![Page 1: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/1.jpg)
CSC 252: Computer Organization Spring 2018: Lecture 2
Instructor: Yuhao Zhu
Department of Computer ScienceUniversity of Rochester
Action Items: • Programming Assignment 1 is out • Trivia 1 is due on Friday, midnight
Slide Credits: Randal E. Bryant, David R. O'Hallaron
![Page 2: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/2.jpg)
Carnegie Mellon
2
Announcement• Programming Assignment 1 is out
• Details: http://cs.rochester.edu/courses/252/spring2018/labs/assignment1.html
• Due on Feb 2, 11:59 PM • Trivia due Friday, 1/26, 11:59 PM • You have 3 slip days (not for trivia)
• Ask the TAs if you have any questions regarding programming assignments
![Page 3: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/3.jpg)
Carnegie Mellon
3
Previously in 252…
Problem
Algorithm
Program
Instruction SetArchitecture
Microarchitecture
Circuit
• How is a human-readable program translated to a representation that computers can understand?
• How does a modern computer execute that program?
}Scope ofComputer Systems(CSC 252)
![Page 4: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/4.jpg)
Carnegie Mellon
4
Previously in 252…Assembly program
movl $1, -4(%rbp) movl $2, -8(%rbp) movl -4(%rbp), %eax addl -8(%rbp), %eax
C Program
void add() { int a = 1; int b = 2; int c = a + b; }
Pre-processorCompiler
![Page 5: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/5.jpg)
Carnegie Mellon
4
Previously in 252…Assembly program
movl $1, -4(%rbp) movl $2, -8(%rbp) movl -4(%rbp), %eax addl -8(%rbp), %eax
Executable Binary
00011001 … 01101010 … 11010101 … 01110001 …
Pre-processorCompiler
AssemblerLinker
![Page 6: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/6.jpg)
Carnegie Mellon
5
Previously in 252…Assembly program
movl $1, -4(%rbp) movl $2, -8(%rbp) movl -4(%rbp), %eax addl -8(%rbp), %eax
Executable Binary
00011001 … 01101010 … 11010101 … 01110001 …
• Is ISA referring to assembly or binary?• They are the same thing; different representations.
• Instruction = Operator + Operand(s)
![Page 7: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/7.jpg)
Carnegie Mellon
5
Previously in 252…Assembly program
movl $1, -4(%rbp) movl $2, -8(%rbp) movl -4(%rbp), %eax addl -8(%rbp), %eax
Executable Binary
00011001 … 01101010 … 11010101 … 01110001 …
• Is ISA referring to assembly or binary?• They are the same thing; different representations.
• Instruction = Operator + Operand(s)
![Page 8: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/8.jpg)
Carnegie Mellon
5
Previously in 252…Assembly program
movl $1, -4(%rbp) movl $2, -8(%rbp) movl -4(%rbp), %eax addl -8(%rbp), %eax
Executable Binary
00011001 … 01101010 … 11010101 … 01110001 …
• Is ISA referring to assembly or binary?• They are the same thing; different representations.
• Instruction = Operator + Operand(s)
![Page 9: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/9.jpg)
Carnegie Mellon
6
Today: Representing Information in Binary
• Why Binary (bits)?• Bit-level manipulations• Integers
• Representation: unsigned and signed• Conversion, casting• Expanding, truncating• Addition, negation, multiplication, shifting• Summary
• Representations in memory, pointers, strings
![Page 10: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/10.jpg)
Carnegie Mellon
7
Everything is bits
![Page 11: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/11.jpg)
Carnegie Mellon
7
Everything is bits• Each bit is 0 or 1• By encoding/interpreting sets of bits in various ways
• Computers determine what to do (instructions)
![Page 12: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/12.jpg)
Carnegie Mellon
7
Everything is bits• Each bit is 0 or 1• By encoding/interpreting sets of bits in various ways
• Computers determine what to do (instructions)
• Why bits? Electronic Implementation
![Page 13: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/13.jpg)
Carnegie Mellon
7
Everything is bits• Each bit is 0 or 1• By encoding/interpreting sets of bits in various ways
• Computers determine what to do (instructions)
• Why bits? Electronic Implementation
Power (Voltage)
![Page 14: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/14.jpg)
Carnegie Mellon
7
Everything is bits• Each bit is 0 or 1• By encoding/interpreting sets of bits in various ways
• Computers determine what to do (instructions)
• Why bits? Electronic Implementation
Power (Voltage)
Transistor
![Page 15: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/15.jpg)
Carnegie Mellon
8
Everything is bits• Each bit is 0 or 1• By encoding/interpreting sets of bits in various ways
• Computers determine what to do (instructions)
• Why bits? Electronic Implementation• Transistor has two states: presence of a high voltage (“1”); presence
of a low voltage (“0”)
0.0V0.2V
0.9V1.1V
0 1 0
![Page 16: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/16.jpg)
Carnegie Mellon
9
Everything is bits• Each bit is 0 or 1• By encoding/interpreting sets of bits in various ways
• Computers determine what to do (instructions)
• Why bits? Electronic Implementation
![Page 17: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/17.jpg)
Carnegie Mellon
9
Everything is bits• Each bit is 0 or 1• By encoding/interpreting sets of bits in various ways
• Computers determine what to do (instructions)
• Why bits? Electronic Implementation
![Page 18: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/18.jpg)
Carnegie Mellon
10
Aside: Why Limit Ourselves Only to Binary?
![Page 19: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/19.jpg)
Carnegie Mellon
10
Aside: Why Limit Ourselves Only to Binary?
• Voltage is continuous. We can interpret it however we want.
![Page 20: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/20.jpg)
Carnegie Mellon
10
Aside: Why Limit Ourselves Only to Binary?
