1. Binary Addition (1 of 2) Two 1-bit values A B A + B 11
1 Carry one to next

3. Binary Subtraction

6. d

7. Example2

9. Subtraction using 2’s complements
How to find 2’s Complement
Step1: Find 1’s complement
Step2: ADD 1 to find 2’s complements

12. d
256
128 64 32 16 8 4 2 1 54 22

15. y

16. Example –
2nd ks 2 complement
+ 1
Find 2’s Complements of Ans
If No carry remaining

17. Example
– Ans:
2nd ks 2 complement
+ 1
+ 1
Find 2’s Complements of Ans
If No carry remaining

18. 11001
256
128 64 32 16 8 4 2 1
=25

19. Binary numbers? Computers work only on two states
Off
Basic memory elements hold only two states
Zero / One
Thus a number system with two elements {0,1}
A binary digit – bit !

20. Common Number Systems System Base Symbols Used by humans?
Used in computers?
Decimal 10
0, 1, … 9
Yes No
Binary 2
0, 1
Octal 8
0, 1, … 7
Hexa- decimal 16
0, 1, … 9,
A, B, … F

21. Quantities/Counting (1 of 3)
Decimal
Binary
Octal
Hexa- decimal 1 2 10 3 11 4
100 5
101 6
110 7
111
p. 33

22. Quantities/Counting (2 of 3)
Decimal
Binary
Octal
Hexa- decimal 8
1000 10 9
1001 11
1010 12 A
1011 13 B
1100 14 C
1101 15 D
1110 16 E
1111 17 F

23. Quantities/Counting (3 of 3)
Decimal
Binary
Octal
Hexa- decimal 16
10000 20 10 17
10001 21 11 18
10010 22 12 19
10011 23 13
10100 24 14
10101 25 15
10110 26
10111 27
Etc.

24. Decimal to Binary
Decimal to Octal
Decimal to Hexadecimal

25. Decimal to Binary
Decimal to Octal
Decimal to
Hexadecimal

26. Decimal to Binary Examples: 7.7510 = (?)2 7 / 2 = 3 (Q), 1 (R) 
710 = 1112
0.75 x 2 =1.50  extract 1
0.5 x 2 =  extract = 0.112
 stop
Ans

27. Exercise Exercise 1: Convert (0.625)10 to its binary form
Solution:
Solution: x 2 = 1.25  extract 1
0.25 x 2 = 0.5  extract 0
0.5 x 2 = 1.0  extract 1
0.0  stop
 (0.625)10 = (0.101)2
0.6 x 2 = 1.2  extract 1
0.2 x 2 = 0.4  extract 0
0.4 x 2 = 0.8  extract 0
0.8 x 2 = 1.6  extract 1
0.6 x 2 = 
 (0.6)10 = ( …)2

28. Exercise Examples: try 5.625B
Exercise 3: Convert (0.8125)10 to its binary form
Solution: x 2 =  extract 1
0.625 x 2 = 1.25  extract 1
0.25 x 2 = 0.5  extract 0
0.5 x 2 = 1.0  extract 1
0.0  stop
 (0.8125)10 = (0.1101)2
Examples: try 5.625B

29. Fractions Decimal to binary
x x x x x x
etc.
p. 50

30. Binary to Decimal
Octal to Decimal
Hexadecimal to Decimal

31. Binary Decimal 1101 = 1 x 23 + 1 x 22 + 0 x 21 + 1 x 20=
(1101)2 = (13)10
1, 2, 4, 8, 16, 32, 64, 128, 256, 512, ….

32. Decimal Binary 2 13
LSB 1 2 6 2 3 1 2 1 1
MSB
(13)10 = (1101)2

33. Octal Decimal 137 = 1 x 82 + 3 x 81 + 7 x 80 = 1 x 64 + 3 x 8 + 7 x 1=
(137)8 = (95)10
Digits used in Octal number system – 0 to 7

34. Decimal Octal 8 95
LSP 7 8 11 3 8 1 1
MSP
(95)10 = (137)8

35. Hex Decimal A = 10, B = 11, C = 12, D = 13, E = 14, F = 15
BAD = 11 x x x 160
= 11 x x x 1 =
(BAD)16 = (2989)10
A = 10, B = 11, C = 12, D = 13, E = 14, F = 15

36. Decimal Hex 16
2989
LSP 13 16
186 10 16 11 11
MSP
(2989)10 = (BAD)16

37. Binary to Decimal Octal to Decimal 1010112 =>
7248 =>
4 x 80 = x 81 = x 82 = 448 Ans: 46810
=>
1 x 20 = 1 1 x 21 = 2 0 x 22 = 0 1 x 23 = 8 0 x 24 = 0 1 x 25 =32
Ans. 4310
Hexadecimal to Decimal
ABC16 =>
C x 160 = 12 x 1 = 12
B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560
274810

38. Fractions Binary to decimal 10.1011 =>
1 x 2-4 = x 2-3 = x 2-2 = x 2-1 = x 20 = x 21 =

39. Binary to Hexadecimal

40. Binary to Hexadecimal Technique Group bits in fours, starting on right
Convert to hexadecimal digits

41. d

42. d