The document discusses binary numbers and arithmetic. It covers topics like addition, subtraction, multiplication in binary, and different methods for representing signed integers like two's complement. It explains how two's complement works by using bitwise operations to represent negative numbers. For example, it shows that adding two positive 8-bit binary numbers in two's complement is simply the bitwise addition, while subtraction can be performed by adding the number and the two's complement of the subtrahend. The document also discusses issues like carry vs overflow that can occur during binary arithmetic operations.
Logic Circuits Design - "Chapter 1: Digital Systems and Information"Ra'Fat Al-Msie'deen
Logic Circuits Design: This material is based on chapter 1 of “Logic and Computer Design Fundamentals” by M. Morris Mano, Charles R. Kime and Tom Martin
Introduction
Number Systems
Types of Number systems
Inter conversion of number systems
Binary addition ,subtraction, multiplication and
division
Complements of binary number(1’s and 2’s
complement)
Grey code, ASCII, Ex
3,BCD
Logic Circuits Design - "Chapter 1: Digital Systems and Information"Ra'Fat Al-Msie'deen
Logic Circuits Design: This material is based on chapter 1 of “Logic and Computer Design Fundamentals” by M. Morris Mano, Charles R. Kime and Tom Martin
Introduction
Number Systems
Types of Number systems
Inter conversion of number systems
Binary addition ,subtraction, multiplication and
division
Complements of binary number(1’s and 2’s
complement)
Grey code, ASCII, Ex
3,BCD
FYBSC IT Digital Electronics Unit I Chapter II Number System and Binary Arith...Arti Parab Academics
Binary Arithmetic:
Binary addition, Binary subtraction, Negative number representation,
Subtraction using 1’s complement and 2’s complement, Binary
multiplication and division, Arithmetic in octal number system,
Arithmetic in hexadecimal number system, BCD and Excess – 3
arithmetic.
In this slide we have discussed, different arithmetic operations like addition, subtraction, multiplication and division for binary numbers. Addition and subtraction operation is achieved using one's complement and two's complement number system.
DIGITAL ELECTRONICS- Number System
Positional Notation
Radix (Base) of a Number System
Radix (Base) of a Number System
Decimal Number Systems
Binary Number System
Decimal to Binary Conversion
Converting Decimal to Binary
FYBSC IT Digital Electronics Unit I Chapter II Number System and Binary Arith...Arti Parab Academics
Binary Arithmetic:
Binary addition, Binary subtraction, Negative number representation,
Subtraction using 1’s complement and 2’s complement, Binary
multiplication and division, Arithmetic in octal number system,
Arithmetic in hexadecimal number system, BCD and Excess – 3
arithmetic.
In this slide we have discussed, different arithmetic operations like addition, subtraction, multiplication and division for binary numbers. Addition and subtraction operation is achieved using one's complement and two's complement number system.
DIGITAL ELECTRONICS- Number System
Positional Notation
Radix (Base) of a Number System
Radix (Base) of a Number System
Decimal Number Systems
Binary Number System
Decimal to Binary Conversion
Converting Decimal to Binary
Reviewing number systems involves understanding various ways in which numbers can be represented and manipulated. Here's a brief overview of different number systems:
Decimal System (Base-10):
This is the most common number system used by humans.
It uses 10 digits (0-9) to represent numbers.
Each digit's position represents a power of 10.
For example, the number 245 in decimal represents (2 * 10^2) + (4 * 10^1) + (5 * 10^0).
Binary System (Base-2):
Used internally by almost all modern computers.
It uses only two digits: 0 and 1.
Each digit's position represents a power of 2.
For example, the binary number 1011 represents (1 * 2^3) + (0 * 2^2) + (1 * 2^1) + (1 * 2^0) in decimal, which equals 11.
Octal System (Base-8):
Less commonly used, but still relevant in some computer programming contexts.
It uses eight digits: 0 to 7.
Each digit's position represents a power of 8.
For example, the octal number 34 represents (3 * 8^1) + (4 * 8^0) in decimal, which equals 28.
