Addition and Subtraction Philipp Koehn 4 September 2019 Philipp - - PowerPoint PPT Presentation

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Addition and Subtraction Philipp Koehn 4 September 2019 Philipp - - PowerPoint PPT Presentation

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Addition and Subtraction Philipp Koehn 4 September 2019 Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019 1 addition Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September

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SLIDE 1

Addition and Subtraction

Philipp Koehn 4 September 2019

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 2

1

addition

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 3

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1-Bit Adder

  • Let’s start simple:

Adding two 1-Bit numbers

  • Truth table

A B A+B 1 1 1 1 1 1 10

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 4

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Really 2 Operations

  • Truth table for "position 0" bit

A B A+B 1 1 1 1 1 1

xor

  • Truth table for carry bit

A B A+B carry 1 1 1 1 1

and

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 5

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Reminder: Basic Gates

AND OR NOT NAND NOR

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 6

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Circuits

  • "Position 0" bit

A B OUT0 1 1 1 1 1 1

xor

NAND AND OR

A B OUT0

  • Carry bit

A B OUTC 1 1 1 1 1

and

A B

AND

OUTC

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 7

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Putting them Together

AND

CARRY

NAND AND OR

A B OUT0

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 8

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N-Bit Addition

011 +11

  • 110
  • 110

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 9

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N-Bit Addition

011 +11

  • 110
  • 110

1+1 = 0, carry the 1

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 10

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N-Bit Addition

011 +11

  • 110
  • 110

1+1+1 = 1, carry the 1

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 11

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N-Bit Addition

011 +11

  • 110
  • 110

copy carry bit

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 12

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1-Bit Adder

AND

CARRY

NAND AND OR

A B OUT0

Our adder cannot handle carry as input yet

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 13

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Half 1-Bit Adder

AND

CARRY

NAND AND OR

A B OUT0

1-BIT HALF ADDER

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 14

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Building a 1-Bit Full Adder

CARRY A B

1-BIT HALF ADDER A B S C

SUM

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 15

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Building a 1-Bit Full Adder

CARRY A B

1-BIT HALF ADDER A B S C

CARRY CARRY SUM

1-BIT HALF ADDER A B S C

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 16

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Building a 1-Bit Full Adder

A B

1-BIT HALF ADDER A B S C

CARRY SUM

1-BIT HALF ADDER A B S C OR

CARRY

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 17

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1-Bit Full Adder

A B

1-BIT HALF ADDER A B S C

CARRY SUM

1-BIT HALF ADDER A B S C OR

CARRY

1-BIT FULL ADDER

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 18

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N-Bit Full Adder

011 +11

  • 110
  • 110

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 19

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N-Bit Full Adder

011 +11

  • 110
  • 110

1-BIT FULL ADDER A B S CO CI

A0 B0 SUM0

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 20

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N-Bit Full Adder

011 +11

  • 110
  • 110

1-BIT FULL ADDER A B S CO CI

A0 B0

1-BIT FULL ADDER A B S CO CI

A1 B1 SUM0 SUM1

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 21

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N-Bit Full Adder

and so

  • n

...

1-BIT FULL ADDER A B S CO CI

A0 B0

1-BIT FULL ADDER A B S CO CI

A1 B1 SUM0 SUM1

1-BIT FULL ADDER A B S CO CI

A2 B2 SUM2

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 22

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subtraction

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 23

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First, a Trick

  • Normally, we subtract like this:

0253

  • 176
  • 0110
  • 0077

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 24

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Computing the Inverse

  • Now we use the inverse of the subtrahend

0999

  • 176
  • 0823

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 25

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Subtraction by Addition

  • This allows us to carry our subtraction by addition

00253 +0823

  • 01076
  • Well, with minor corrections

01076 +0001

  • 1000
  • 0077

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 26

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Also Works in Binary

Original problem 00253

  • 0176
  • 00077

0011111101

  • 010110000
  • 0001001101

Inverse of subtrahend 00823 0001001111 Addition 00253 +0823

  • 01076

0011111101 +001001111

  • 0101001100

Corrections +0001

  • 1000
  • 0077

+000000001

  • 100000000
  • 0001001101

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 27

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Start with N-Bit Adder

4-BIT FULL ADDER A0 B3 S0 CO CI A1 A2 A3 B0 B1 B2 S1 S2 S3

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 28

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Invert Bits of Subtrahend

4-BIT FULL ADDER A0 B3 S0 CO CI A1 A2 A3 B0 B1 B2 S1 S2 S3

B0 B1 B2 B3

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 29

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Add One

4-BIT FULL ADDER A0 B3 S0 CO CI A1 A2 A3 B0 B1 B2 S1 S2 S3

B0 B1 B2 B3 V A0 A1 A2 A3

Trick: add one as carry in

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 30

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Invert Overflow --- DONE

4-BIT FULL ADDER A0 B3 S0 CO CI A1 A2 A3 B0 B1 B2 S1 S2 S3

B0 B1 B2 B3 V A0 A1 A2 A3 OVERFLOW SUM

mm...

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 31

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unifying addition and subtraction machines

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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Goal

  • Not two machines for addition and subtraction

⇒ Combined adder and subtractor

  • Input:

A, B, and subtraction flag SUB

  • Output

– if SUB=0: A+B – if SUB=1: A-B

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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NOT only if SUB

4-BIT FULL ADDER A0 B3 S0 CO CI A1 A2 A3 B0 B1 B2 S1 S2 S3

B0 B1 B2 B3 V A0 A1 A2 A3 OVERFLOW SUM

mm...

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 34

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NOT only if SUB

  • Truth table

SUB X OUT 1 1 1 1 1 1

  • Looks like XOR

XOR

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019

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SLIDE 35

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Combined Machine

4-BIT FULL ADDER A0 B3 S0 CO CI A1 A2 A3 B0 B1 B2 S1 S2 S3

B0 B1 B2 B3 SUB A0 A1 A2 A3 OVERFLOW SUM SUB SUB

Philipp Koehn Computer Systems Fundamentals: Addition and Subtraction 4 September 2019