• Voltage is continuous. We can interpret it however we want.• Classic Example: Camera Sensor
• Photoelectric Effect
![Page 21: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/21.jpg)
Carnegie Mellon
10
Aside: Why Limit Ourselves Only to Binary?
• Voltage is continuous. We can interpret it however we want.• Classic Example: Camera Sensor
• Photoelectric Effect
![Page 22: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/22.jpg)
Carnegie Mellon
10
Aside: Why Limit Ourselves Only to Binary?
• Voltage is continuous. We can interpret it however we want.• Classic Example: Camera Sensor
• Photoelectric Effect
![Page 23: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/23.jpg)
Carnegie Mellon
10
Aside: Why Limit Ourselves Only to Binary?
• Voltage is continuous. We can interpret it however we want.• Classic Example: Camera Sensor
• Photoelectric Effect
![Page 24: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/24.jpg)
Carnegie Mellon
11
Binary Notation
![Page 25: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/25.jpg)
Carnegie Mellon
11
Binary Notation• Base 2 Number Representation
• e.g., 10112 = 1110
![Page 26: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/26.jpg)
Carnegie Mellon
11
Binary Notation• Base 2 Number Representation
• e.g., 10112 = 1110
•Weighted Positional Notation• Each bit has a weight depending on its position
![Page 27: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/27.jpg)
Carnegie Mellon
11
Binary Notation• Base 2 Number Representation
• e.g., 10112 = 1110
•Weighted Positional Notation• Each bit has a weight depending on its position
• b3b2b1b0 = b0*20 + b1*21 + b2*22 + b3*23
![Page 28: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/28.jpg)
Carnegie Mellon
11
Binary Notation• Base 2 Number Representation
• e.g., 10112 = 1110
•Weighted Positional Notation• Each bit has a weight depending on its position
• b3b2b1b0 = b0*20 + b1*21 + b2*22 + b3*23
• 10112 = 1*20 + 1*21 + 0*22 + 1*23 = 1110
![Page 29: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/29.jpg)
Carnegie Mellon
11
Binary Notation• Base 2 Number Representation
• e.g., 10112 = 1110
•Weighted Positional Notation• Each bit has a weight depending on its position
• b3b2b1b0 = b0*20 + b1*21 + b2*22 + b3*23
• 10112 = 1*20 + 1*21 + 0*22 + 1*23 = 1110
Decimal Binary0 00001 00012 00103 00114 01005 01016 01107 01118 10009 100110 101011 101112 110013 110114 111015 1111
![Page 30: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/30.jpg)
Carnegie Mellon
11
Binary Notation• Base 2 Number Representation
• e.g., 10112 = 1110
•Weighted Positional Notation• Each bit has a weight depending on its position
• b3b2b1b0 = b0*20 + b1*21 + b2*22 + b3*23
• 10112 = 1*20 + 1*21 + 0*22 + 1*23 = 1110
• Binary Arithmetic
Decimal Binary0 00001 00012 00103 00114 01005 01016 01107 01118 10009 100110 101011 101112 110013 110114 111015 1111
![Page 31: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/31.jpg)
Carnegie Mellon
11
Binary Notation• Base 2 Number Representation
• e.g., 10112 = 1110
•Weighted Positional Notation• Each bit has a weight depending on its position
• b3b2b1b0 = b0*20 + b1*21 + b2*22 + b3*23
• 10112 = 1*20 + 1*21 + 0*22 + 1*23 = 1110
• Binary Arithmetic
Decimal Binary0 00001 00012 00103 00114 01005 01016 01107 01118 10009 100110 101011 101112 110013 110114 111015 1111
0110 + 0101
1011
![Page 32: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/32.jpg)
Carnegie Mellon
11
Binary Notation• Base 2 Number Representation
• e.g., 10112 = 1110
•Weighted Positional Notation• Each bit has a weight depending on its position
• b3b2b1b0 = b0*20 + b1*21 + b2*22 + b3*23
• 10112 = 1*20 + 1*21 + 0*22 + 1*23 = 1110
• Binary Arithmetic
Decimal Binary0 00001 00012 00103 00114 01005 01016 01107 01118 10009 100110 101011 101112 110013 110114 111015 1111
0110 + 0101
1011
6 + 5
11
![Page 33: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/33.jpg)
Carnegie Mellon
12
Hexdecimal (Hex) Notation• Base 16 Number Representation• Use characters ‘0’ to ‘9’ and ‘A’ to ‘F’ • Four bits per Hex digit • 111111102 = FE16
• Write FA1D37B16 in C as• 0xFA1D37B • 0xfa1d37b
Hex Decimal Binary0 0 00001 1 00012 2 00103 3 00114 4 01005 5 01016 6 01107 7 01118 8 10009 9 1001A 10 1010B 11 1011C 12 1100D 13 1101E 14 1110F 15 1111
![Page 34: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/34.jpg)
Carnegie Mellon
13
Bit, Byte, Word• Byte = 8 bits• Binary 000000002 to 111111112; Decimal: 010 to 25510; Hex: 0016 to FF16 • Least Significant Bit (LSb) vs. Most Significant Bit (MSb)
10110111LSbMSb
![Page 35: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/35.jpg)
Carnegie Mellon
13
Bit, Byte, Word• Byte = 8 bits• Binary 000000002 to 111111112; Decimal: 010 to 25510; Hex: 0016 to FF16 • Least Significant Bit (LSb) vs. Most Significant Bit (MSb)
10110111LSbMSb
• Word = 4 Bytes (32-bit machine) / 8 Bytes (64-bit machine)• Least Significant Byte (LSB) vs. Most Significant Byte (MSB)
![Page 36: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/36.jpg)
Carnegie Mellon
13
Bit, Byte, Word• Byte = 8 bits• Binary 000000002 to 111111112; Decimal: 010 to 25510; Hex: 0016 to FF16 • Least Significant Bit (LSb) vs. Most Significant Bit (MSb)
10110111LSbMSb
• Word = 4 Bytes (32-bit machine) / 8 Bytes (64-bit machine)• Least Significant Byte (LSB) vs. Most Significant Byte (MSB)
Questions?
