This document discusses computer arithmetic, focusing on binary addition and the representation of both unsigned and signed numbers. It explains the methods of representing signed numbers, including sign-and-magnitude, 1's complement, and 2's complement, along with examples of complementary arithmetic and its application in subtraction. The document also illustrates how to perform addition with both positive and negative numbers using 2's complement.

1. Lecture No. 06
Computer Arithmetic

2. Introduction
• ADDITION
– Like decimal numbers, two numbers can be
added by adding each pair of digits together
with carry propagation.
(647)10
+ (537)10
(1184)10

3. Binary Addition
• Two 1-bit values
A B A + B
0 0 0
0 1 1
1 0 1
1 1 10
“two”

4. Binary Addition
• Two n-bit values
– Add individual bits
– Propagate carries
– E.g.,
10101 21
+ 11001 + 25
101110 46
11

5. Try it: 0011 + 1010

6. 1 1 1 1 0 1
+ 1 0 1 1 1
---------------------
0
1
0
1
1
1111
1 1 00
carries
Example Add (11110)2 to
(10111)2
carry
Binary Addition

7. Negative Numbers Representation
 Unsigned numbers: only non-negative values.
 Signed numbers: include all values (positive and negative).
 Till now, we have only considered how unsigned (non-
negative) numbers can be represented. There are three
common ways of representing signed numbers (positive and
negative numbers) for binary numbers:
 Sign-and-Magnitude
 1s Complement
 2s Complement

8. Negative Numbers:
Sign-and-Magnitude
 Negative numbers are usually written by writing a
minus sign in front.
 Example:
– - (12)10 , - (1100)2
 In sign-and-magnitude representation, this sign is
usually represented by a bit:
– 0 for +
– 1 for -

9. Negative Numbers:
Sign-and-Magnitude
 Example: an 8-bit number can have 1-bit sign and 7-bit
magnitude.
sign magnitude

10. Negative Numbers:
Sign-and-Magnitude
 Largest Positive Number: 0 1111111 +(127)10
 Largest Negative Number: 1 1111111 -(127)10
 Zeroes: 0 0000000 +(0)10
1 0000000 -(0)10
 Range: -(127)10 to +(127)10

11. Compliments
• Subtraction in any number system can be accomplished
through the use of complements.
• A complement is a number that is used to represent the
negative of a given number.

12. 1s and 2s Complement
 Two other ways of representing signed numbers
for binary numbers are:
 1s-complement
 2s-complement

13. Complementary Arithmetic
• 1’s complement
– Switch all 0’s to 1’s and 1’s to 0’s
Binary # 10110011
1’s complement 01001100

14. Complementary Arithmetic
• 2’s complement
– Step 1: Find 1’s complement of the number
Binary # 11000110
1’s complement 00111001
– Step 2: Add 1 to the 1’s complement
00111001
+ 00000001
00111010

15. 2’s Complement Examples
Example #1
Complement Digits
Add 1
5 = 00000101
-5 = 11111011

11111010
+1

16. 2’s Complement Examples
Complement Digits
Add 1
-13 = 11110011
13 = 00001101

00001100
+1

17. Using The 2’s Compliment Process
9
+ (-5)
4
(-9)
+ 5
- 4
(-9)
+ (-5)
- 14
9
+ 5
14
POS
+ POS
POS

POS
+ NEG
POS

NEG
+ POS
NEG

NEG
+ NEG
NEG

Use the 2’s complement process to add together the
following numbers.

18. POS + POS → POS Answer
If no 2’s complement is needed, use regular binary addition.
000010019
+ 5
14

 00001110
00000101 +

19. POS + NEG → POS Answer
Take the 2’s complement of the negative number and use
regular binary addition.
000010019
+ (-5)
4


11111011+
00000101

11111010
+1
11111011
2’s
Complement
Process
1]00000100
8th Bit = 0: Answer is Positive
Disregard 9th Bit

20. POS + NEG → NEG Answer
Take the 2’s complement of the negative number and use
regular binary addition.
11110111(-9)
+ 5
-4


00000101+
00001001

11110110
+1
11110111
2’s
Complement
Process
11111100
8th Bit = 1: Answer is Negative
11111100

00000011
+1
00000100
To Check:
Perform 2’s
Complement
On Answer

21. NEG + NEG → NEG Answer
Take the 2’s complement of both negative numbers and use
regular binary addition.
11110111(-9)
+ (-5)
-14


11111011 +
2’s Complement
Numbers, See
Conversion Process
In Previous Slides
1]11110010
8th Bit = 1: Answer is Negative
Disregard 9th Bit
11110010

00001101
+1
00001110
To Check:
Perform 2’s
Complement
On Answer

22. Thank you