This document discusses 2's complement arithmetic in digital electronics. It explains that subtracting one number from another is the same as making one number negative and adding them. It then demonstrates how to represent negative numbers in binary by taking the 2's complement of a number, which involves complementing all its digits and adding 1. Various examples are provided of adding positive and negative binary numbers by taking the 2's complement of negative terms before adding. The most significant bit is identified as the sign bit that determines if a number is positive or negative.
Description of all types of Loaders from System programming subjects.
eg. Compile-Go Loader
General Loader
Absolute Loader
Relocating Loader
Practical Relocating Loader
Linking Loader
Linker Vs. Loader
general relocatable loader
A macro processor is a system software. Macro is that the Section of code that the programmer writes (defines) once, and then can use or invokes many times.
Description of all types of Loaders from System programming subjects.
eg. Compile-Go Loader
General Loader
Absolute Loader
Relocating Loader
Practical Relocating Loader
Linking Loader
Linker Vs. Loader
general relocatable loader
A macro processor is a system software. Macro is that the Section of code that the programmer writes (defines) once, and then can use or invokes many times.
Topics Covered:
Linker: Types of Linker:
Loaders : Types of loader
Example of Translator, Link and Load Time Address
Object Module
Difference between Static and Dynamic Binding
Translator, Link and Load Time Address
Program Relocatability
In these slides the registration organization and stack organization have discussed in detail. Stack organization is discussed with the aid of animation to let the user understand it in a better and easy way.
CPU REGISTERS
what is cpu registers
types of cpu registers
function of cpu registers
explanation of cpu registers
categories of cpu registers
5 major categories of cpu registers
pipelining is the concept of decomposing the sequential process into number of small stages in which each stage execute individual parts of instruction life cycle inside the processor.
Assemblers Elements of Assembly Language Programming, Design of the Assembler, Assembler Design Criteria, Types of Assemblers, Two-Pass Assemblers, One-Pass Assemblers, Single pass Assembler for Intel x86 , Algorithm of Single Pass Assembler, Multi-Pass Assemblers, Advanced Assembly Process, Variants of Assemblers Design of two pass assembler
Topics Covered:
Linker: Types of Linker:
Loaders : Types of loader
Example of Translator, Link and Load Time Address
Object Module
Difference between Static and Dynamic Binding
Translator, Link and Load Time Address
Program Relocatability
In these slides the registration organization and stack organization have discussed in detail. Stack organization is discussed with the aid of animation to let the user understand it in a better and easy way.
CPU REGISTERS
what is cpu registers
types of cpu registers
function of cpu registers
explanation of cpu registers
categories of cpu registers
5 major categories of cpu registers
pipelining is the concept of decomposing the sequential process into number of small stages in which each stage execute individual parts of instruction life cycle inside the processor.
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Binary addition, Binary subtraction, Negative number representation, Subtraction using 1’s complement and 2’s complement, Binary multiplication and division, Arithmetic in octal, hexadecimal number system, BCD and Excess – 3 arithmetic
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2. 2’s Complement Arithmetic
This presentation will demonstrate
• That subtracting one number from another is
the same as making one number negative
and adding.
• How to create negative numbers in the binary
number system.
• The 2’s Complement Process.
• How the 2’s complement process can be use
to add (and subtract) binary numbers.
2
3. Negative Numbers?
• Digital electronics requires frequent addition and
subtraction of numbers. You know how to design
an adder, but what about a subtract-er?
• A subtract-er is not needed with the 2’s
complement process. The 2’s complement
process allows you to easily convert a positive
number into its negative equivalent.
• Since subtracting one number from another is
the same as making one number negative and
adding, the need for a subtract-er circuit has
been eliminated.
3
4. How To Create A Negative Number
• In digital electronics you cannot simply put a
minus sign in front of a number to make it
negative.
• You must represent a negative number in a fixedlength binary number system. All signed
arithmetic must be performed in a fixed-length
number system.
• A physical fixed-length device (usually memory)
contains a fixed number of bits (usually 4-bits, 8bits, 16-bits) to hold the number.
4
5. 3-Digit Decimal Number System
A bicycle odometer with only
three digits is an example of a
fixed-length decimal number
system.
The problem is that without a
negative sign, you cannot tell
a +998 from a -2 (also a 998).
Did you ride forward for 998
miles or backward for 2 miles?
Note: Car odometers do not work this way.
999
forward (+)
998
997
001
000
999
998
002
001
backward (-)
5
6. Negative Decimal
How do we represent
negative numbers in this 3digit decimal number system
without using a sign?
+499
499
+498
498
+497
497
Cut the number system in half.
+001
001
Use 001 – 499 to indicate
positive numbers.
000
000
-001
999
-002
998
-499
501
-500
500
Use 500 – 999 to indicate
negative numbers.
Notice that 000 is not positive
or negative.
pos(+)
neg(-)
6
8. Complex Problems
• The previous examples demonstrate that this
process works, but how do we easily convert a
number into its negative equivalent?
• In the examples, converting the negative
numbers into the 3-digit decimal number system
was fairly easy. To convert the (-3), you simply
counted backward from 1000 (i.e., 999, 998,
997).
• This process is not as easy for large numbers
(e.g., -214 is 786). How did we determine this?
• To convert a large negative number, you can use
the 10’s Complement Process.