![Page 37: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/37.jpg)
Carnegie Mellon
14
Today: Representing Information in Binary
• Why Binary (bits)?• Bit-level manipulations• Integers
• Representation: unsigned and signed• Conversion, casting• Expanding, truncating• Addition, negation, multiplication, shifting• Summary
• Representations in memory, pointers, strings
![Page 38: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/38.jpg)
Carnegie Mellon
15
Boolean Algebra• Developed by George Boole in 19th Century• Algebraic representation of logic
• Encode “True” as 1 and “False” as 0
And ■ A&B = 1 when both A=1 and B=1
Or ■ A|B = 1 when either A=1 or B=1
Not ■ ~A = 1 when A=0
Exclusive-‐Or (Xor) ■ A^B = 1 when either A=1 or B=1, but not both
![Page 39: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/39.jpg)
Carnegie Mellon
16
General Boolean Algebras• Operate on Bit Vectors• Operations applied bitwise
• All of the Properties of Boolean Algebra Apply
01101001 & 01010101 01000001
01101001 | 01010101 01111101
01101001 ^ 01010101 00111100
~ 01010101 10101010
![Page 40: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/40.jpg)
Carnegie Mellon
16
General Boolean Algebras• Operate on Bit Vectors• Operations applied bitwise
• All of the Properties of Boolean Algebra Apply
01101001 & 01010101 01000001
01101001 | 01010101 01111101
01101001 ^ 01010101 00111100
~ 01010101 1010101001000001
![Page 41: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/41.jpg)
Carnegie Mellon
16
General Boolean Algebras• Operate on Bit Vectors• Operations applied bitwise
• All of the Properties of Boolean Algebra Apply
01101001 & 01010101 01000001
01101001 | 01010101 01111101
01101001 ^ 01010101 00111100
~ 01010101 1010101001000001 01111101
![Page 42: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/42.jpg)
Carnegie Mellon
16
General Boolean Algebras• Operate on Bit Vectors• Operations applied bitwise
• All of the Properties of Boolean Algebra Apply
01101001 & 01010101 01000001
01101001 | 01010101 01111101
01101001 ^ 01010101 00111100
~ 01010101 1010101001000001 01111101 00111100
![Page 43: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/43.jpg)
Carnegie Mellon
16
General Boolean Algebras• Operate on Bit Vectors• Operations applied bitwise
• All of the Properties of Boolean Algebra Apply
01101001 & 01010101 01000001
01101001 | 01010101 01111101
01101001 ^ 01010101 00111100
~ 01010101 1010101001000001 01111101 00111100 10101010
![Page 44: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/44.jpg)
Carnegie Mellon
17
Bit-Level Operations in C• Operations &, |, ~, ^ Available in C• Apply to any “integral” data type
• long, int, short, char, unsigned• View arguments as bit vectors • Arguments applied bit-‐wise
• Examples (Char data type)• ~0x41 ➙ 0xBE
• ~010000012 ➙ 101111102• ~0x00 ➙ 0xFF
• ~000000002 ➙ 111111112• 0x69 & 0x55 ➙ 0x41
• 011010012 & 010101012 ➙ 010000012• 0x69 | 0x55 ➙ 0x7D
• 011010012 | 010101012 ➙ 011111012
![Page 45: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/45.jpg)
Carnegie Mellon
18
Contrast: Logic Operations in C• Contrast to Logical Operators• &&, ||, !
• View 0 as “False” • Anything nonzero as “True” • Always return 0 or 1 • Early termination
• Examples (char data type)• !0x41 ➙ 0x00• !0x00 ➙ 0x01• !!0x41 ➙ 0x01
• 0x69 && 0x55 ➙ 0x01• 0x69 || 0x55 ➙ 0x01• p && *p (avoids null pointer access)
![Page 46: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/46.jpg)
Carnegie Mellon
18
Contrast: Logic Operations in C• Contrast to Logical Operators• &&, ||, !
