Binary Arithmetic
Outline
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Binary Addition
2’s complement
Binary Subtraction
Half Adder
Logisim
Binary Addition
• How to add two binary numbers?
• Single bit:
0
+0
0
0
+1
1
1
+0
1
1
+1
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carry bit
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Binary Addition
• What is a carry bit?
• Same idea as “carry the 1” in decimal arithmetic:
13
+9
3
10
13
+9
2
10
13
+9
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Binary Addition
• carry bit gets “carried over” to the next bit (to the left..)
• Just like decimal addition
0
+0
0
0
+1
1
1
+0
1
1
+1
10
carry bit
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Binary Addition
• Example #1
0100 0011
+ 0000
1010
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Binary Addition
• Example #1
carry bit
0100 0011
+ 0000
1010
0100 1101
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Binary Addition
• Example #1
• You can always check your work by converting to decimal
0100 0011
+ 0000
1010
0100 1101
7
67
+ 10
77
Binary Addition
• Example #2
1100 0110
+ 0100
1010
8
Binary Addition
• Example #2
1001 1100
1100 0110
+ 0100
1010
1 0001
0000
9th bit!?!
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Binary Addition
• Example #2
• If we must store the answer in a single byte, 9th bit gets ignored…
• This is known as overflow
1100 0110
+ 0100
1010
1 0001
0000
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Binary Addition
• Example #2
1100 0110
+ 0100
1010
1 0001
0000
198
+ 74
16
16 !?!
Shouldn’t the answer be 272?
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Binary Addition
• What’s the largest number that can be stored in 8-bits?
1111
1111
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???
Binary Addition
• What’s the largest number that can be stored in 8-bits?
1111
1111
• 272 is greater than 255
• 272 cannot be represented using 8 bits
• Has this ever happened to you?
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255
Outline
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Binary Addition
2’s complement
Binary Subtraction
Half Adder
Logisim
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..”
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2’s Complement
• 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
• 0000 0001 => 1
• 0111 1111 => 127
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2’s Complement
• When the leftmost bit is 1, we do the following to obtain the
signed decimal representation:
1.
2.
3.
4.
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Complement (invert) the binary digits (0 => 1; 1 => 0)
Convert binary digits to decimal number
Multiply by -1
Subtract 1
2’s Complement
• Example #1: What is the 2’s complement value of 1100 0110?
1100 0110
1.
2.
3.
4.
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Invert bits: 0011 1001
Convert to decimal: 57
Multiple by -1: -57
Subtract 1: -58
2’s Complement
• Example #2: What is the 2’s complement value of 1001 1001?
• Your turn!
1001 1001
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2’s Complement
• Example #2: What is the 2’s complement value of 1001 1001?
1001 1001
1.
2.
3.
4.
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Invert bits: 0110 0110
Convert to decimal: 102
Multiple by -1: -102
Subtract 1: -103
2’s Complement
• Decimal to 2’s complement
• If the number is positive:
1.
2.
leftmost bit is 0
remaining bits identical to unsigned binary number
• E.g., Represent the number 97 using 8-bits, 2’s complement
97
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= 64 + 32 + 1
= 26 + 25 + 20
= 0110 0001
2’s Complement
• Decimal to 2’s complement
• If decimal number is negative:
1.
2.
3.
4.
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add 1
multiply by -1 (to create positive number)
create binary sequence
invert bits
2’s Complement
• Example #3: represent -97 using 8-bits, 2’s complement
1. add 1:
2. multiply by -1:
3. get binary:
0000
4. invert bits:
1111
-97 + 1 = -96
-96 × -1 = 96
96 = 0110
-97 => 1001 1111
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1001
2’s Complement
• Example #4: represent -123 using 8-bits, 2’s complement
1. add 1:
2. multiply by -1:
3. get binary:
1010
4. invert bits:
0101
-123 + 1 = -122
-122 × -1 = 122
122 = 0111
-123 => 1000 0101
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1000
Break Time!!!
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Outline
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Binary Addition
2’s complement
Binary Subtraction
Half Adder
Logisim
Binary Subtraction
• To subtract two binary numbers, X – Y,
1.
2.
3.
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
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Binary Subtraction
• Example #5:
1. invert Y
00001010 => 11110101
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0010 0001
- 0000 1010
2. add 1 to Y’
11110101 + 00000001 =
11110110
0010 0001
3. add Y’ to X
+ 1111 0110
1 0001
this 9 bit 1 gets ignored
0111
th
Binary Subtraction
• It’s always good to verify your work…
0010 0001
- 0000
1010
0001 0111
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33
- 10
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Binary Subtraction
• Example #6
0001 1001
- 1110
0010
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Binary Subtraction
• Example #6
1. invert Y
1110 0010 => 0001 1101
0001 1001
- 1110
0010
2. add 1 to Y’
0001 1101 + 0000 0001 = 0001 1110
3. add Y’ to X
31
0001 1001
+ 0001 1110
0011 0111
Binary Subtraction
• Verify…
0001 1001
- 1110
0010
0011 0111
32
25
- -30
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Outline
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Binary Addition
2’s complement
Binary Subtraction
Half Adder
Logisim
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
B
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S
Half Adder
C
Half Adder
• Truth table for Half Adder
S : sum of A + B
A B S C
C : carry bit
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
A
B
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S
Half Adder
C
Half Adder
• Logic gate circuit for Half Adder?
• A, B inputs
• S, C outputs
A B S C
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
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Half Adder
• Logic gate circuit for Half Adder?
• A, B inputs
• S, C outputs
A B S C
0 0 0 0
0 1 1 0
1 0 1 0
1 1 0 1
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Half Adder
• How would you expand the Half Adder to create a:
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2-bit adder?
4-bit adder?
8-bit adder?
…?
• This will be part of your homework…
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Outline
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Binary Addition
2’s complement
Binary Subtraction
Half Adder
Logisim
Logisim
• Logisim is FREE logic gate simulation software
• Please download / install this software
• http://www.cburch.com/logisim/index.html
• It should be on the lab machines, too
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Next Steps
• Download / install Logisim
• Complete homework #2
• Data representation
• Binary arithmetic
• Create (useful) logic gate circuits using Logisim
• Next lecture: Micro-architecture
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