1. CSCI206 - Computer Organization & Programming
Integer Addition and Subtraction
zyBook: 10.1, 10.2

2. Review Positional Number Systems
n-th digit
Base

3. Binary to Decimal (b=2)
110011
543210
Digit count (n), or position

4. Decimal to Binary (b=2) (left to right)
42 = ??
First power of two <= N is the first 1, e.g., 2^5 which is 32
Subtract the value from the original, e.g., == 10
Repeat for remaining digits
insert zero if greater than remaining value
else insert 1 and subtract value

5. Decimal to Binary (b=2) (left to right)
insert 0’s in where the value is too small, e.g.,
42-32 == 10, no 2^4 == 16 b/c 16 > 10
42 =
first power of 2 <= 42
subtract, have 10 remaining
next power of 2 <= 10, subtract 2 remain
2 remains, 2^1 is the last 1

6. Decimal to Binary (b=2) (right to left)
Let N be the base 10 number
while N > 0
If N is odd (N%2==1)
add a 1
else
add a 0
N = N / 2 (integer divide)

7. Decimal to Binary (b=2) (right to left)
42 = even 0, 42/2 = 21R0
21 = odd 1, 21/2 = 10R1
10 = even 0, 10/2 = 5R0
5 = odd 1, 5/2 = 2R1
2 = even 0, 2/2 = 1R0
1 = odd 1, 1/2 = 0R1

8. Decimal to Binary (b=2) (right to left)
42 = even 0, 42/2 = 21R0
21 = odd 1, 21/2 = 10R1
10 = even 0, 10/2 = 5R0
5 = odd 1, 5/2 = 2R1
2 = even 0, 2/2 = 1R0
1 = odd 1, 1/2 = 0R1
Generate result right to left:
101010

9. Binary addition (zyBook 10.2)
same as elementary base-10 arithmetic 1
529
+ 742
1271
Sum each place value from right to left and add carry bits if result requires an additional digit.

10. Binary Addition
101101 1 A B
Sum
Carry 1

11. Binary Addition
101101 11 A B
Sum
Carry 1

12. Binary Addition 1
101101
011 A B
Sum
Carry 1

13. Binary Addition 1
101101
011 A B
Sum
Carry 1

14. Binary Addition 1
101101
011 A B
Sum
Carry 1

15. Binary Addition
1 1
101101
1011 A B
Sum
Carry 1

16. Binary Addition
1 1
101101
11011 A B
Sum
Carry 1

17. Binary Addition
1 1
101101
111011 A B
Sum
Carry 1

18. Binary Addition
= 45
= 14
= 59 A B
Sum
Carry 1

19. Ripple Carry Adder Add bits sequentially
Carry out goes into next carry in
ALU Symbol

20. Ripple Carry Adder
1001
+ 0101
1110
Simple to understand, but slow because the result has to be computed 1 bit at a time
Faster, more complex, circuits exist!

21. Subtraction
2’s complement negation is done with bitwise inverse plus one:
Easily implemented by computing the bitwise inverse and setting carry in to be 1 on bit 0, i.e., Bi = not(yi)

22. Multiplexer (Mux)
By Lenehey at en.wikipedia Later version(s) were uploaded by Fresheneesz at en.wikipedia. (Transferred from en.wikipedia) [Public domain], from Wikimedia Commons

23. Adder/subtractor

24. Overflow Only possible when signs are the same of both operands.
Differing signs always produce a smaller absolute value.
Same signs produce a larger absolute value.
Overflow if positive + positive = negative or negative + negative = positive.
carry out of MSB (most significant bit) is not a valid test for overflow!

25. Absolute Value
if (x > 0) x
else -x

26. Absolute Value
if (x > 0) x
else -x

27. Absolute Value 1

28. Absolute Value Overflow?
if result is negative
only a problem with the C standard library implementation
NAME
abs, labs, llabs, imaxabs - compute the absolute value of an integer
SYNOPSIS
#include <stdlib.h>
int abs(int j);
Result of abs is a signed integer!

29. Consider the representations …
Assuming an 8-bit integer
INT_MIN : // -128
INT_MAX: // +127
If unsigned, == 128
If signed, 2’s comp: =
because it is signed, it is now -128

30. Absolute Value Overflow?
the invert plus 1 will overflow if the inverted value is all 1’s
is the most negative 2’s complement value (INT_MIN)
Detect if the number is negative and its absolute value is still negative! (INT_MIN!)