1. CS 286 Computer Architecture & Organization
Computer Arithmetic
Integers and Two’s Complement
Department of Computer Science
Southern Illinois University Edwardsville
Fall, 2019
Dr. Hiroshi Fujinoki
Arithmetic/000

2. CS 286 Computer Architecture & Organization
Arithmetic systems used in computers
Unsigned
Integers
Signed
Popularly used
Integers
Fractions
Two’s
complement
Sign
Magnitude
Not popularly used
Arithmetic systems
used in computers
Popularly used
Floating-point numbers
Fixed-point numbers
Popularly used
Not popularly used
Computers use binary implementation for these numbers
Arithmetic/001

3. CS 286 Computer Architecture & Organization
32(or 64)-bit
2’s complement
integer
8-bit unsigned
integer
IEEE-754
single-precision
floating-point
32(or 64)-bit
unsigned integer
IEEE-754
double-precision
floating-point
Arithmetic/002

4. CS 286 Computer Architecture & Organization
Unsigned Integers
“unsigned” assumed to be “positive”
Example: 8-bit unsigned integers
(=1)
(=2)
(=4)
(=8)
(=16)
(=32)
(=64)
(=128)
Weight 20 21 22 23 24 25 26 27
MSB
LSB 1
Bit #’s 1 2 3 4 5 6 7
Convert it to a decimal number
(27  1) + (26  0) + (25  1) + (24  1) + (23  0) + (22  0) + (21  0) + (20  1)
= (27  1) + ((25  1) + (24  1) + (20  1) =
= 177
Arithmetic/003

5. CS 286 Computer Architecture & Organization
Properties of unsigned integers
The minimum number: 20 21 22 23 24 25 26 27
(=1)
(=2)
(=4)
(=8)
(=16)
(=32)
(=64)
(=128)
LSB
MSB
For “0”
The maximum number (for 8-bit unsigned numbers):
255
= 28-1 20 21 22 23 24 25 26 27
(=1)
(=2)
(=4)
(=8)
(=16)
(=32)
(=64)
(=128)
LSB
MSB 1
The maximum number (for N-bit unsigned numbers):
2N-1
The value range of N-bit unsigned numbers: between 0 and 2N-1
Arithmetic/004

6. CS 286 Computer Architecture & Organization
Properties of unsigned integers
Interval of two numbers: 1 1 1
MAX.
MIN.
2N-1
2N-2 1 1
Are all intervals between two consecutive numbers exactly 1?
Analysis
Given N bits, the number of possible different combinations of 0’s
and 1’s will be 2N.
Between 0 and 2N-1 (inclusive), there are 2N consecutive distinct
integers, for which the interval between the two consecutive
integers is always exactly 1.
Arithmetic/005

7. CS 286 Computer Architecture & Organization
Example: Addition of two 8-bit unsigned integers
as the “bit-wise operation” 20 21 22 23 24 25 26 27
MSB 1 97 + + + + + + + + +
MSB
104 1 1 = 11 = 10 = = 1 = = = 1 =
201
MSB 1 1 1 1
= 201
Binary + Operations 1 + + 1 + 1 + 1 + 1 1 10 11
Arithmetic/006

8. CS 286 Computer Architecture & Organization
Example: Addition of two 8-bit unsigned integers
as the “bit-wise operation” 27 26 25 24 23 22 21 20
MSB 1
225 + + + + + + + + +
329
104
You need
“+256”
MSB 1 10 = 11 = 10 = = 1 = = = 1 =
Carry
Flag 1
MSB 1 1 1
= 73
+ 256
Binary + Operations
= 329  1 + 11 + 1 + 1 + 1 + 10
Arithmetic/007

9. CS 286 Computer Architecture & Organization
Example: Subtraction of two 8-bit unsigned integers
as the “bit-wise operation” 27 26 25 24 23 22 21 20
MSB 1 1 1
104 1 1 1
borrow 1 1 1 - - 7
MSB 97 1 = = = = = 1 = 1 = 1 =
= 7
Binary - Operations - 1 - 1 - 1 -
Arithmetic/008

