The Power of Symbols

1. The Power of Symbols MEETING THE CHALLENGES OF DISCRETE MATHEMATICS FOR COMPUTER SCIENCE

2. One CS Goal • Syntax • Semantics

3. 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.

4. 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 =

5. Who Chose our Symbols and Why? • 3 minute student presentations • Sources: books, google • Some choices: carefully thought out • Some: serendipitous

6. 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

7. 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.

8. Some Examples • ~ as an abstraction for “is related to” • 0 for place value •  perpendicular, undefined •  print availability • 

9. Cool Facts about “1” • Natural Number • Smallest Positive Odd Integer • Multiplicative / Division identity • Exponentiation

10. 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

11. The Symbol for Percent

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

13. 20 p 100 • An anonymous Italian manuscript of 1425 • By 1650

14. 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.

15. 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.

16. The 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.

17. Historyof 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:

18. 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”:

19. Venn Diagrams

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

21. 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.

22. Symmetric Venn DiagramsInvolving Higher Number Sets

23. 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

24. 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 number 15 + 5 + 1 = 21

25. Happy Number • Reduces to one when the following pattern is repeated: • Square the number • Take the sum of the squares of the digits • Repeat

26. 72 = 49 • 42 + 92 = 97 • 92 + 72 = 130 • 12 + 32 + 02 = 10 • 12 + 02 = 1

27.  • 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.

28. 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

29. Future Symbol Use • Formal Specifications • Unicode