• View 0 as “False” • Anything nonzero as “True” • Always return 0 or 1 • Early termination
• Examples (char data type)• !0x41 ➙ 0x00• !0x00 ➙ 0x01• !!0x41 ➙ 0x01
• 0x69 && 0x55 ➙ 0x01• 0x69 || 0x55 ➙ 0x01• p && *p (avoids null pointer access)
Watch out for && vs. & (and || vs. |)… one of the more common oopsies in C programming
![Page 47: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/47.jpg)
Carnegie Mellon
19
Shift Operations• Left Shift: x << y • Shift bit-‐vector x left y positions
• Throw away extra bits on left • Fill with 0’s on right
• Right Shift: x >> y • Shift bit-‐vector x right y positions
• Throw away extra bits on right • Logical shift
• Fill with 0’s on left • Arithmetic shift
• Replicate most significant bit on left
• Undefined Behavior• Shift amount < 0 or ≥ word size
01100010Argument x
00010000<< 3
00011000Log. >> 2
00011000Arith. >> 2
10100010Argument x
00010000<< 3
00101000Log. >> 2
11101000Arith. >> 2
![Page 48: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/48.jpg)
Carnegie Mellon
19
Shift Operations• Left Shift: x << y • Shift bit-‐vector x left y positions
• Throw away extra bits on left • Fill with 0’s on right
• Right Shift: x >> y • Shift bit-‐vector x right y positions
• Throw away extra bits on right • Logical shift
• Fill with 0’s on left • Arithmetic shift
• Replicate most significant bit on left
• Undefined Behavior• Shift amount < 0 or ≥ word size
01100010Argument x
00010000<< 3
00011000Log. >> 2
00011000Arith. >> 2
10100010Argument x
00010000<< 3
00101000Log. >> 2
11101000Arith. >> 2
00010000
![Page 49: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/49.jpg)
Carnegie Mellon
19
Shift Operations• Left Shift: x << y • Shift bit-‐vector x left y positions
• Throw away extra bits on left • Fill with 0’s on right
• Right Shift: x >> y • Shift bit-‐vector x right y positions
• Throw away extra bits on right • Logical shift
• Fill with 0’s on left • Arithmetic shift
• Replicate most significant bit on left
• Undefined Behavior• Shift amount < 0 or ≥ word size
01100010Argument x
00010000<< 3
00011000Log. >> 2
00011000Arith. >> 2
10100010Argument x
00010000<< 3
00101000Log. >> 2
11101000Arith. >> 2
0001000000010000
![Page 50: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/50.jpg)
Carnegie Mellon
19
Shift Operations• Left Shift: x << y • Shift bit-‐vector x left y positions
• Throw away extra bits on left • Fill with 0’s on right
• Right Shift: x >> y • Shift bit-‐vector x right y positions
• Throw away extra bits on right • Logical shift
• Fill with 0’s on left • Arithmetic shift
• Replicate most significant bit on left
• Undefined Behavior• Shift amount < 0 or ≥ word size
01100010Argument x
00010000<< 3
00011000Log. >> 2
00011000Arith. >> 2
10100010Argument x
00010000<< 3
00101000Log. >> 2
11101000Arith. >> 2
0001000000010000
00011000
![Page 51: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/51.jpg)
Carnegie Mellon
19
Shift Operations• Left Shift: x << y • Shift bit-‐vector x left y positions
• Throw away extra bits on left • Fill with 0’s on right
• Right Shift: x >> y • Shift bit-‐vector x right y positions
• Throw away extra bits on right • Logical shift
• Fill with 0’s on left • Arithmetic shift
• Replicate most significant bit on left
• Undefined Behavior• Shift amount < 0 or ≥ word size
01100010Argument x
00010000<< 3
00011000Log. >> 2
00011000Arith. >> 2
10100010Argument x
00010000<< 3
00101000Log. >> 2
11101000Arith. >> 2
0001000000010000
0001100000011000
![Page 52: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/52.jpg)
Carnegie Mellon
19
Shift Operations• Left Shift: x << y • Shift bit-‐vector x left y positions
• Throw away extra bits on left • Fill with 0’s on right
• Right Shift: x >> y • Shift bit-‐vector x right y positions
• Throw away extra bits on right • Logical shift
• Fill with 0’s on left • Arithmetic shift
• Replicate most significant bit on left
• Undefined Behavior• Shift amount < 0 or ≥ word size
01100010Argument x
00010000<< 3
00011000Log. >> 2
00011000Arith. >> 2
10100010Argument x
00010000<< 3
00101000Log. >> 2
11101000Arith. >> 2
0001000000010000
0001100000011000
00011000
![Page 53: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/53.jpg)
Carnegie Mellon
19
Shift Operations• Left Shift: x << y • Shift bit-‐vector x left y positions
• Throw away extra bits on left • Fill with 0’s on right
• Right Shift: x >> y • Shift bit-‐vector x right y positions
• Throw away extra bits on right • Logical shift
• Fill with 0’s on left • Arithmetic shift
• Replicate most significant bit on left
• Undefined Behavior• Shift amount < 0 or ≥ word size
01100010Argument x
00010000<< 3
00011000Log. >> 2
00011000Arith. >> 2
10100010Argument x
00010000<< 3
00101000Log. >> 2
11101000Arith. >> 2
0001000000010000
0001100000011000
0001100000011000
![Page 54: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/54.jpg)
Carnegie Mellon
19
Shift Operations• Left Shift: x << y • Shift bit-‐vector x left y positions
• Throw away extra bits on left • Fill with 0’s on right
• Right Shift: x >> y • Shift bit-‐vector x right y positions
• Throw away extra bits on right • Logical shift
• Fill with 0’s on left • Arithmetic shift
• Replicate most significant bit on left
• Undefined Behavior• Shift amount < 0 or ≥ word size
01100010Argument x
00010000<< 3
00011000Log. >> 2
00011000Arith. >> 2
10100010Argument x
00010000<< 3
00101000Log. >> 2
11101000Arith. >> 2
0001000000010000
0001100000011000
0001100000011000
00010000
![Page 55: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/55.jpg)
Carnegie Mellon
19
Shift Operations• Left Shift: x << y • Shift bit-‐vector x left y positions
• Throw away extra bits on left • Fill with 0’s on right
• Right Shift: x >> y • Shift bit-‐vector x right y positions
• Throw away extra bits on right • Logical shift
• Fill with 0’s on left • Arithmetic shift
• Replicate most significant bit on left
• Undefined Behavior• Shift amount < 0 or ≥ word size
01100010Argument x
00010000<< 3
00011000Log. >> 2
00011000Arith. >> 2
10100010Argument x
00010000<< 3
00101000Log. >> 2
11101000Arith. >> 2