8
9. 10’s Complement Process
The 10’s Complement process uses base-10
(decimal) numbers. Later, when we’re working with
base-2 (binary) numbers, you will see that the 2’s
Complement process works in the same way.
First, complement all of the digits in a number.
– A digit’s complement is the number you add to the digit to
make it equal to the largest digit in the base (i.e., 9 for
decimal). The complement of 0 is 9, 1 is 8, 2 is 7, etc.
Second, add 1.
– Without this step, our number system would have two
zeroes (+0 & -0), which no number system has.
9
11. 8-Bit Binary Number System
Apply what you have learned to the
binary number systems. How do you
represent negative numbers in this 8-bit
binary system?
Use 00000001 – 01111111 to indicate
positive numbers.
Use 10000000 – 11111111 to indicate
negative numbers.
Notice that 00000000 is not positive or
negative.
01111111
+126
01111110
+125
01111101
+1
00000001
0
00000000
-1
11111111
-2
Cut the number system in half.
+127
11111110
-127
10000001
-128
10000000
pos(+)
neg(-)
11
12. Sign Bit
•
•
•
What did do you notice about the
most significant bit of the binary
numbers?
The MSB is (0) for all positive
numbers.
+127
01111111
+126
01111110
+125
01111101
+1
00000001
The MSB is (1) for all negative
numbers.
0
00000000
-1
11111111
•
The MSB is called the sign bit.
-2
11111110
•
In a signed number system, this
allows you to instantly determine
whether a number is positive or
negative.
-127
10000001
-128
10000000
pos(+)
neg(-)
12
13. 2’S Complement Process
The steps in the 2’s Complement process are similar
to the 10’s Complement process. However, you will
now use the base two.
First, complement all of the digits in a number.
– A digit’s complement is the number you add to the digit to
make it equal to the largest digit in the base (i.e., 1 for
binary). In binary language, the complement of 0 is 1, and
the complement of 1 is 0.
Second, add 1.
– Without this step, our number system would have two
zeroes (+0 & -0), which no number system has.
13
15. Using The 2’s Compliment Process
Use the 2’s complement process to add together
the following numbers.
POS
+ POS
POS
9
⇒+ 5
14
NEG
+ POS
NEG
⇒
POS
+ NEG
POS
9
⇒ + (-5)
4
NEG
+ NEG
NEG
(-9)
⇒ + (-5)
- 14
(-9)
+ 5
-4
15
16. POS + POS → POS Answer
If no 2’s complement is needed, use regular binary
addition.
9
+ 5
14
→
→
00001001
+ 00000101
←
00001110
16
17. POS + NEG → POS Answer
Take the 2’s complement of the negative number and
use regular binary addition.
9
+ (-5)
4
→
←
00001001
+ 11111011
1]00000100
8th Bit = 0: Answer is Positive
Disregard 9th Bit
00000101
↓↓↓↓↓↓↓↓
11111010
+1
11111011
2’s
Complement
Process
17
18. POS + NEG → NEG Answer
Take the 2’s complement of the negative number and
use regular binary addition.
(-9)
+ 5
-4
→
←
11110111
+ 00000101
11111100
8th Bit = 1: Answer is Negative
To Check:
Perform 2’s
Complement
On Answer
11111100
↓↓↓↓↓↓↓↓
00000011
+1
00000100
00001001
↓↓↓↓↓↓↓↓
11110110
+1
11110111
2’s
Complement
Process
18
19. NEG + NEG → NEG Answer
Take the 2’s complement of both negative numbers and
use regular binary addition.
(-9)
+ (-5)
-14
→
→
←
11110111
+ 11111011
1]11110010
2’s Complement
Numbers, See
Conversion Process
In Previous Slides
8th Bit = 1: Answer is Negative
Disregard 9th Bit
To Check:
Perform 2’s
Complement
On Answer
11110010
↓↓↓↓↓↓↓↓
00001101
+1
00001110
19
Editor's Notes
2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
Introductory Slide / Overview of Presentation 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
This slide explains the need to negative number and why we don’t have subtract-er circuits. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
This slide explains why we need a way to represent negative numbers. It introduces the concept of a fixed-length number system. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
Before we examine negative binary numbers, let’s first look at something we understand very well: the decimal number system. This slide uses a fixed-length decimal number system to illustrate the limitation of not being able to use a minus sign. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
This slide explains how to represent negative numbers in a fix-length decimal number system. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
Here are four example of adding two numbers in the odometer number system (i.e., 3-digit decimal number system). 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
This slide describes the 10’s complement conversion process. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
Examples of the 10’s Complement Process. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
Introduction to the 8-Bit Binary Number system and how negative numbers are represented. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
Explanation of the sign bit. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
This slide describes the 2’s complement conversion process. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
Examples of the 2’s Complement Process. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
This slide show that there are only four possible combinations for adding together two signed numbers. The next four slides demonstrate each of these examples. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
Addition of two Positive numbers. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
This example shows the addition of one positive and one negative numbers. Note that this is done in the same way as subtracting a positive number from a positive number. In this case, the answer is positive. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
This slide demonstrates the addition of one positive and one negative number. Again, this is is the same a subtracting a positive number from a positive number. In this case the answer happens to be negative. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009
This slide demonstrates the addition of two negative numbers. 2's Complement Arithmetic Digital Electronics Lesson 2.4 – Specific Comb Circuit & Misc Topics Project Lead The Way, Inc. Copyright 2009