Hexadecimal System (Base-16):
Widely used in computer science and programming.
It uses sixteen digits: 0 to 9 followed by A to F (representing 10 to 15).
Each digit's position represents a power of 16.
Often used to represent memory addresses and binary data more compactly.
For example, the hexadecimal number 2F represents (2 * 16^1) + (15 * 16^0) in decimal, which equals 47.
Each number system has its own advantages and applications. Decimal is intuitive for human comprehension, binary is fundamental in computing due to its simplicity for electronic systems, octal and hexadecimal are often used for human-readable representations of binary data in programming, particularly when dealing with memory addresses and byte-oriented data.
Understanding these number systems is essential for various fields such as computer science, electrical engineering, and mathematics, as they provide different perspectives on how numbers can be represented and manipulated.
A digital system can understand positional number system only where there are only a few symbols called digits and these symbols represent different values depending on the position they occupy in the number.
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June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
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• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
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Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
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It is possible to hide or invisible some fields in odoo. Commonly using “invisible” attribute in the field definition to invisible the fields. This slide will show how to make a field invisible in odoo 17.
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This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
12. Multiplication (binary)
1101
1011
1101
11010
1101000
10001111
It’s interesting to note
that binary multiplication
is a sequence of shifts
and adds of the first
term (depending on the
bits in the second term.
110100 is missing here
because the
corresponding bit in the
second terms is 0.
Admission.edhole.com
14. Representing numbers (ints)
Fixed, finite number of bits.
bits bytes C/C++ Intel Sun
8 1 char [s]byte byte
16 2 short [s]word half
32 4 int or long [s]dword word
64 8 long long [s]qword xword
15. Representing numbers (ints)
Fixed, finite number of bits.
bits Intel signed unsigned
8 [s]byte -27..+27-1 0..+28-1
16 [s]word -215..+215-1 0..+216-1
32 [s]dword -231..+231-1 0..+232-1
64 [s]qword -263..+263-1 0..+264-1
In general, for k bits, the unsigned range is [0..+2k-1] and
the signed range is [-2k-1..+2k-1-1].
16. Methods for representing signed ints.
1. signed magnitude
2. 1’s complement (diminished radix complement)
3. 2’s complement (radix complement)
4. excess bD-1
17. Signed magnitude
Ex. 4-bit signed magnitude
1 bit for sign
3 bits for magnitude
N N
0 0000 1000
1 0001 1001
2 0010 1010
3 0011 1011
4 0100 1100
5 0101 1101
6 0110 1110
7 0111 1111
18. Signed magnitude
Ex. 4-bit signed magnitude
1 bit for sign
3 bits for magnitude
N N
0 0000 1000
1 0001 1001
2 0010 1010
3 0011 1011
4 0100 1100
5 0101 1101
6 0110 1110
7 0111 1111
19. 1’s complement
(diminished radix complement)
Let x be a non-negative number.
Then –x is represented by bD-1+(-x) where
b = base
D = (total) # of bits (including the sign bit)
Ex. Let b=2 and D=4.
Then -1 is represented by 24-1-1 = 1410 or 11102.
20. 1’s complement
(diminished radix complement)
Let x be a non-negative number.
Then –x is represented by bD-1+(-x) where
b = base & D = (total) # of bits (including the sign bit)
Ex. What is the 9’s complement of 1238910?
Given b=10 and D=5. Then the 9’s complement of 12389
= 105 – 1 – 12389
= 100000 – 1 – 12389
= 99999 – 12389
= 87610
21. 1’s complement
(diminished radix complement)
Let x be a non-negative number.
Then –x is represented by bD-1+(-x)
where
b = base
D = (total) # of bits (including the sign
bit)
Shortcut for base 2?
All combinations used, but 2 zeros!
N N
0 0000 1111
1 0001 1110
2 0010 1101
3 0011 1100
4 0100 1011
5 0101 1010
6 0110 1001
7 0111 1000
22. 2’s complement
(radix complement)
Let x be a non-negative number.
