1. Computer Organization COMP 210
Integers
Topics
Numeric Encodings
Unsigned & Two’s complement
Programming Implications
C promotion rules
Basic operations
Addition, negation, multiplication
Consequences of overflow
Using shifts to perform power-of-2 multiply/divide

2. C Puzzles Answers on the last slide
Assume machine with 32 bit word size, two’s complement integers
For each of the following C expressions, either:
Argue that is true for all argument values
Give example where not true
x < 0  ((x*2) < 0)
ux >= 0
x & 7 == 7  (x<<30) < 0
ux > -1
x > y  -x < -y
x * x >= 0
x > 0 && y > 0  x + y > 0
x >= 0  -x <= 0
x <= 0  -x >= 0
Initialization
int x = foo();
int y = bar();
unsigned ux = x;
unsigned uy = y;

3. Real Life Ariane 5 – European missile
On June 4, 1996 an unmanned Ariane 5 rocket launched by the European Space Agency exploded just forty seconds after its lift-off from Kourou, French Guiana.
first voyage,
a decade of development costing $7 billion.
rocket and its cargo were valued at $500 million
Cause:
a 64 bit floating point number relating to the horizontal velocity of the rocket with respect to the platform was converted to a 16 bit signed integer.
The number was larger than 32,767, the largest integer storeable in a 16 bit signed integer,
the conversion failed.
See

4. How do we get to bits?
We write programs in an abstract language like C How do we get to the binary representation? Compilers & Assemblers! We write x = 5. The compiler changes it into mov 5, 0x005F The assembler changes it into: 80483c7: 8b 45 5F
Compiler figures out that this is an integer and changes it into 2’s Complement

5. Negative numbers Problem: how do we represent negative numbers?
Sign-magnitude.
Wastes space (range decreases)
Not used anymore (was used on a few IBM mainframes in the dark ages)
2’s complement
Space efficient
Arithmetic works well

6. The structure of a signed integer (if we’re using sign-magnitude)
This is no longer used!

7. Instead of using sign magnitude…
We’ll break the number line in the middle

8. …and shift the right part of the number line to the left side to get…
two’s complement
…and shift the right part of the number line to the left side to get…
Note that negative numbers start with a 1, but starting with a 1 is not what makes the number negative.
Result is called 2’s complement.
There is an easy arithmetic way to convert numbers to their negative equivalent with 2’s complement.

9. This is the absolute value of 110
2’s complement
Rule:
To find the value represented by a binary number that starts with a 1 (i.e., a negative number)
Take the complement of the binary number.
Add 1 to the result.
determine the decimal value
To find the value represented by 110
Take the Complement:
add 1 to the complement]
find the decimal value = +2
To find the value represented by a binary number that starts with a 0 (i.e., a positive number)
Do nothing!
This is the absolute value of 110

10. 2’s complement range
There will be more negative numbers than positive numbers since 0 takes a positive space:

11. The signed integers for a four-bit cell

12. 2’s complement range Range for any cell size:
Smallest number is 1000… 0
Largest number is 0111…..1
The magnitude of the smallest negative number is 1 greater than the magnitude of the largest positive number.
Example: 6-bit cell: -32 to 31

13. Encoding Integers Unsigned Two’s Complement Sign Bit
short int x = ;
short int y = ;
Sign
Bit
C short 2 bytes long
Sign Bit
For 2’s complement, most significant bit indicates sign
0 for nonnegative
1 for negative

14. Numeric Ranges Unsigned Values UMin = 0 UMax = 2w – 1
000…0
UMax = 2w – 1
111…1
Two’s Complement Values
TMin = –2w–1
100…0
TMax = 2w–1 – 1
011…1
Other Values
Minus 1
111…1
Values for W = 16

15. Values for Different Word Sizes
C Programming
 #include <limits.h>
K&R App. B11
Declares constants, e.g.,
 ULONG_MAX
 LONG_MAX
 LONG_MIN
Values platform-specific
Observations
|TMin | = TMax + 1
Asymmetric range
UMax = 2 * TMax + 1