1. CSE 20: Discrete Mathematics for Computer Science Prof. Shachar Lovett

2. Today’s Topics: 1. Number representations in different bases 2. Converting between bases 2

3. 1. Number representations all your base are belong to us 3

4. Numbers are building blocks

5. Bases 5

6. Values in different bases 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.

7. Values in different bases 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.

8. Values in different bases 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.

9. Uniqueness Is it possible to have two different representations for an integer in base 2? That is, 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.

10. Parity and shift

11. Shifts

12. Values in different bases 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.

13. Values in different bases 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?

14. 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 binary=“” i=n While i>0 Do If (i is even) Then binary=“0”+binary End If (i is odd) Then binary=“1”+binary End i=i DIV 2 Output binary End.

15. Other numbers?  Fractional components  Negative numbers aka how to subtract … first, how do we add? A. 111 B. 100 C. 1011 D. 1111 E. Other

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

17. 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 ]

18. 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?

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

20. How to add binary numbers? ? ? ? ? ? ? ? ? ? carry 1 1 0 0 1 0 1 0 1 +1 0 1 1 0 1 1 0 1 ? ? ? ? ?

21. 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 ? ? ? ? ?

22. Numbers … logic … circuits