This document provides an overview of binary and other number systems used in computing. It discusses how computers use binary digits for operations while programmers find decimal more convenient. It then covers the basic binary, decimal, hexadecimal, and octal number systems including their digits and bases. The document also summarizes how to convert between these different number systems through examples of multiplication, division, and place value. Finally, it briefly mentions signed arithmetic, BCD, Gray code, logic gates, and flip-flops.

1. 1

2. 2
2
+1
=3
HOW THE COMPUTER GETS
THE ANSWER

3. 3
A computer understands
information composed of only zeros and
ones.
The decimal number system is
convenient for the programmer.
The computer uses binary digits for its
operation.
0
1
0
1
1
0
1
1
0
0
1
0

4. BASIC
NUMBER
SYSTEM
DECIMAL
HEXA
DECIMAL
OCTAL
BINARY
4

5. DECIMAL NUMBER SYSTEM
5
• 0,1,2,3,4,5,6,7,8,9.
DIGITS
• 10
BASE

6. 6
8*10
9*10
5*10
4*10
DECIMAL NUMBER 4598
0
1
2
3

7. 7
BINARY NUMBER SYSTEM
•0,1
DIGITS
•2
BASE

8. 8
BINARY NUMBER 1011
1*2
1*2
0*2
1*2
0
1
2
3

9. 9
HEXADECIMAL NUMBER
SYSTEM
•0,1,2,3,4,5,6,7,
•8,9,A,B,C,D,E,F.
DIGITS
•16
BASE

10. 10
HEXADECIMAL NUMBER 1A5D
D*16
5*16
A*16
1*16
0
1
2
3

11. 11
OCTAL NUMBER SYSTEM
•0,1,2,3,4,
•5,6,7,8.
DIGITS
•8
BASE

12. 12
OCTAL NUMBER 5273
3*8
7*8
2*8
5*8
0
1
2
3

13. 13
CONVERSIONS IN BASIC
NUMBER SYSTEM

14. 14
BINARY TO DECIMAL
• 1 0 1 0 1
BINARY
• 5 4 3 2 1
BIT
POSITION • 1*2 +0*2 +1*2
+0*2 +1*2.
MUL WITH
BASE
• 21
DECIMAL
4 3 2
1 0

15. 15
HEXADECIMAL TO DECIMAL
•5A9
HEX
•3 2 1
BIT POSITION
• 5*16
+A*16
+9*16
MUL WITH
BASE
•1449
DECIMAL
2
1
0

16. 16
OCTAL TO DECIMAL
•645
OCTAL
•3 2 1
BIT
POSITION
•6*8 +4*8
+5*8.
MUL WITH
BASE
•421
DECIMAL
2 1
0

17. 17
DECIMAL TO BINARY
• Q=19
• R=1
39 • Q=9
• R=1
19 • Q=4
• R=1
9 • Q=2
• R=0
4 • Q=1
• R=0
2 • Q=0
• R=1
1
LSB MSB
Divide through out by 2
DECIMAL = 39
BINARY = 100111

18. 18
DECIMAL TO HEX
• Q=2
• R=3
35 • Q=0
• R=2
2
Divide through out by 16
LSB MSB
DECIMAL = 35
HEX = 23

19. 19
DECIMAL TO OCTAL
• Q=57
• R=5
461
• Q=7
• R=1
57
• Q=0
• R=7
7
Divide through out by 8
LSB MSB
DECIMAL = 461
OCTAL = 715

20. 20
BINARY TO HEXADECIMAL
BINARY
• (010111011001)
4BITS DIV
• (0101)(1101)(1001)
HEX
• (5) (D) (9) =(5D9)
2 16

21. 21
BINARY TO OCTAL
BINARY
• (101111100)
3BIT DIV
• (101)(111)(100)
OCTAL
• (5) (7) (4)
=(574)
2
8

22. 22
HEXADECIMAL TO BINARY
HEX
• (5C)
EXPANSION
• (0101)(1100)
BINARY
• (01011100)
16 2

23. 23
OCTAL TO BINARY
OCTAL
• (436)
EXPANSION
• (100)(011)
BINARY
• (100011)
8 2

24. 24
HEXADECIMAL TO OCTAL
• (4DF)
HEX
• (0100)(1101)(1111)
EXP
• (010011010000)
BINARY
• (010)(011)(011)(111)
3BIT DIV
• (2337)
OCTAL
16
2
8

