0. Binary Arithmetic

1. Outline
Binary Addition
2’s complement
Binary Subtraction
Half Adder
Logisim

2. Binary Addition
How to add two binary numbers?
Single bit:
+ 0
+ 1 1 1
+ 0 1
+ 1 10
carry bit

3. 10 13 + 9 2 10 13 + 9 22 13 + 9 Binary Addition What is a carry bit?
Same idea as “carry the 1” in decimal arithmetic: 10 13
+ 9 2 10 13
+ 9 22 13
+ 9

4. Binary Addition
carry bit gets “carried over” to the next bit (to the left..)
Just like decimal addition
+ 0
+ 1 1 1
+ 0 1
+ 1 10
carry bit

5. Binary Addition
Example #1

6. Binary Addition
Example #1
carry bit

7. 0100 0011 + 0000 1010 0100 1101 67 + 10 77 Binary Addition Example #1
You can always check your work by converting to decimal 67
+ 10 77

8. Binary Addition
Example #2

9. 1001 1100 1100 0110 + 0100 1010 1 0001 0000 Binary Addition Example #2
9th bit!?!

10. 1100 0110 + 0100 1010 1 0001 0000 Binary Addition Example #2
If we must store the answer in a single byte, 9th bit gets ignored…
This is known as overflow

11. Shouldn’t the answer be 272?
Binary Addition
Example #2
198
+ 74 16
16 !?!
Shouldn’t the answer be 272?

12. Binary Addition
What’s the largest number that can be stored in 8-bits?
???

13. Binary Addition
What’s the largest number that can be stored in 8-bits?
272 is greater than 255
272 cannot be represented using 8 bits
Has this ever happened to you?
255

14. Outline
Binary Addition
2’s complement
Binary Subtraction
Half Adder
Logisim

15. 8-bits can be used to represent numbers between 0 and 255
2’s Complement
8-bits can be used to represent numbers between 0 and 255
How do we represent negative numbers in binary?
Drumroll…. 2’s complement!!
Makes addition, subtraction, multiplication easier
Most common way to represent signed numbers
Signed: positive AND negative numbers
And no, it’s not 2’s compliment
“Hey 2, your hair looks nice today..”

16. In 2’s complement system, the leftmost bit indicates the sign
0 for positive
1 for negative
When the leftmost bit is 0, the remaining bits are interpreted as before
=> 1
=> 127

17. 2’s Complement
When the leftmost bit is 1, we do the following to obtain the signed decimal representation:
Complement (invert) the binary digits (0 => 1; 1 => 0)
Convert binary digits to decimal number
Multiply by -1
Subtract 1

18. 1100 0110 Invert bits: 0011 1001 Convert to decimal: 57
2’s Complement
Example #1: What is the 2’s complement value of ?
Invert bits:
Convert to decimal: 57
Multiple by -1: -57
Subtract 1: -58

19. 2’s Complement
Example #2: What is the 2’s complement value of ?
Your turn!

20. 1001 1001 Invert bits: 0110 0110 Convert to decimal: 102
2’s Complement
Example #2: What is the 2’s complement value of ?
Invert bits:
Convert to decimal: 102
Multiple by -1: -102
Subtract 1: -103

21. Decimal to 2’s complement If the number is positive:
leftmost bit is 0
remaining bits identical to unsigned binary number
E.g., Represent the number 97 using 8-bits, 2’s complement
97 = = =

22. Decimal to 2’s complement If decimal number is negative:
add 1
multiply by -1 (to create positive number)
create binary sequence
invert bits

23. add 1: -97 + 1 = -96 multiply by -1: -96 × -1 = 96
2’s Complement
Example #3: represent -97 using 8-bits, 2’s complement
add 1: = -96
multiply by -1: -96 × -1 = 96
get binary: =
invert bits:
-97 =>

24. add 1: -123 + 1 = -122 multiply by -1: -122 × -1 = 122
2’s Complement
Example #4: represent -123 using 8-bits, 2’s complement
add 1: = -122
multiply by -1: × -1 = 122
get binary: =
invert bits:
-123 =>

25. Break Time!!!

26. Outline
Binary Addition
2’s complement
Binary Subtraction
Half Adder
Logisim

27. To subtract two binary numbers, X – Y,
Binary Subtraction
To subtract two binary numbers, X – Y,
Invert Y => Y’
Add 1 to Y’
Add X + Y’
We are basically taking the 2’s complement of Y (Y’) before adding it to X

28. 0010 0001 - 0000 1010 invert Y 00001010 => 11110101 add 1 to Y’
Binary Subtraction
Example #5:
invert Y
=>
add 1 to Y’ =
add Y’ to X
this 9th bit 1 gets ignored

29. Binary Subtraction
It’s always good to verify your work… 33
- 10 23

30. Binary Subtraction
Example #6

31. Binary Subtraction
Example #6
invert Y
=>
add 1 to Y’ =
add Y’ to X

32. Binary Subtraction
Verify… 25
- -30 55

33. Outline
Binary Addition
2’s complement
Binary Subtraction
Half Adder
Logisim

34. Can use logic gates to construct adder circuit
Half Adder
Can use logic gates to construct adder circuit
Circuit is capable of binary addition
Half Adder has two inputs (A, B) and two outputs (S, C)
S: sum
C: carry A
Half Adder S B C

35. Truth table for Half Adder
S : sum of A + B
C : carry bit A B S C 1
Half Adder A B S C

36. Logic gate circuit for Half Adder?
A, B inputs
S, C outputs A B S C 1

37. Logic gate circuit for Half Adder?
A, B inputs
S, C outputs A B S C 1

38. How would you expand the Half Adder to create a:
2-bit adder?
4-bit adder?
8-bit adder? …?
This will be part of your homework…

39. Outline
Binary Addition
2’s complement
Binary Subtraction
Half Adder
Logisim

40. Logisim is FREE logic gate simulation software
Please download / install this software
It should be on the lab machines, too

41. Download / install Logisim Complete homework #2
Next Steps
Download / install Logisim
Complete homework #2
Data representation
Binary arithmetic
Create (useful) logic gate circuits using Logisim
Next lecture: Micro-architecture