Obaidur Rahman CSE-100: Introduction to Computer Systems Lecture 06 computer arithmetic Department of Computer Science and Engineering European University of Bangladesh

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