25. 25
OCTAL TO HEXADECIMAL
• (456)
OCTAL
• (100)(101)(110)
EXP
• (100101110)
BINARY
• (0001)(0010)(1110)
4BIT DIV
• (12E)
HEX
8
2
16

26. 26
REPRESENTATION OF NEGATIVE
NUMBER
9’S & 10’S COMPLIMENT
• DECIMAL NUMBER SYSTEM
1’S & 2’S COMPLIMENT
• BINARY NUMBER SYSTEM

27. 27
BINARY ARITHMETIC
1
• BINARY ADDITION
2
• BINARY SUBTRACTION
3
• BINARY MULTIPLICATION
4
• BINARY DIVISION

28. 28
0 + 0 = 0
1 + 0 = 1
0 + 1 = 1
1 + 1 = 0 1 (Carry bit)
BINARYADDITION
1 1 0 1 (13 decimal)
+0 0 0 1 (+1 decimal)
1 1 1 0 (14 decimal)

29. 29
BINARY SUBTRACTION
0 ‐ 0 = 0
1 ‐ 0 = 1
0 ‐ 1 = 1 1 (Carry bit)
1 ‐ 1 = 0
1 1 0 1 (13 decimal)
+ 0 0 1 1 (-3 decimal)
1 0 1 0 (10 decimal)
Borrow

30. 30
1 0 0 0 =810
X 0 1 1 0 =610
0 0 0 0
+ 1 0 0 0
+ 1 0 0 0
+ 0 0 0 0
0 1 1 0 0 0 0 = 4810
BINARY MULTIPLICATION

31. 31
BINARY DIVISION
011 ) 0 1 1 0 0 1 0 ( 1
0 1 1
0 0 0 (0
0 0 0
0 0 0 (0
0 0 0
0 0 1 (0
0 0 0
0 1 0
Q=1000=1610
R=10= 210

32. 32
SIGNED ARITHMETIC OPERATION
MSB bit is reserved to represent the sign of
the number.
When the number is negative, the sign bit is
kept one.
When the number is positive, the sign bit is 0.
In 8-bit processor, MSB = sign bit & other 7
bits = number.
In 16-bit processor, MSB = sign bit & other
15 bits = number.

33. 33
EXAMPLES
0 0 0 0 0 1 0 1 (+5 decimal)
0 0 0 0 0 1 0 0 (+4 decimal)
0 0 0 0 1 0 0 1 (+9 decimal)
0 0 0 0 0 1 0 1 (+5 decimal)
1 0 0 0 0 0 1 0 (-2 decimal)
1 1 1 1 1 1 0 1 (1’comp of -2)
1 1 1 1 1 1 1 0 (2’s comp of -2)
0 0 0 0 0 0 1 1 (+3 decimal)
1
2
1 2 1

34. 34
BCD AND GRAY CODE

35. 35
LOGIC GATES
Logic gates perform basic logical functions.
They are fundamental building blocks of
digital integrated circuits.
Most logic gates take an input of two binary
values, and output a single value of a 1 or 0.
 Some circuits may have only a few logic
gates, while others, such as microprocessors,
may have millions of them.
There are seven different types of logic gates,
which are outlined.

36. 36

37. 37

38. 38

39. 39

40. 40

41. 41

42. 42

43. 43
FLIP-FLOPS
A flip-flop or latch is a circuit that has two
stable states and can be used to store state
information.
Each flip-flop stores one bit of information

44. 44
THANK YOU