0001000000010000
0001100000011000
0001100000011000
0001000000010000
![Page 56: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/56.jpg)
Carnegie Mellon
19
Shift Operations• Left Shift: x << y • Shift bit-‐vector x left y positions
• Throw away extra bits on left • Fill with 0’s on right
• Right Shift: x >> y • Shift bit-‐vector x right y positions
• Throw away extra bits on right • Logical shift
• Fill with 0’s on left • Arithmetic shift
• Replicate most significant bit on left
• Undefined Behavior• Shift amount < 0 or ≥ word size
01100010Argument x
00010000<< 3
00011000Log. >> 2
00011000Arith. >> 2
10100010Argument x
00010000<< 3
00101000Log. >> 2
11101000Arith. >> 2
0001000000010000
0001100000011000
0001100000011000
00010000
00101000
00010000
![Page 57: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/57.jpg)
Carnegie Mellon
19
Shift Operations• Left Shift: x << y • Shift bit-‐vector x left y positions
• Throw away extra bits on left • Fill with 0’s on right
• Right Shift: x >> y • Shift bit-‐vector x right y positions
• Throw away extra bits on right • Logical shift
• Fill with 0’s on left • Arithmetic shift
• Replicate most significant bit on left
• Undefined Behavior• Shift amount < 0 or ≥ word size
01100010Argument x
00010000<< 3
00011000Log. >> 2
00011000Arith. >> 2
10100010Argument x
00010000<< 3
00101000Log. >> 2
11101000Arith. >> 2
0001000000010000
0001100000011000
0001100000011000
00010000
00101000
00010000
00101000
![Page 58: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/58.jpg)
Carnegie Mellon
19
Shift Operations• Left Shift: x << y • Shift bit-‐vector x left y positions
• Throw away extra bits on left • Fill with 0’s on right
• Right Shift: x >> y • Shift bit-‐vector x right y positions
• Throw away extra bits on right • Logical shift
• Fill with 0’s on left • Arithmetic shift
• Replicate most significant bit on left
• Undefined Behavior• Shift amount < 0 or ≥ word size
01100010Argument x
00010000<< 3
00011000Log. >> 2
00011000Arith. >> 2
10100010Argument x
00010000<< 3
00101000Log. >> 2
11101000Arith. >> 2
0001000000010000
0001100000011000
0001100000011000
00010000
00101000
11101000
00010000
00101000
![Page 59: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/59.jpg)
Carnegie Mellon
19
Shift Operations• Left Shift: x << y • Shift bit-‐vector x left y positions
• Throw away extra bits on left • Fill with 0’s on right
• Right Shift: x >> y • Shift bit-‐vector x right y positions
• Throw away extra bits on right • Logical shift
• Fill with 0’s on left • Arithmetic shift
• Replicate most significant bit on left
• Undefined Behavior• Shift amount < 0 or ≥ word size
01100010Argument x
00010000<< 3
00011000Log. >> 2
00011000Arith. >> 2
10100010Argument x
00010000<< 3
00101000Log. >> 2
11101000Arith. >> 2
0001000000010000
0001100000011000
0001100000011000
00010000
00101000
11101000
00010000
00101000
11101000
![Page 60: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/60.jpg)
Carnegie Mellon
20
Today: Representing Information in Binary
• Why Binary (bits)?• Bit-level manipulations• Integers
• Representation: unsigned and signed• Conversion, casting• Expanding, truncating• Addition, negation, multiplication, shifting• Summary
• Representations in memory, pointers, strings
![Page 61: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/61.jpg)
Carnegie Mellon
21
Representing Numbers in Binary• What numbers do we need to represent in bits?
• Integer (Negative and Non-negative) • Fractions • Irrationals
0 1 2 3 4 5 6 7-1-2-3-4-5-6-7 ……
![Page 62: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/62.jpg)
Carnegie Mellon
21
Representing Numbers in Binary• What numbers do we need to represent in bits?
• Integer (Negative and Non-negative) • Fractions • Irrationals
0 1 2 3 4 5 6 7-1-2-3-4-5-6-7 ……
![Page 63: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/63.jpg)
Carnegie Mellon
22
Encoding Negative Numbers
![Page 64: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/64.jpg)
Carnegie Mellon
22
Encoding Negative Numbers• So far we have been discussing non-negative numbers: so
called unsigned. How about negative numbers?
![Page 65: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/65.jpg)
Carnegie Mellon
22
Encoding Negative Numbers• So far we have been discussing non-negative numbers: so
called unsigned. How about negative numbers?• Solution 1: Sign-magnitude
• First bit represents sign; 0 for positive; 1 for negative • The rest represents magnitude
![Page 66: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/66.jpg)
Carnegie Mellon
22
Encoding Negative Numbers• So far we have been discussing non-negative numbers: so
called unsigned. How about negative numbers?• Solution 1: Sign-magnitude
• First bit represents sign; 0 for positive; 1 for negative • The rest represents magnitude
0 1 2 3 4 5 6 7
![Page 67: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/67.jpg)
Carnegie Mellon
22
Encoding Negative Numbers• So far we have been discussing non-negative numbers: so
called unsigned. How about negative numbers?• Solution 1: Sign-magnitude
• First bit represents sign; 0 for positive; 1 for negative • The rest represents magnitude
0 1 2 3 4 5 6 7
000 001 010 011100 101 110 111
![Page 68: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/68.jpg)
Carnegie Mellon
22
Encoding Negative Numbers• So far we have been discussing non-negative numbers: so
called unsigned. How about negative numbers?• Solution 1: Sign-magnitude
• First bit represents sign; 0 for positive; 1 for negative • The rest represents magnitude
0 1 2 3-1-2-3
000 001 010 011111 110 101 100
![Page 69: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/69.jpg)
Carnegie Mellon
23
Sign-Magnitude Implications• Bits have different semantics
• Two zeros… • Normal arithmetic doesn’t work • Make hardware design harder
Signed Value
Binary
0 0001 0012 0103 011-0 100-1 101-2 110-3 111
![Page 70: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/70.jpg)
Carnegie Mellon
23
Sign-Magnitude Implications• Bits have different semantics
• Two zeros… • Normal arithmetic doesn’t work • Make hardware design harder
