Lecture Overview Lecture Topics How computers encode information Bits and Bytes How

Lecture Overview Lecture Topics How computers encode information Bits and Bytes How to quantify information and memory How to represent and communicate binary data Computer Literacy Lecture 4 The aim is to be able to reason

250 views • 5 slides
Data (Some repeating CS1083, ECE course) bits and bit sequences integers (signed and

Data (Some repeating CS1083, ECE course) bits and bit sequences integers (signed and

Data (Some repeating CS1083, ECE course) bits and bit sequences integers (signed and unsigned) bit vectors strings and characters floating point numbers hexadecimal and octal notations Bits and Bit Sequences

885 views • 64 slides
Data (Some repeating CS1083, ECE course) bits and bit sequences integers (signed and

Data (Some repeating CS1083, ECE course) bits and bit sequences integers (signed and

Data (Some repeating CS1083, ECE course) bits and bit sequences integers (signed and unsigned) bit vectors strings and characters floating point numbers hexadecimal and octal notations Bits and Bit Sequences

887 views • 65 slides
DATA STORAGE COMS W1001 Introduction to Information Science Boyi Xie 2 Todays Topics

DATA STORAGE COMS W1001 Introduction to Information Science Boyi Xie 2 Todays Topics

1 DATA STORAGE COMS W1001 Introduction to Information Science Boyi Xie 2 Todays Topics Bits and Logic Gates Bits, Bytes and the Binary System Binary, Decimal, Octal, Hexadecimal Representing Numbers Representing Text 3

614 views • 20 slides
Representing Data with Bits bits, bytes, numbers, and notation positional number representation =

Representing Data with Bits bits, bytes, numbers, and notation positional number representation =

Representing Data with Bits bits, bytes, numbers, and notation positional number representation = 2 x 10 2 + 4 x 10 1 + 0 x 10 0 2 4 0 100 10 1 weight 10 2 10 1 10 0 position 2 1 0 Base determines: Maximum digit (base 1).

472 views • 23 slides
File Types Session 5 INST 346 Agenda Some examples of file types Text Images

File Types Session 5 INST 346 Agenda Some examples of file types Text Images

File Types Session 5 INST 346 Agenda Some examples of file types Text Images Video Audio | 0 NUL | 32 SPACE | 64 @ | 96 ` | | 1 SOH | 33 ! | 65 A | 97 a | | 2 STX | 34 " | 66 B | 98 b | ASCII

754 views • 30 slides
CSCI341 Lecture 27, ASCII &Unicode, Addressing Modes ASCII American Standard Code for

CSCI341 Lecture 27, ASCII &Unicode, Addressing Modes ASCII American Standard Code for

CSCI341 Lecture 27, ASCII &Unicode, Addressing Modes ASCII American Standard Code for Information Interchange One byte = One character How many possible characters? COMMON INSTRUCTIONS lb (as signed) sb lbu (used in C

819 views • 18 slides
What's a string? Characters enclosed by double quotes "this is a String" " this

What's a string? Characters enclosed by double quotes "this is a String" " this

What's a string? Characters enclosed by double quotes "this is a String" " this String starts with spaces" "12345" "the above String is made up of digit characters" Print Strings to the Console using

458 views • 22 slides
Bit Basics Eric McCreath Bit Basics A bit (Binary digIT) is single unit of binary storage. A bit

Bit Basics Eric McCreath Bit Basics A bit (Binary digIT) is single unit of binary storage. A bit

Bit Basics Eric McCreath Bit Basics A bit (Binary digIT) is single unit of binary storage. A bit is normally grouped with other bits to represent data or information. 4 bits form a nibble 8 bits form a byte (e.g. 10100001 or 0xA1 in hex) Bytes

845 views • 9 slides
Good Evening! INT1005 Introduction to Computer Systems Ulrich Werner Bits, Bytes, and Code

Good Evening! INT1005 Introduction to Computer Systems Ulrich Werner Bits, Bytes, and Code

Good Evening! INT1005 Introduction to Computer Systems Ulrich Werner Bits, Bytes, and Code Tables The simple foundation of computing Data Representation The circuitry in a computer or Eight bits grouped together as a mobile device

129 views • 11 slides
DATA aka. The class where you learned to count in powers of 2. REPRESENTATION 321 ...in

DATA aka. The class where you learned to count in powers of 2. REPRESENTATION 321 ...in

DATA aka. The class where you learned to count in powers of 2. REPRESENTATION 321 ...in decimal (That is, Base 10) How did we get these? x100 x10 x1 Therefore, base 10 10 2 10 1 10 0

897 views • 50 slides
C Language Elements CSCI 112: Programming in C A simple program to convert miles to kilometers

C Language Elements CSCI 112: Programming in C A simple program to convert miles to kilometers

C Language Elements CSCI 112: Programming in C A simple program to convert miles to kilometers Ask the user for a number of miles Convert that number to kilometers Display the result to the user #include <stdio.h> #define

828 views • 29 slides
Conditional statements (1) +variable scope Based on slides by Asst. Prof. Kitsana Waiyamai,

Conditional statements (1) +variable scope Based on slides by Asst. Prof. Kitsana Waiyamai,

Conditional statements (1) +variable scope Based on slides by Asst. Prof. Kitsana Waiyamai, Ph.D. Dept. of Computer Engineering Faculty of Engineering, Kasetsart University Bangkok, Thailand http://kdl.cpe.ku.ac.th If-Then-Else Statement 1

599 views • 33 slides

More recommend