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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
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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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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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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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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