Then –x is represented by bD+(-x).
Ex. Let b=2 and D=4. Then -1 is represented by 24-
1 = 15 or 11112.
Ex. Let b=2 and D=4. Then -5 is represented by 24
– 5 = 11 or 10112.
Ex. Let b=10 and D=5. Then the 10’s complement
of 12389 = 105 – 12389 = 100000 – 12389 = 87611.
23. 2’s complement
(radix complement)
Let x be a non-negative number.
Then –x is represented by bD+(-x).
Ex. Let b=2 and D=4. Then -1 is
represented by 24-1 = 15 or 11112.
Ex. Let b=2 and D=4. Then -5 is
represented by 24 – 5 = 11 or 10112.
Shortcut for base 2?
N N
0 0000 0000
1 0001 1111
2 0010 1110
3 0011 1101
4 0100 1100
5 0101 1011
6 0110 1010
7 0111 1001
24. 2’s complement
(radix complement)
Shortcut for base 2?
Yes! Flip the bits and add 1.
N N
0 0000 0000
1 0001 1111
2 0010 1110
3 0011 1101
4 0100 1100
5 0101 1011
6 0110 1010
7 0111 1001
25. 2’s complement
(radix complement)
Are all combinations of 4 bits used?
No. (Now we only have one zero.)
1000 is missing!
What is 1000?
Is it positive or negative?
Does -8 + 1 = -7 work in 2’s complement?
N N
0 0000 0000
1 0001 1111
2 0010 1110
3 0011 1101
4 0100 1100
5 0101 1011
6 0110 1010
7 0111 1001
26. excess bD-1 (biased
representation)
For pos, neg, and 0, x is represented by
bD-1 + x
Ex. Let b=2 and D=4. Then the excess 8 (24-1)
representation for 0 is 8+0 = 8 or 10002.
Ex. Let b=2 and D=4. Then excess 8 for -1 is 8 –
1 = 7 or 01112.
27. excess bD-1
For pos, neg, and 0, x is represented
by
bD-1 + x.
Ex. Let b=2 and D=4. Then the
excess 8 (24-1) representation for 0 is
8+0 = 8 or 10002.
Ex. Let b=2 and D=4. Then excess
8 for -1 is 8 – 1 = 7 or 01112.
N N
0 1000 1000
1 1001 0111
2 1010 0110
3 1011 0101
4 1100 0100
5 1101 0011
6 1110 0010
7 1111 0001
28. 2’s complement vs. excess bD-1
In 2’s, positives start with 0; in
excess, positives start with 1.
Both have one zero (positive).
Remaining bits are the same.
N N
0 1000 1000
1 1001 0111
2 1010 0110
3 1011 0101
4 1100 0100
5 1101 0011
6 1110 0010
7 1111 0001
29. Summary of methods for
representing signed ints.
signedMag sComp sComp excess
1 2 8
N n n n n n
n
0 0000 1000 1111 0000 1000 1000
1 0001 1001 1110 1111 0111 1001
2 0010 1010 1101 1110 0110 1010
3 0011 1011 1100 1101 0101 1011
4 0100 1100 1011 1100 0100 1100
5 0101 1101 1010 1011 0011 1101
6 0110 1110 1001 1010 0010 1110
7 0111 1111 1000 1001 0001 1111
1000=-8| 0000 unused
32. Addition w/ signed magnitude
algorithm
For A - B, change the sign of B and perform addition of
A + (-B) (as in the next step)
For A + B:
if (Asign==Bsign) then { R = |A| + |B|; Rsign = Asign; }
else if (|A|>|B|) then { R = |A| - |B|; Rsign = Asign; }
else if (|A|==|B|) then { R = 0; Rsign = 0; }
else { R = |B| - |A|; Rsign = Bsign; }
Complicated?
