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 < 2 < 0 < 1 Step 1: 2 < 0 This expression evaluates to

false and is converted to 0, since PL/1 represents false as 0.

Step 2: 0 < 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

The Symbol for Percent

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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 Є X and y Є Y}.

Is the basis for the Cartesian coordinate system.

The History of ZeroThe History of Zero

Babylonian’s 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:

2rArea

1 1 12

1

2

114

k k kk

81256

825

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 Euler’s number

•One of the five most important numbers in mathematics along with 0, 1, i, and pi.

Euler eventually related all five of math’s most important numbers in his famous “Euler’s 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 can’t be generated by adding any integer to the sum of its digits

Ex: 21 is not a self number15 + 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