COS 111 Review Session 1

1. COS 111 Review Session 1 Friday, March 4, 2005

2. Outline • All About Numbers • Boolean/Logic Circuits • Assignment 4 • Questions

3. Can you say five ?

4. Say five Dutch – vijf German – fünf French – cinq Spanish – cinco Hindi – paanch Slang -- Lincoln Math -- 5

5. Say five Dutch – vijf German – fünf French – cinq Spanish – cinco Hindi – paanch Slang – Lincoln Math -- 5 Spoken form

6. Say five Dutch – vijf German – fünf French – cinq Spanish – cinco Hindi – paanch Slang – Lincoln Math -- 5 Visual form

7. When is five not five • When using different langauges • GM called one of their small cars "Nova". They didn't sell too many in Spain where 'NoVa' means “doesn’t go” • Math has many sub-dialects – binish, tertiarist, octalish, hexadecimalish, AnyNish (I am making the names up but that’s not the point :))

8. How much is 10 ? • You need to know what language it is being spoken in • V in roman numerals refers to decimal 5 but refers to decimal 31 in hexatridecimalish • How do we translate from one dialect to another ? • We need to understand the structure of math-dialects

9. Closer look at Roman Numerals • Pick a few agreed upon quantities – I, V, X, L, C, D, M • Express all other numbers as sums and differences of above – 7 is VII, 19 is XIX, 10000 is MMMMMMMMMM • Not very convenient as numbers become large • Structure also cumbersome – 41 is XLI or IXL

10. Penta System • Instead of sums and differences, can we use multiplication to provide structure to number ? • MMMMMMMMMM can be X-M • But a odd collection I, V, X, L, C, D, M wont do • Pick 5 symbols – 0, 1, 2, 3, 4. Why 5 ? • Its arbitrary. • It doesn’t matter what the base is as long as its fixed

11. Lets count… 0 1 2 3 4 What now ? We need to combine our symbols to come up write bigger numbers

12. Lets count… 0 1 2 3 4 What now ? We have made one pass over all symbols. So lets note down that fact. One pass and no more.

13. Lets count… 0 1 2 3 4 10 – lets call this a fif We now use position of a symbol in a number to hold its value.

14. Lets count… 0 1 2 3 4 10 10 – fif 11 – fif one 12 – fif two 13 – fif three 14 – fif four 20 -- twofif

15. Lets count… 0 1 2 3 4 10 10 11 12 13 14 20 20 21 22 23 24 30 30 31 32 33 34 40 40 – fourfif 41 – fourfif one 42 – fourfif two 43 – fourfif three 44 – fourfif four 100 – fiffif We now use position of a symbol in a number to hold its value

16. Penta System • A number ABCDE is hence • A fif-fif-fif-fif + • B fif-fif-fif + • C fif-fif + • D fif + • E • A*fif^4 + B*fif^3 + C*fif^2 + D*fif + E

17. b System • A number Xk-1….X0 in base b is • Sum of Xi-1*b^i for i from 0 to k-1 • All rules of multiplication, addition, subtraction are similar to what we normally do in base 10 numbers

18. Lets do some practice • Conversion from one base to another • Subtraction, addition, multiplication in any base • Suggest numbers and operations and we work it out together.

19. Before we move to next topic… • Old number systems joke – • Why is Christmas like Halloween ? • Because 31 oct = 25 dec

20. Outline • All About Numbers • Boolean/Logic Circuits • Assignment 4 • Questions

21. Boolean Algebra • Shorthand for writing and thinking about logic circuits • Notation • ' is a NOT • . is an AND • + is an OR • 1 represents TRUE • 0 represents FALSE

22. Some simple rules • (A ') ' = A • (A ' + A) = 1 • A + 0 = A • A + 1 = 1 • (A '.A) = 0 • A.0 = 0 • A.1 = A • A + A = A • A.A = A

23. Distributive Laws • E +(E1.E2...En) = (E+E1).(E+E2)...(E+En) • E.(E1+E2+...En) = (E.E1) + (E.E2)... + (E.En)

24. DeMorgan’s Laws • (E1 + E2 + ... + En)' = E1'.E2'....En' • (E1.E2...En)' = E1' + E2' + ... + En'

25. Lets try some examples • x'.y + x.y + x • x.y.z + x'.y.z + x'.y'.z + x'.y'.z + x.y'.z' + x.y'.z • x'.y + x'.y' + x.y' + x.y

26. Outline • All About Numbers • Boolean/Logic Circuits • Assignment 4 • Questions