34. Representing numbers (ints)
using 2’s complement
Fixed, finite number of bits.
bits Intel signed
8 sbyte -27..+27-1
16 sword -215..+215-1
32 sdword -231..+231-1
64 sqword -263..+263-1
In general, for k bits, the signed range is [-2k-1..+2k-1-1].
So where does the extra negative value come from?
35. Representing numbers (ints)
Fixed, finite number of bits.
bits Intel signed
8 sbyte -27..+27-1
16 sword -215..+215-1
32 sdword -231..+231-1
64 sqword -263..+263-1
In general, for k bits, the signed range
is
[-2k-1..+2k-1-1].
So where does the extra negative value
come from?
n n
0 0000 0000
1 0001 1111
2 0010 1110
3 0011 1101
4 0100 1100
5 0101 1011
6 0110 1010
7 0111 1001
8 1000
36. Addition of 2’s complement
binary numbers
Consider 8-bit 2’s complement binary numbers.
Then the msb (bit 7) is the sign bit. If this bit is 0,
then this is a positive number; if this bit is 1, then
this is a negative number.
Addition of 2 positive numbers.
Ex. 40 + 58 = 98
1 1 1
00101000
00111010
01100010
37. Addition of 2’s complement
binary numbers
Consider 8-bit 2’s complement
binary numbers.
Addition of a negative to a
positive.
What are the values of these 2
terms?
-88 and 122
-88 + 122 = 34
1 1 1 1
10101000
01111010
1 00100010
41. Addition of 2’s complement
binary numbers
Carry vs. overflow when adding A + B
If A and B are of opposite sign, then overflow cannot
occur.
If A and B are of the same sign but the result is of
the opposite sign, then overflow has occurred (and
the answer is therefore incorrect).
Overflow occurs iff the carry into the sign bit differs from the
carry out of the sign bit.
42. Addition of 2’s complement
binary numbers
class test {
public static void main ( String
args[] )
{
byte A = 127;
byte B = 127;
byte result = (byte)(A + B);
System.out.println( "A + B = "
+ result );
}
}
#include <stdio.h>
int main ( int argc, char* argv[] )
{
char A = 127;
char B = 127;
char result = (char)(A + B);
printf( "A + B = %d n", result );
return 0;
} Result = -2 in both
Java (left) and C++
(right). Why?
43. Addition of 2’s complement
binary numbers
class test {
public static void main ( String
args[] )
{
byte A = 127;
byte B = 127;
byte result = (byte)(A + B);
System.out.println( "A + B = "
+ result );
}
}
Result = -2 in both Java and
C++.
Why?
What’s 127 as a 2’s
complement binary number?
01111111
01111111
11111110
What is 111111102?
Flip the bits: 00000001.
Then add 1: 00000010.
45. Addition with 1’s complement
Note: 1’s complement has two 0’s!
1’s complement addition is tricky
(end-around-carry).
N N
0 0000 1111
1 0001 1110
2 0010 1101
3 0011 1100
4 0100 1011
5 0101 1010
6 0110 1001
7 0111 1000
46. 8-bit 1’s complement addition
Ex. Let X = A816 and Y = 8616.
Calculate Y - X using 1’s complement.
47. 8-bit 1’s complement addition
Ex. Let X = A816 and Y = 8616.
Calculate Y - X using 1’s complement.
Y = 1000 01102 = -12110
X = 1010 10002 = -8710
~X = 0101 01112
(Note: C=0 out of msb.)
1000 0110
0101 0111
1101 1101
Y - X = -121 + 87 = -34 (base 10)
48. 8-bit 1’s complement addition
Ex. Let X = A816 and Y = 8616.
Calculate X - Y using 1’s complement.
49. 8-bit 1’s complement addition
Ex. Let X = A816 and Y = 8616.
Calculate X - Y using 1’s complement.
X = 1010 10002 = -8710
Y = 1000 01102 = -12110
~Y = 0111 10012
(Note: C=1 out of msb.)
1010 1000
0111 1001
1 0010 0001
1
end around
carry
0010 0010
X - Y = -87 + 121 = 34 (base 10)