I Real world systems and processes

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I Real world systems and processes Mostly
continuous (at the macroscopic level) time,
acceleration, chemical reactions Sometimes
discrete quantum states, mass ( of
atoms) Mathematics to represent physical systems
is continuous (calculus) Mathematics for number
theory, counting, approximating physical systems
can be discrete
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• II Representation of information
• A. Continuousrepresented analogously as a value
  of a continuously variable parameter
• position of a needle on a meter
• rotational angle of a gear
• amount of water in a vessel
• electric charge on a capacitor
• B. Discretedigitized as a set of discrete
  values corresponding to a finite number
• of states
• 1. digital clock
• 2. painted pickets

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• III Representation of continuous processes
• Analogous to the process itself
• Great Brass Braina geared machine to simulate
  the tides
• Slide rulean instrument which does
  multiplication by adding lengths which correspond
  to the logarithms of numbers.
• Differential analyzer (Vannevar
  Bush)variable-size friction wheels to simulate
  the behavior of differential equations

Vannevar Bush integrator
Tide calculator
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Brass Brain was the equal of 100 mathematicians,
weighted a mere 2500 lbs Imagine the fearful
gnashings of mathematicians in November, 1928
upon reading this account of the USGS's new
"brass brain," which could "do the work of 100
trained mathematicians" in calculating tides
The machine weighs 2,500 pounds. It is 11 feet
long, 2 feet wide, and 6 feet high. Its whirring
cogs are enclosed in a housing of mahogany and
glass. Earthquakes, fresh-water floods, and
strong winds that cannot be predicted affect the
accuracy of the Brass Brain to a degree.
Nevertheless 70 of the predicted tides agree
within five minutes of the observed tide. The
Coast and Geodetic Survey issues an annual
bulletin in which it lists the forthcoming tides
in 84 ports of the world. The report contains
upwards of a million figures, all compiled by the
Brass Brain. It has been estimated that the Brass
Brain saves the government 125,000 each year in
salaries of mathematicians who would be required
to take its place.
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From Instruments of Science an historical
encyclopedia Great Brass Brain It remained in
use until the late 1960s,when an IBM 7090
computer took over the job. Even when digital
computers finally took over from analog
instruments, the amount of arithmetic needed to
properly evaluate the cosine series was so vast
that the output had to be limited to simply times
of high and low tide for any particular area.
This was overcome only when, during the 1970s,
digital computers became powerful enough. . .
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Discrete representations
What is it??
What is it??
What is it??
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Babbage difference engine to calculuate
polynomials
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• Electronic analog computerscircuitry connected
  to simulate differential equations
• Phonograph recordwiggles in grooves to represent
  sound oscillations
• Electric clocks
• Mercury thermometers/barometers

Stereo phonograph record
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IV Manipulation A. Analog
1. adding the length-equivalents of
logarithms to obtain a multiply, e.g., a
slide-rule 2. adjusting the
volume on a stereo 3. sliding a
weight on a balance-beam scale 4. adding
charge to an electrical capacitor B.
Discrete 1. countingpush-button
counters 2. digital operationsmechanical
calculators 3. switchingopen/clos
ing relays 4. logic circuitstrue/false
determination
Marble binary counter
Marchant mechanical calculator
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V Analog vs. Discrete Note "Digital"
is a form of representation for discrete
A. Analog 1.
infinitely variable--information density high
2. limited resolution--to what
resolution can you read a meter? 3.
irrecoverable data degradation--sandpaper a vinyl
record B. Discrete/Digital
1. limited states--information density low,
e.g., one decimal digit
can represent only one of ten values
2. arbitrary resolution--keep adding states (or
digits) 3. mostly recoverable data
degradation, e.g., if information is
encoded as painted/not-painted
pickets, repainting can
perfectly restore data
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Decimal Binary 21 Hexa-decimal 24
0 0000 0
1 0001 1
2 0010 2
3 0011 3
4 0100 4
5 0101 5
6 0110 6
7 0111 7
8 1000 8
9 1001 9
10 1010 A
11 1011 B
12 1100 C
13 1101 D
14 1110 E
15 1111 F
VI Digital systems A. decimal--not so
good, because there are few 10-state devices
that could be used to store information
fingers. . .? B. binary--excellent for
hardware lots of 2-state devices
switches, lights, magnetics--poor for
communication 2-state devices
require many digits to represent values with
reasonable resolution--excellent for
logic systems whose states are true
and false. But binary is king because
components are so easy (and cheap)
to fabricate. C. octal --base 8 used to
conveniently represent binary data
almost as efficient as decimal D.
hexadecimal--base 16 more efficient than
decimal more practical than octal
because of binary digit groupings in
computers
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VII Binary logic and arithmetic A.
Background 1. George Boole(1854)
linked arithmetic, logic, and binary number
systems by showing how a binary system could be
used to simplify complex logic problems
2. Claude Shannon(1938) demonstrated that any
logic problem could be represented by a system
of series and parallel switches and that binary
addition could be done with electric
switches 3. Two branches of binary logic
systems a) Combinatorialin which the output
depends only on the present state of the
inputs b) Sequentialin which the output may
depend on a previous state of the
inputs, e.g., the flip-flop circuit
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AND gate
A
B
A B C 0 0 0 1 0 0 0 1 0 1 1 1
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AND gate
A
B
A B C 0 0 0 1 0 0 0 1 0 1 1 1
Simple AND Circuit
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OR gate
A B C 0 0 0 1 0 1 0 1 1 1 1 1
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OR gate
Simple OR circuit
A B C 0 0 0 1 0 1 0 1 1 1 1 1
A
B
C
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NOT gate
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NOT gate
Simple NOT circuit
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NAND gate
A
B
A B C 0 0 1 1 0 1 0 1 1 1 1 0
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NAND gate
A
B
A B C 0 0 1 1 0 1 0 1 1 1 1 0
Simple NAND Circuit
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3. Control systems e.g., car will start only
if doors are locked, seat belts are on, key
is turned D S K I 0 0 0 0 0 0 1
0 0 1 0 0 0 1 1 0 1 0 0
0 1 0 1 0 1 1 0 0 1 1 1 1

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3. Control systems e.g., car will start only
if doors are locked, seat belts are on, key
is turned D S K I 0 0 0 0 0 0 1
0 0 1 0 0 0 1 1 0 1 0 0
0 1 0 1 0 1 1 0 0 1 1 1 1

I D AND S AND K
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Binary arithmetic e.g., adding two binary
digits A B R C 0 0 0 0
0 1 1 0 1 0 1 0
1 1 0 1


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Binary arithmetic e.g., adding two binary
digits A B R C 0 0 0 0
0 1 1 0 1 0 1 0
1 1 0 1

R (A OR B) AND NOT (A AND B) C A AND B

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Boolean algebra properties
AND rules OR rules
AA A A A A
AA' 0 A A' 1 0A
0 0A A 1A A
1 A 1 AB BA
A B BA A(BC) (AB)C
A(BC) (AB)C A(BC) ABBC
ABC (AB)(AC) A'B' (AB)'
A'B' (AB) (DeMorgans theorem)
Notation AND OR
NOT
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Computing power has been growing at an
exponential rate Note graph is a semi-log
plotthe best way to indicate a function
y(t)aekt.