Binary Arithmetic

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Outline

• Binary Addition• 2’s complement• Binary Subtraction• Half Adder • Logisim

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Binary Addition

• How to add two binary numbers?• Single bit:

0+ 0

0

0+ 1

1

1+ 0

1

1+ 110

carry bit

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Binary Addition

• What is a carry bit?• Same idea as “carry the 1” in decimal arithmetic:

13+ 9

1013

+ 92

1013

+ 922

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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+ 110

carry bit

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Binary Addition

• Example #1

0100 0011+ 0000

1010

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Binary Addition

• Example #1

0100 0011+ 0000

10100100 1101

carry bit

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Binary Addition

• Example #1• You can always check your work by converting to decimal

0100 0011+ 0000

10100100 1101

67+

1077

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Binary Addition

• Example #2

1100 0110+ 0100

1010

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Binary Addition

• Example #2

1001 11001100 0110

+ 0100 1010

1 0001 00009th

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

10101 0001

0000

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Binary Addition

• Example #2

198+ 74

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16 !?! Shouldn’t the answer be 272?

1100 0110+ 0100

10101 0001

0000

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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?

• 272 is greater than 255• 272 cannot be represented using 8 bits• Has this ever happened to you?

1111 1111

255

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Outline

• Binary Addition• 2’s complement• Binary Subtraction• Half Adder • Logisim

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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. Complement (invert) the binary digits (0 => 1; 1 => 0)2. Convert binary digits to decimal number3. Multiply by -14. Subtract 1

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2’s Complement

• Example #1: What is the 2’s complement value of 1100 0110?

1100 0110

1. Invert bits: 0011 1001

2. Convert to decimal: 57

3. Multiple by -1: -574. Subtract 1: -58

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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. Invert bits: 0110 0110

2. Convert to decimal: 102

3. Multiple by -1: -1024. Subtract 1: -103

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2’s Complement

• Decimal to 2’s complement• If the number is positive:

1. leftmost bit is 02. remaining bits identical to unsigned binary number

• E.g., Represent the number 97 using 8-bits, 2’s complement

97 = 64 + 32 + 1= 26 + 25 + 20

= 0110 0001

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2’s Complement

• Decimal to 2’s complement

• If decimal number is negative:1. add 12. multiply by -1 (to create positive number)3. create binary sequence4. invert bits

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2’s Complement

• Example #3: represent -97 using 8-bits, 2’s complement

1. add 1: -97 + 1 = -96

2. multiply by -1: -96 × -1 = 96

3. get binary: 96 = 0110 0000

4. invert bits: 1001 1111

-97 => 1001 1111

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2’s Complement

• Example #4: represent -123 using 8-bits, 2’s complement

1. add 1: -123 + 1 = -122

2. multiply by -1: -122 × -1 = 122

3. get binary: 122 = 0111 1010

4. invert bits: 1000 0101

-123 => 1000 0101

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Break Time!!!

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Outline

• Binary Addition• 2’s complement• Binary Subtraction• Half Adder • Logisim

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Binary Subtraction

• To subtract two binary numbers, X – Y,1. Invert Y => Y’2. Add 1 to Y’3. 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: 0010 0001- 0000

10101. invert Y 00001010 => 11110101

2. add 1 to Y’11110101 + 00000001

= 11110110

3. add Y’ to X0010 0001

+ 1111 0110

1 0001 0111

this 9th bit 1 gets ignored

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Binary Subtraction

• It’s always good to verify your work…

0010 0001- 0000

10100001 0111

33- 10

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Binary Subtraction

• Example #6

0001 1001- 1110

0010

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Binary Subtraction

• Example #6 0001 1001- 1110

00101. invert Y 1110 0010 => 0001 1101

2. add 1 to Y’0001 1101 + 0000 0001 =

0001 1110

3. add Y’ to X0001 1001

+ 0001 1110 0011 0111

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Binary Subtraction

• Verify…

0001 1001- 1110

00100011 0111

25- -30

55

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Outline

• 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

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Half Adder

A

B

S

C

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Half Adder

• Truth table for Half Adder

Half Adder

A

B

S

C

A B S C

0 0 0 0

0 1 1 0

1 0 1 0

1 1 0 1

S : sum of A + B

C : carry bit

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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

• 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:• 2-bit adder?• 4-bit adder?• 8-bit adder?• …?

• This will be part of your homework…

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Outline

• Binary Addition• 2’s complement• Binary Subtraction• Half Adder • Logisim

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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