1. CSE 20 DISCRETE MATH Prof. Shachar Lovett http://cseweb.ucsd.edu/classes/wi15/cse20-a/ Clicker frequency: CA

2. Todays topics Algorithms: number systems, binary representation Section 1.3 in Jenkyns, Stephenson

3. Numbers are building blocks

4. Positional representation JS p. 22 What’s the (decimal) value of 10001 {2}? A. 5 {10} B. 17 {10} C. -1 {10} D. 10001 {10} E. None of the above / more than one of the above.

5. Positional representation JS p. 22 What’s the base 2 representation of the (decimal) number 42 {10}? A. 111111 {2} B. 100001 {2} C. 101010 {2} D. 110011 {2} E. None of the above / more than one of the above.

6. Positional representation JS p. 22 What’s the biggest integer value whose binary representation has 4 bits? A. 2 4 = 16 {10} B. 2 3 = 8 {10} C. 4 {10} D. 1000 {10} E. None of the above / more than one of the above.

7. Uniqueness Is it possible to have ? A. No. B. Yes, but m has to be the same as n. C. Yes, and m,n can be different but for each kind of coefficient that appears in both, it has to agree. That is, a 0 = b 0, a 1 = b 1, etc. D. Yes, if m=n and all the coefficients agree. E. More than one of the above / none of the above.

8. Parity and shift

9. Shifts

10. Positional representation JS p. 22 What’s the base 2 representation of the (decimal) number 2014 {10}? A. 11111011110 {2} B. 10000000000 {2} C. 10101010101 {2} D. 1000000001 {2} E. None of the above / more than one of the above.

11. Positional representation JS p. 22 What’s the base 2 representation of the (decimal) number 2014 {10}? A. 11111011110 {2} B. 10000000000 {2} C. 10101010101 {2} D. 1000000001 {2} E. None of the above / more than one of the above. Is there a systematic way (aka algorithm) to do it?

12. Decimal to Binary conversion Right to left Questions to ask: Does it always terminate? Does it give the correct answer? What is the time complexity? toBinary(pos int n) Begin x  “”; i  n; While i>0 Do If (i is even) Then x  “0”.x; End; If (i is odd) Then x  “1”.x; End; i  i/2; Output x End.

13. Other numbers? Fractional components Negative numbers aka how to subtract … first, how do we add? A. 111 B. 100 C. 1011 D. 1111 E. None of the above.

14. One bit addition 1 0 1 + 1 1 0 1 0 0 1 1 Carry:

15. Subtraction JS p. 6 Borrowing A – B = (A – 10) + (10 – B) Carrying A – B = (A+10) – (B+10) Complementation A – B = A + B c = A + [ (99-B) - 99 ] = A + [ (100-B) – 100 ]

16. 2’s complement 0000 0001 0010 1111 1110 1000 0111 0 1 2 -2 -7 -8 7 -3 1101 -4 1100 0011 3 1001 -5 -6 4 5 6 Complete the wheel of numbers! How many numbers are we representing with 4 bits?

17. How to add binary numbers? 1 1 0 0 1 0 1 0 1 + 1 0 1 1 0 1 1 0 1 ? ? ? ? ?

18. How to add binary numbers? ? ? ? ? ? ? ? ? (carry) 1 1 0 0 1 0 1 0 1 + 1 0 1 1 0 1 1 0 1 ? ? ? ? ?

19. How to add binary numbers? Two basic operations: One-Bit-Addition(bit1, bit2, carry) Next-carry(bit1, bit2, carry) ? ? ? ? ? ? ? ? (carry) 1 1 0 0 1 0 1 0 1 + 1 0 1 1 0 1 1 0 1 ? ? ? ? ?

20. Numbers … logic … circuits

21. Next class Boolean circuits and truth tables Read sections 3.2-3.4 in Jenkyns, Stephenson