Signed Value
Binary
0 0001 0012 0103 011-0 100-1 101-2 110-3 111
010 +) 101
111
![Page 71: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/71.jpg)
Carnegie Mellon
23
Sign-Magnitude Implications• Bits have different semantics
• Two zeros… • Normal arithmetic doesn’t work • Make hardware design harder
Signed Value
Binary
0 0001 0012 0103 011-0 100-1 101-2 110-3 111
010 +) 101
111
2 +) -1
-3
![Page 72: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/72.jpg)
Carnegie Mellon
23
Sign-Magnitude Implications• Bits have different semantics
• Two zeros… • Normal arithmetic doesn’t work • Make hardware design harder
Signed Value
Binary
0 0001 0012 0103 011-0 100-1 101-2 110-3 111
010 +) 101
111
2 +) -1
-3
![Page 73: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/73.jpg)
Carnegie Mellon
24
Encoding Negative Numbers• Solution 2: Two’s Complement
![Page 74: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/74.jpg)
Carnegie Mellon
24
Encoding Negative Numbers• Solution 2: Two’s Complement
0 1 2 3 4 5 6 7
Unsigned Binary0 0001 0012 0103 0114 1005 1016 1107 111
![Page 75: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/75.jpg)
Carnegie Mellon
24
Encoding Negative Numbers• Solution 2: Two’s Complement
0 1 2 3-1-2-3-4
Unsigned Binary0 0001 0012 0103 0114 1005 1016 1107 111
![Page 76: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/76.jpg)
Carnegie Mellon
24
Encoding Negative Numbers• Solution 2: Two’s Complement
0 1 2 3-1-2-3-4
Unsigned Binary0 0001 0012 0103 0114 1005 1016 1107 111
Signed0123-4-3-2-1
![Page 77: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/77.jpg)
Carnegie Mellon
24
Encoding Negative Numbers• Solution 2: Two’s Complement
Signed Weight
Unsigned Weight
Bit Position
20 20 021 21 12-2 22 2
0 1 2 3-1-2-3-4
Unsigned Binary0 0001 0012 0103 0114 1005 1016 1107 111
Signed0123-4-3-2-1
![Page 78: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/78.jpg)
Carnegie Mellon
24
Encoding Negative Numbers• Solution 2: Two’s Complement
Signed Weight
Unsigned Weight
Bit Position
20 20 021 21 12-2 22 2
0 1 2 3-1-2-3-4
Unsigned Binary0 0001 0012 0103 0114 1005 1016 1107 111
Signed0123-4-3-2-1
![Page 79: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/79.jpg)
Carnegie Mellon
24
Encoding Negative Numbers• Solution 2: Two’s Complement
Signed Weight
Unsigned Weight
Bit Position
20 20 021 21 12-2 22 2
0 1 2 3-1-2-3-4
Unsigned Binary0 0001 0012 0103 0114 1005 1016 1107 111
Signed0123-4-3-2-11012 = 1*20 + 0*21 + 1*2-2 = -310
![Page 80: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/80.jpg)
Carnegie Mellon
25
Two-Complement Encoding Example x = 15213: 00111011 01101101 y = -15213: 11000100 10010011
Weight' 15213' ,15213'1' 1' 1' 1' 1'2' 0' 0' 1' 2'4' 1' 4' 0' 0'8' 1' 8' 0' 0'16' 0' 0' 1' 16'32' 1' 32' 0' 0'64' 1' 64' 0' 0'128' 0' 0' 1' 128'256' 1' 256' 0' 0'512' 1' 512' 0' 0'1024' 0' 0' 1' 1024'2048' 1' 2048' 0' 0'4096' 1' 4096' 0' 0'8192' 1' 8192' 0' 0'16384' 0' 0' 1' 16384',32768' 0' 0' 1' ,32768'
Sum$ $ 15213$ $ )15213$
![Page 81: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/81.jpg)
Carnegie Mellon
26
Two-Complement Implications• Only 1 zero• Usual arithmetic still works• There is a bit that represents sign!• Most widely used in today’s machines
Signed Binary0 0001 0012 0103 011-4 100-3 101-2 110-1 111
![Page 82: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/82.jpg)
Carnegie Mellon
26
Two-Complement Implications• Only 1 zero• Usual arithmetic still works• There is a bit that represents sign!• Most widely used in today’s machines
010 +) 101
111
Signed Binary0 0001 0012 0103 011-4 100-3 101-2 110-1 111
![Page 83: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/83.jpg)
Carnegie Mellon
26
Two-Complement Implications• Only 1 zero• Usual arithmetic still works• There is a bit that represents sign!• Most widely used in today’s machines
010 +) 101
111
2 +) -3
-1
Signed Binary0 0001 0012 0103 011-4 100-3 101-2 110-1 111
![Page 84: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/84.jpg)
Carnegie Mellon
27
Numeric Ranges
![Page 85: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/85.jpg)
Carnegie Mellon
27
Numeric Ranges• Unsigned Values
• UMin = 0 000…0
• UMax = 2w – 1 111…1
![Page 86: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/86.jpg)
Carnegie Mellon
27
Numeric Ranges• Unsigned Values
• UMin = 0 000…0
• UMax = 2w – 1 111…1
• Two’s Complement Values▪ TMin = –2w–1
100…0 ▪ TMax = 2w–1 – 1
011…1
![Page 87: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/87.jpg)
Carnegie Mellon
27
Numeric Ranges• Unsigned Values
• UMin = 0 000…0
• UMax = 2w – 1 111…1
• Two’s Complement Values▪ TMin = –2w–1
100…0 ▪ TMax = 2w–1 – 1
011…1
Decimal Hex Binary UMax 65535 FF FF 11111111 11111111 TMax 32767 7F FF 01111111 11111111 TMin -32768 80 00 10000000 00000000 -1 -1 FF FF 11111111 11111111 0 0 00 00 00000000 00000000
Values for W = 16
![Page 88: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/88.jpg)
Carnegie Mellon
27
Numeric Ranges• Unsigned Values
• UMin = 0 000…0
• UMax = 2w – 1 111…1
• Two’s Complement Values▪ TMin = –2w–1
100…0 ▪ TMax = 2w–1 – 1
011…1• Other Values
▪ Minus 1 111…1
Decimal Hex Binary UMax 65535 FF FF 11111111 11111111 TMax 32767 7F FF 01111111 11111111 TMin -32768 80 00 10000000 00000000 -1 -1 FF FF 11111111 11111111 0 0 00 00 00000000 00000000
Values for W = 16
![Page 89: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/89.jpg)
Carnegie Mellon
28
Data Representations in C (in Bytes)• By default variables are signed• Unless explicitly declared as unsigned (e.g., unsigned int)• Signed variables use two-complement encoding
C Data Type 32-‐bit 64-‐bit
char 1 1
short 2 2
int 4 4
long 4 8
![Page 90: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/90.jpg)
Carnegie Mellon
29
Data Representations in C (in Bytes)
C Data Type 32-‐bit 64-‐bit
char 1 1
short 2 2
int 4 4
long 4 8
! W"! 8" 16" 32" 64"
UMax" 255! 65,535! 4,294,967,295! 18,446,744,073,709,551,615!TMax" 127! 32,767! 2,147,483,647! 9,223,372,036,854,775,807!TMin" -128! -32,768! -2,147,483,648! -9,223,372,036,854,775,808!
