How computers calculate

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How computers calculate

• How binary operations yield complex capabilities

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Activity Part 1

• Answer the questions in the Before You Start
  section

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Why Digital Not Analog?

• What is difference?
• Digital is discrete number analog varies
  continuously
• Analog signals prone to distortion because of
  large range
• Analog signals lose information with every
  replication and over time

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Discussion Question

• What are some applications for which you would
  prefer an analog device? When would a digital
  device be preferable?

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Why Binary not Decimal?

• Two options is a lot easier to manage than 10
• Switch can be on/off
• Less confusion between different settings
• Faster

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What is Binary?

• Only uses digits 1 and 0
• So 1110 (112)
• 10111 (213)
• 111100 (314)
• 10011010? (182819)
• 10011 (1216)

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Activity Part 2

• Hook up the equipment as shown by your
  instructor, and answer the first 3 questions in
  the What type of signal is it? section

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What Good is Binary?

• Can use numbers to represent letters
• Standard code ASCII has 7 binary digits (bits)
  per character
• e 1100101
• ! 0100001
• Del 1111111
• E 1000101

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Activity Part 3

• Answer questions 4-8 in the What are the
  properties of the signal? section

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Activity Part 4

• Answer questions 9-15 in the How do signals from
  different keys compare? section

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Boolean Logic

• NOT the simplest outputs the opposite of the
  input.
• AND outputs a 1 only if both of its two inputs
  are 1.
• NAND outputs a 1 unless both of its two inputs
  are 1.
• OR outputs 1 if either input is 1.
• XOR outputs 1 if only one input is 1

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So how do these do math?

• Can create a half-adder and a full adder
• Half-adders add two digits and output the unit
  output and the carry digit
• Full -adders add two digits and a carried digit
  and give the two outputs

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Creating a half-adder

• An XOR gate can give the unit output
• 0 0 ? 0
• 0 1 ? 1
• 1 0 ? 1
• 1 1 ? 0
• An AND gate can give the carry digit
• 0 0 ? 0
• 0 1 ? 0
• 1 0 ? 0
• 1 1 ? 1

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Creating a full-adder

• Combine 2 half-adders and 1 OR
• First half-adder adds two input digits, giving
  unit and carry digit
• Second adds new unit digit with carry input,
  giving final unit output and carry digit
• OR gives carry digit if either half-adder has a
  carry digit (you will never have carry digits
  from both)

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Adding 1111
Ones (20)
Twos (21)
1
1
1
1
1
1
1
0
1
1
So answer is 110, or 6 in decimal representation
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Adding large numbers

• Start with right-most digits and work left
• Number of gates needed grows quickly

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An exercise

• How many gates needed to add any possible 2-digit
  decimal integers?
• 9916413201608041211
• So need to add 2 7-digit binary numbers
• Need one half-adder and 6 full-adders
• Half-adder has 1 XOR and 1 AND
• Full-adders have 2 XOR, 2 AND, 1 OR
• Total is 13 XOR, 13 AND, 6 OR

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Discussion Question

• Think about different mathematical functions you
  are familiar with. Could you perform each of
  them using only adders and half-adders? How
  would you do it?

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The Rest of Mathematics

• Subtraction, Multiplication, and Division can be
  performed by combos of Addition and data shifting
• Mathematical features such as Taylor sums can be
  used to express complicated functions using
  addition.
• Data processing boils down to nothing but
  addition (and storage and shifting).

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So what have we learned today?

• Computers use digital binary numbers to
  communicate and calculate
• One convention for expressing text as binary
  numbers is ASCII
• Keyboards do not use ASCII, but use a quite
  different format
• Boolean operators can process binary information
• AND, NAND, and XOR can be used to create
  half-adders and full-adders
• All mathematical functions can be performed by a
  series of addition and data shifting

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Evaluation

• Log on to WebCT and answer the 6 questions in the
  Act02eval quiz