1. Positional Number Systems

2. Decimal Review 5049 = 5(1000) + 0(100) + 4(10) + 9(1)
5049 = 5· · · ·100
place
digit

3. Binary Representation
27 =
27 = 1·24 + 1·23 + 0·22 + 1·21 + 1·20
27 =
place
digit

4. Some Binary Representations
010 ?2
110
210
310
410
510
610
710
810
910

5. Some Binary Representations
010 02
110 12
210
102
310
112
410
1002
510
1012
610
1102
710
1112
810
10002
910
10012

6. Powers of Two

7. Convert Binary to Decimal

8. Convert Binary to Decimal
= 53

9. Convert Decimal to Binary
20910 = smaller number
= 1(128) + 81
= 1(128) smaller number
= 1(128) + 1(64) + 17
= 1(128) + 1(64) + 0(32) + 1(16) + 1
1(128)+1(64)+0(32)+1(16)+0(8)+0(4)+0(2)+1(1) =

10. Binary Addition

11. Binary Addition
carry

12. Binary Addition 12 + 12 1 02 12 12 + 12 1 12 1 1 carry
carry

13. Binary Addition 12 + 12 1 02 12 12 + 12 1 12 1 1 1 carry
carry

14. Binary Addition 12 + 12 1 02 12 12 + 12 1 12 1 1 1 carry
carry

15. Subtraction in Decimal System

16. Subtraction in Decimal System
borrowing

17. Subtraction in Decimal System
borrowing

18. Subtraction in Binary System

19. Subtraction in Binary System
borrowing

20. Subtraction in Binary System
borrowing

21. Two’s Complement Arithmetic
Computers often use 2’s complement arithmetic for working with signed numbers
2’s complement of a in n-bit arithmetic is the binary representation of 2n – a

22. Two’s Complement Example
The 8 bit representation of -27 is
( 28 – 27)10 = =
Or flip the bits and add one
-27 = = =

23. Two’s Complement Arithmetic
To subtract, take the two’s complement and then add.
Otherwise just add the binary numbers and throw away any positions greater than 2n-1.
If -2n-1  result < 2n-1 then everything is fine.
Otherwise you have an overflow.

24. Hexadecimal Representations
Binary 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15

25. Hexadecimal Representations
Binary 1 2 3 4 5 6 7 8 9 10 A 11 B 12 C 13 D 14 E 15 F

26. Hexadecimal Representations
Binary
0000 1
0001 2
0010 3
0011 4
0100 5
0101 6
0110 7
0111 8
1000 9
1001 10 A
1010 11 B
1011 12 C
1100 13 D
1101 14 E
1110 15 F
1111

27. Convert Hexadecimal to Decimal
3CF16
= 3(162) + 12(161) + 15(160)
= 97510

28. Convert Hexadecimal to Binary
C50A16
C A

29. Convert Binary to Hexadecimal
D A