!!
![Page 91: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/91.jpg)
Carnegie Mellon
29
Data Representations in C (in Bytes)
C Data Type 32-‐bit 64-‐bit
char 1 1
short 2 2
int 4 4
long 4 8
! W"! 8" 16" 32" 64"
UMax" 255! 65,535! 4,294,967,295! 18,446,744,073,709,551,615!TMax" 127! 32,767! 2,147,483,647! 9,223,372,036,854,775,807!TMin" -128! -32,768! -2,147,483,648! -9,223,372,036,854,775,808!
!! • C Language
•#include <limits.h>•Declares constants, e.g.,
•ULONG_MAX•LONG_MAX•LONG_MIN
•Values platform specific
![Page 92: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/92.jpg)
Carnegie Mellon
30
Can We Represent Fractions in Binary?• What does 10.012 mean?• C.f., Decimal
• 12.45 = 1*101 + 2*100 + 4*10-1 + 5*10-2
• 10.012 = 1*21 + 0*20 + 0*2-1 + 1*2-2 = 2.2510
![Page 93: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/93.jpg)
Carnegie Mellon
30
Can We Represent Fractions in Binary?• What does 10.012 mean?• C.f., Decimal
• 12.45 = 1*101 + 2*100 + 4*10-1 + 5*10-2
• 10.012 = 1*21 + 0*20 + 0*2-1 + 1*2-2 = 2.2510
Decimal Binary0 00.000.25 00.010.5 00.100.75 00.111 01.001.25 01.011.5 01.101.75 01.112 10.002.25 10.012.5 10.102.75 10.113 11.003.25 11.013.5 11.103.75 11.11
![Page 94: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/94.jpg)
Carnegie Mellon
30
Can We Represent Fractions in Binary?• What does 10.012 mean?• C.f., Decimal
• 12.45 = 1*101 + 2*100 + 4*10-1 + 5*10-2
• 10.012 = 1*21 + 0*20 + 0*2-1 + 1*2-2 = 2.2510
Decimal Binary0 00.000.25 00.010.5 00.100.75 00.111 01.001.25 01.011.5 01.101.75 01.112 10.002.25 10.012.5 10.102.75 10.113 11.003.25 11.013.5 11.103.75 11.11
0 1 2 3 4 5 6 7 …. 15
![Page 95: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/95.jpg)
Carnegie Mellon
30
Can We Represent Fractions in Binary?• What does 10.012 mean?• C.f., Decimal
• 12.45 = 1*101 + 2*100 + 4*10-1 + 5*10-2
• 10.012 = 1*21 + 0*20 + 0*2-1 + 1*2-2 = 2.2510
Decimal Binary0 00.000.25 00.010.5 00.100.75 00.111 01.001.25 01.011.5 01.101.75 01.112 10.002.25 10.012.5 10.102.75 10.113 11.003.25 11.013.5 11.103.75 11.11
0 1 2 3
![Page 96: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/96.jpg)
Carnegie Mellon
30
Can We Represent Fractions in Binary?• What does 10.012 mean?• C.f., Decimal
• 12.45 = 1*101 + 2*100 + 4*10-1 + 5*10-2
• 10.012 = 1*21 + 0*20 + 0*2-1 + 1*2-2 = 2.2510
Decimal Binary0 00.000.25 00.010.5 00.100.75 00.111 01.001.25 01.011.5 01.101.75 01.112 10.002.25 10.012.5 10.102.75 10.113 11.003.25 11.013.5 11.103.75 11.11
0 1 2 3
01.10 + 01.01
10.11
1.50 + 1.25
2.75
![Page 97: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/97.jpg)
Carnegie Mellon
30
Can We Represent Fractions in Binary?• What does 10.012 mean?• C.f., Decimal
• 12.45 = 1*101 + 2*100 + 4*10-1 + 5*10-2
• 10.012 = 1*21 + 0*20 + 0*2-1 + 1*2-2 = 2.2510
Decimal Binary0 00.000.25 00.010.5 00.100.75 00.111 01.001.25 01.011.5 01.101.75 01.112 10.002.25 10.012.5 10.102.75 10.113 11.003.25 11.013.5 11.103.75 11.11
0 1 2 3
01.10 + 01.01
10.11
1.50 + 1.25
2.75
Integer Arithmetic Still Works!
