The Power of Symbols

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The Power of Symbols

• MEETING THE CHALLENGES OF DISCRETE MATHEMATICS
  FOR COMPUTER SCIENCE

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One CS Goal

• Syntax
• Semantics

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Kurt Godel

• greatest single piece of work in the whole
  history of mathematical logic
• Incompleteness result
• 120 pages
• Theory of Computation students can do in one page
  using reduction.

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The Role of Symbols in How We Think

• The meaning in math (symmetric)
• The meaning in Java and C (not symmetric)
• ? not symmetric
• not symmetric
• unnecessary if assignment operator is not

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Who Chose our Symbols and Why?

• 3 minute student presentations
• Sources books, google
• Some choices carefully thought out
• Some serendipitous

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Overloading

• In Math , -, , etc. for a variety of number
  systems and more abstract systems
• In CS built-in for numbers in most languages
• User-defined allowed in C, not allowed in Java

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Symbol Anomaly

• PL1 use of lt
• 2 lt 0 lt 1
• Step 1 2 lt 0 This expression evaluates to
  false and is converted to 0, since PL/1
  represents false as 0.
• Step 2 0 lt 1 This expression evaluates to
  true and is converted to 1, since PL/1 represents
  true as 1.
• So the overall evaluation is true.

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Some Examples

• as an abstraction for is related to
• 0 for place value
• ? perpendicular, undefined
• ? print availability
• ?

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Cool Facts about 1

• Natural Number
• Smallest Positive Odd Integer
• Multiplicative / Division identity
• Exponentiation

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i

• Girolamo Cardano 1545
• Ars Magna
• Equations with solutions not on the real line
• Imaginary numbers
• Earlier recognition of such equations by the
  Greek Heron in 1 AD, but no name given

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The Symbol for Percent
12

• Roman Emperor Augustus levied a tax on all goods
  sold at auction
• The rate of it was 1/100

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• An anonymous Italian manuscript of 1425
• By 1650

20 p 100
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Square Root

• First approximation was by Babylonians of the
  was
• 1 24/60 51/60² 10/60³ 1.41421296
• The symbol ( ) was first used in the 16th
  century. It was suppose to represent a lowercase
  r, for the Latin word radix.

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Cartesian Products

• Created by French philosopher René Descartes in
  the 17th century.
• X x Y (x,y) x ? X and y ? Y.
• Is the basis for the Cartesian coordinate system.

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The History of Zero
Babylonians had no concept of the number zero
????? 2 ????? 120
Europe
-Not used until Fibonacci, who was introduced to
zero because of the Spanish Moors adopting the
Arabic Numeral system.
-Hindu-Arabic numerals until the late 15th
century seem to have predominated among
mathematicians, while merchants preferred to use
the abacus. It was only from the 16th century
that they became common knowledge in Europe.
Mayans
Had concept of zero as early as 36 B.C. on their
Long Count calendar.
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History of p

• First Introduced by William Jones
• Made Standard by Leonard Euler
• Greeks, Babylonians, Egyptians and Indian
  slightly more than 3
• Indian and Greek
• Madhava of Sangamagrama
• Ahmes
• Babylonians

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e 2.71828 18284 59045 23536
e can be expressed as

• The constant was first discovered by Jacob
  Bernoulli when attempting a continuous interest
  problem
• Was originally written as b
• Euler called it e in his book Mechanica
• Is also called Eulers number
• One of the five most important numbers in
  mathematics along with 0, 1, i, and pi.

Euler eventually related all five of maths most
important numbers in his famous Eulers
Identity
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Venn Diagrams
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Uses

• Show logical relationships between sets in set
  theory.
• Compare and contrast two ideas.

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History

• Developed by John Venn, logician and
  mathematician.
• Introduced in 1880 in a paper called On the
  Diagrammatic and Mechanical Representation of
  Propositions and Reasonings.
• His paper first appeared in the Philosophical
  Magazine and Journal of Science.

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Symmetric Venn DiagramsInvolving Higher Number
Sets
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Facts About 7

• Most picked random number 1-10
• A self number
• Smallest happy number
• 999,999/7 142,857
• 1/7 0.142857142857142857
• Most magical number Albus Dumbledore

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Self Numbers

• A number such that cant be generated by adding
  any integer to the sum of its digits
• Ex 21 is not a self number
• 15 5 1 21

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Happy Number

• Reduces to one when the following pattern is
  repeated
• Square the number
• Take the sum of the squares of the digits
• Repeat

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• 72 49
• 42 92 97
• 92 72 130
• 12 32 02 10
• 12 02 1

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?

• The mathematical symbol for infinity is called
  the lemniscate. 1655 by John Wallis, and named
  lemniscus (latin, ribbon) by Bernoulli about
  forty years later.
• The lemniscate is patterned after the device
  known as a mobius (named after a nineteenth
  century mathemetician Mobius) strip, a strip of
  paper which is twisted and attached at the ends,
  forming an 'endless' two dimensional surface.

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Lessons Learned

• For Programming choice of variable names and
  symbols is important.
• For Language Design ditto
• For Documentation ditto
• For Reasoning ditto
• Human Computer Interaction ditto

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Future Symbol Use

• Formal Specifications
• Unicode