10. CS 286 Computer Architecture & Organization
Three problems in unsigned integers
Major problems
 No negative integers
 Huge numbers require a large # of bits
 Different operations for additions and subtractions
Signed Integers (to solve  and )
Sign-magnitude integers
Two’s complement integers
- Easy to understand for human users 
Signed integers
- Difficult to handle for computer hardware 
Most of the computer
systems use 2’s
complement integers
- Difficult to understand for human users 
- Easy to handle for computer hardware 
Arithmetic/009

11. CS 286 Computer Architecture & Organization
Sign-magnitude integers
Use the left-most bit as the sign-bit
(0 = positive, 1 = negative)
The other bits are treated and used in the exactly the same way
as unsigned integers 20 21 22 23 24 25 26 27
MSB 1
104
SIGN 20 21 22 23 24 25 26
MSB 1
+104
MSB 1
-104
Arithmetic/010

12. CS 286 Computer Architecture & Organization
Properties of sign-magnitude integers
The maximum number (for 8-bit sign magnitude integers):
+127
MSB 1
+127 20 21 22 23 24 25 26
SIGN
The minimum number (for 8-bit sign magnitude integers):
-127
MSB 20 21 22 23 24 25 26
SIGN 1
-127
For 8-bit sign magnitude integers,
Interval of two consecutive
numbers: 1
Maximum:
Minimum:
+2(N-1) - 1
-2(N-1) - 1
Arithmetic/011

13. CS 286 Computer Architecture & Organization
Two problems in sign-magnitude integers
 Two different expressions for “0”
MSB +0 20 21 22 23 24 25 26
SIGN 1 -0
Arithmetic/012

14. CS 286 Computer Architecture & Organization
Two problems in sign-magnitude integers
 Adding –N to +N does not result in 0 20 21 22 23 24 25 26
SIGN
MSB 1
+44 + 1
-44 +
-88 1 1 1 1
This does not
make sense  1 + 10
Binary + Operations
Arithmetic/013

15. CS 286 Computer Architecture & Organization
Two’s complement integers
 The left-most bit is the sign bit
0 = positive
1 = negative
Same as “sign-magnitude integers”
 For any positive integer, the rest of bits are treated as unsigned (and sign-
magnitude integers)
Sign-Bit
MSB
LSB 1 32
+106 64 8 2
 “0” is represented by all binary zero’s (e.g., “0000”)
 The bit pattern of a negative integer can be found by:
Obtain the bit pattern of the positive counterpart
Invert the bit pattern
Add one to it
Arithmetic/014

16. CS 286 Computer Architecture & Organization
Two’s complement integers: Bit pattern for negative integers
(8 bits)
Example
- 85
Step 1
Obtain the bit pattern of positive counterpart
+85 = 20 21 22 23 24 25 26
SIGN
MSB 1
Step 2
Invert the bit pattern
MSB 1 1 1 1
Step 3
Add one to it
MSB 1
Arithmetic/015

17. Properties in two’s complement integers POSITIVE
POSITIVE -1 7 6 -2 1 5 2 4 -3 3 3 2 1
- 4 4 -1
NEGATIVE -2 5
- 5 -3 -4
- 6 6 -5 -7 7 -6
- 8 -7 -8
Arithmetic/016

18. CS 286 Computer Architecture & Organization
Properties in two’s complement integers 7
The maximum (for 4-bit 2’s complement integers):
The minimum (for 4-bit 2’s complement integers): 7 6 -8 5 4 3
The maximum (for N-bit 2’s complement integers):
The minimum (for N-bit 2’s complement integers):
2(N-1)-1 2 1
2(N-1)
Only one representation for “0” -1 -2
Do we need two procedures for addition and subtraction? -3 -4 -5 -6 -7 -8
Arithmetic/017

19. CS 286 Computer Architecture & Organization
Conversions between two’s complement integers and decimal integers
Positive Decimal  Binary 2’s complement
To indicate
“positive”
SIGN 26 25 24 23 22 21 20
MSB 1
Example
Decimal 85
Decimal 85 = =
Arithmetic/018