![Page 98: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/98.jpg)
Carnegie Mellon
31
Fixed-Point RepresentationDecimal Binary0 00.000.25 00.010.5 00.100.75 00.111 01.001.25 01.011.5 01.101.75 01.112 10.002.25 10.012.5 10.102.75 10.113 11.003.25 11.013.5 11.103.75 11.11
• Fixed interval between two representable numbers as long as the binary point stays fixed
• Each bit represents 0.2510 • Fixed-point representation of numbers
• Integer is one special case of fixed-point
0 1 2 3
01.10 + 01.01
10.11
1.50 + 1.25
2.75
![Page 99: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/99.jpg)
Carnegie Mellon
32
Aside: Quantization
8 129 10 1176543210 13 14 15
![Page 100: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/100.jpg)
Carnegie Mellon
32
Aside: Quantization
• Representing all integers precisely requires 4 bits
8 129 10 1176543210 13 14 15
![Page 101: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/101.jpg)
Carnegie Mellon
32
Aside: Quantization
• Representing all integers precisely requires 4 bits•What if we can tolerate some imprecisions
• 1, 2, 3 are approximated by 0 • 5, 6, 7 are approximated by 4… • We would only need 2 bits
8 129 10 1176543210 13 14 15
![Page 102: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/102.jpg)
Carnegie Mellon
32
Aside: Quantization
• Representing all integers precisely requires 4 bits•What if we can tolerate some imprecisions
• 1, 2, 3 are approximated by 0 • 5, 6, 7 are approximated by 4… • We would only need 2 bits
8 129 10 1176540 13 14 15
![Page 103: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/103.jpg)
Carnegie Mellon
32
Aside: Quantization
• Representing all integers precisely requires 4 bits•What if we can tolerate some imprecisions
• 1, 2, 3 are approximated by 0 • 5, 6, 7 are approximated by 4… • We would only need 2 bits
8 129 10 1140 13 14 15
![Page 104: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/104.jpg)
Carnegie Mellon
32
Aside: Quantization
• Representing all integers precisely requires 4 bits•What if we can tolerate some imprecisions
• 1, 2, 3 are approximated by 0 • 5, 6, 7 are approximated by 4… • We would only need 2 bits
8 1240 13 14 15
![Page 105: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/105.jpg)
Carnegie Mellon
32
Aside: Quantization
• Representing all integers precisely requires 4 bits•What if we can tolerate some imprecisions
• 1, 2, 3 are approximated by 0 • 5, 6, 7 are approximated by 4… • We would only need 2 bits
8 1240
![Page 106: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/106.jpg)
Carnegie Mellon
32
Aside: Quantization
• Representing all integers precisely requires 4 bits•What if we can tolerate some imprecisions
• 1, 2, 3 are approximated by 0 • 5, 6, 7 are approximated by 4… • We would only need 2 bits
8 1240
00 01 10 11
![Page 107: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/107.jpg)
Carnegie Mellon
32
Aside: Quantization
• Representing all integers precisely requires 4 bits•What if we can tolerate some imprecisions
• 1, 2, 3 are approximated by 0 • 5, 6, 7 are approximated by 4… • We would only need 2 bits
• That is, 1 bit represents 410 • 102 becomes 4 * (1 * 21) = 8 • Every time we increment a bit, the value is incremented by 4 • 1, 2, 3 are represented approximately by 102
8 1240
00 01 10 11
![Page 108: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/108.jpg)
Carnegie Mellon
32
Aside: Quantization
• Representing all integers precisely requires 4 bits•What if we can tolerate some imprecisions
• 1, 2, 3 are approximated by 0 • 5, 6, 7 are approximated by 4… • We would only need 2 bits
• That is, 1 bit represents 410 • 102 becomes 4 * (1 * 21) = 8 • Every time we increment a bit, the value is incremented by 4 • 1, 2, 3 are represented approximately by 102
8 1240
00 01 10 11
Note that this is different from “base 4”• 104 = 1 * 41 + 0 * 40 = 4 • Every increment still only increments 1
![Page 109: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/109.jpg)
Carnegie Mellon
33
Aside: Quantization
8 1240
00 01 10 11
![Page 110: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/110.jpg)
Carnegie Mellon
33
Aside: Quantization
• Saves storage space and improves computation speed• 50% space saving • 4-bit arithmetic becomes 2-bit arithmetic
8 1240
00 01 10 11
![Page 111: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/111.jpg)
Carnegie Mellon
33
Aside: Quantization
• Saves storage space and improves computation speed• 50% space saving • 4-bit arithmetic becomes 2-bit arithmetic
• Many real-world applications can tolerate imprecisions• Image processing • Computer vision • Real-time graphics • Machine learning (Neural networks)
8 1240
00 01 10 11
![Page 112: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/112.jpg)
Carnegie Mellon
33
Aside: Quantization
• Saves storage space and improves computation speed• 50% space saving • 4-bit arithmetic becomes 2-bit arithmetic
• Many real-world applications can tolerate imprecisions• Image processing • Computer vision • Real-time graphics • Machine learning (Neural networks)
8 1240
00 01 10 11
![Page 113: CSC 252: Computer Organization Spring 2018: Lecture 2 · Announcement •Programming ... • Addition, negation, multiplication, shifting ... •All of the Properties of Boolean Algebra](https://reader034.vdocuments.us/reader034/viewer/2022050422/5f917db6bbad246b860a219a/html5/thumbnails/113.jpg)
Carnegie Mellon
33
Aside: Quantization
• Saves storage space and improves computation speed• 50% space saving • 4-bit arithmetic becomes 2-bit arithmetic
• Many real-world applications can tolerate imprecisions• Image processing • Computer vision • Real-time graphics • Machine learning (Neural networks)
8 1240
00 01 10 11Questions?