1. Bits, Data types, and Operations: Chapter 2 COMP 2610 Dr. James Money COMP 2610 1

2. Addition and Subtraction We will now use only two’s complement representation for integers The arithmetic proceeds as we introduced before using long addition At each stage there is a sum digit and a carry digit

3. Example Using 5 bit notation, what is 11+3? Note – (11) 10 = (01011) 2 – (3) 10 = (00011) 2 – Sum is (01110) 2 = (14) 10

4. Subtraction Recall the definition of subtraction from algebra: For all real numbers a, b, we define subtraction by a-b = a+(-b)

5. Negation How do we form –b? We take b and form the the two’s complement of the number What is the two’s complement form in 5 bit notation for – 5 – -10

6. Subtraction What is 14-9? – (14) 10 = (01110) 2 – (9) 10 = (01001) 2 – (-9) 10 = (10111) 2 – Thus 14-9 = 14+-9 = (00101) 2 = (5) 10

7. Addition of same number What happens when you add a number to itself, that is x+x? Using 8 bits, consider x=(59) 10 = (00111011) 2 We get x+x=2x = (118) 10 = (01110110) 2 This is the bits shifted to the left by one!

8. Addition of same number Why? 59 = 0x2 6 +1x2 5 +1x2 4 +1x2 3 +0x2 2 +0x2 1 +1x2 0 So 59+59 = 2 x 59 = 2(0x2 6 +1x2 5 +1x2 4 +1x2 3 +0x2 2 +0x2 1 +1x2 0 ) = 0x2 7 +1x2 6 +1x2 5 +1x2 4 +0x2 3 +0x2 2 +1x2 1 This is shifting the bits to the left by one

9. Sign Extension Many times we represent a number with fewer bits to save space For example, (5) 10 = (00000000000101) 2 And it is the same as (000101) 2 For positive numbers, we can zero extend the bits to the left to get the same number

10. Sign Extension How about negative numbers? For (-5) 10 = (111011) 2 in 8 bits In 16 bits, we get (11111111111011) 2 We add leading ones for negative numbers

11. Sign Extension Suppose we had to add the numbers 000000000000110113 + 111011-5

12. Sign Extension In either case, we extend the bits using Sign- EXTension, abbreviated SEXT That is we extend by the correct sign: – By 0 if positive – By 1 if negative

13. Overflow So far we have assumed that the sum of the two integers is small enough to be represented in the number of available bits What happens if this is not the case? We get a condition known as overflow

14. Overflow This situation is analogous to the old odometers which had only 5 digits What happens after 99999? It rolls back to 00000 and looks like a new car with no miles The carry of ten-thousands digit is lost

15. Overflow For two’s complement arithmetic, this can be more subtle Consider 5 bits numbers and add 9+11 01001 +01011 10100

16. Overflow However, this number is -12! We say this result has overflowed the capacity of the representation since we added to positive numbers and got a negative one

17. Overflow Consider -12+-6: 10100 +11010 01110