20. CS 286 Computer Architecture & Organization
Conversions between two’s complement integers and decimal integers
Negative Decimal  Binary 2’s complement 20 21 22 23 24 25 26
-27 1 2 4 8 16 32 64
-128
MSB 1
ANOTHER TECHNIQUE
Example
Convert a binary #
(- 113)
to a 2’s complement integer
The decimal # = = (-128) + 15
= (-128)
Arithmetic/019

21. CS 286 Computer Architecture & Organization
Conversions between two’s complement integers and decimal integers
Positive Binary 2’s complement  Decimal
- 27 26 25 24 23 22 21 20
MSB 1
Example
Convert a binary 2’s complement # to a decimal #
( )
The decimal # =
= 85
Arithmetic/020

22. CS 286 Computer Architecture & Organization
Conversions between two’s complement integers and decimal integers
Negative Binary 2’s complement  Decimal
SIGN
-128 64 32 16 8 4 2 1
MSB 1
Example
Convert a binary 2’s complement # to a decimal #
( )
The decimal # = (-128)
= (-128) + 85
= - 43
Arithmetic/021

23. CS 286 Computer Architecture & Organization
Conversions a two’s complement integer to a different bit length 7 6 -1 1 5 -2 2 4 3 -3 3 2 1
- 4 4 -1 -2 5
- 5 -3 -4
- 6 6 -5 -7 7 -6
- 8 -7
Overflow
Boundary -8
Arithmetic/022

24. CS 286 Computer Architecture & Organization
Additions and subtractions for two’s complement integers 7
Example #1
Two positive numbers 6
= +3 5 +
= +4 4  1 1 1
= +7 3 2
Example #2
Two negative numbers 1
= -4
 Still correct!  -1
Carry
flag +
= -1 -2 1 1 1 1
= -5 -3 -4
Example #3
Negative and positive numbers -5 -6
= -7 -7 +
= +5 -8  1 1 1
= -2
Arithmetic/023

25. CS 286 Computer Architecture & Organization
Additions and subtractions for two’s complement integers
Example #4
Negative and positive numbers 7 6
= -3
 Still correct! 5
Carry
flag +
= +5 4 1 1
= +2 3 2
Example #5
Negative and positive numbers 1
Carry
flag
= -4
 Still correct! +
= +4 -1 1
= -0 -2 -3 4 5 -4 -5 -6 -7 -8
Arithmetic/024

26. CS 286 Computer Architecture & Organization
Overflow
Flag = 1
Additions and subtractions for two’s complement integers 
Example #6
Two negative numbers 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7 -8
= -7
 Incorrect!
Carry
flag +
= -6 1 1 1
= +3
Overflow
Flag = 1
Example #7
Two positive numbers 
= +5
 Incorrect!
Carry
flag +
= +4 1 1
= -7 6 7 
Arithmetic/025

27. CS 286 Computer Architecture & Organization
Additions and subtractions for two’s complement integers 1 2 3 4 5 6 7 -1 -2 -3 -4 -5 -6 -7 -8
Example #8
Subtract a positive # from a positive #
= +3 -
= +4
= -4
2‘s Complement
= +3 + 1 1 1 1
= -1
For overflow, the same rule applies!
Arithmetic/026

28. CS 286 Computer Architecture & Organization
Conversions a two’s complement integer to a different bit length
This will work for positive numbers
An 8-bit 2’s complement integer 1
MSB
LSB
MSB 1
LSB
A 16-bit 2’s complement integer
This will work for negative numbers
An 8-bit 2’s complement integer 1
MSB
LSB
MSB 1 1
LSB
A 16-bit 2’s complement integer
Arithmetic/027

29. CS 286 Computer Architecture & Organization
Conversions a two’s complement integer to a different bit length
This will work for positive numbers 1
MSB
LSB
An 8-bit 2’s complement integer
MSB
LSB
A 16-bit 2’s complement integer 1 1
MSB
LSB
An 8-bit 2’s complement integer
This won’t work for negative numbers
This indicates a negative integer
MSB
LSB
A 16-bit 2’s complement integer 1
Arithmetic/028

30.
- 8 4
- 4 2
- 6 -2 6 1 3 5 7 -7
- 5 -3 -1