1. Computer Number System
Lecture 3
Computer Number System

2. Number System The Binary Number System 0 (00) 1 (01) 2 (10) 3 (11)
To convert data into strings of numbers, computers use the binary number system.
Humans use the decimal system (“deci” stands for “ten”).
Elementary storage units inside computer are electronic switches. Each switch holds one of two states: on (1) or off (0).
We use a bit (binary digit), 0 or 1, to represent the state. ON
OFF
Farazul H Bhuiyan
Lecturer
Dept. of CSE
BRAC University
Bangladesh
The binary number system works the same way as the decimal system, but has only two available symbols (0 and 1) rather than ten (0, 1, 2, 3, 4, 5, 6, 7, 8, and 9).
0 (00)
1 (01)
2 (10)
3 (11)

3. Number System
Bits and Bytes
A single unit of data is called a bit, having a value of 1 or 0.
Computers work with collections of bits, grouping them to represent larger pieces of data, such as letters of the alphabet.
Eight bits make up one byte. A byte is the amount of memory needed to store one alphanumeric character.
With one byte, the computer can represent one of 256 different symbols or characters.
Farazul H Bhuiyan
Lecturer
Dept. of CSE
BRAC University
Bangladesh

4. Common Number Systems System Base Symbols Decimal 10 0, 1, … 9 Binary2
0, 1
Octal 8
0, 1, … 7
Hexa- decimal 16
0, 1, … 9,
A, B, … F

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

6. Quantities/Counting (2 of 2)
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

7. Conversion Among Bases
The possibilities:
Decimal
Octal
Binary
Hexadecimal

8. Quick Example
2510 = = 318 = 1916
Base

9. Decimal to Decimal (just for fun)
Octal
Binary
Hexadecimal
Next slide…

10. Weight
12510 => 5 x 100 = x 101 = x 102 =
Base

11. Binary to Decimal
Decimal
Octal
Binary
Hexadecimal

12. Binary to Decimal Technique
Multiply each bit by 2n, where n is the “weight” of the bit
The weight is the position of the bit, starting from 0 on the right
Add the results

13. Example
Bit “0”
=> 1 x 20 = x 21 = x 22 = x 23 = x 24 = x 25 = 32
4310

14. Octal to Decimal
Decimal
Octal
Binary
Hexadecimal

15. Octal to Decimal Technique
Multiply each bit by 8n, where n is the “weight” of the bit
The weight is the position of the bit, starting from 0 on the right
Add the results

16. Example
7248 => 4 x 80 = x 81 = x 82 =

17. Hexadecimal to Decimal
Octal
Binary
Hexadecimal

18. Hexadecimal to Decimal
Technique
Multiply each bit by 16n, where n is the “weight” of the bit
The weight is the position of the bit, starting from 0 on the right
Add the results

19. Example
ABC16 => C x 160 = 12 x 1 = B x 161 = 11 x 16 = A x 162 = 10 x 256 = 2560
274810

20. Decimal to Binary
Decimal
Octal
Binary
Hexadecimal

21. Decimal to Binary Repeated Division-by-2 Method (for whole number)
To convert a whole number to binary, use successive division by 2 until the quotient is 0. The remainders form the answer, with the first remainder as the least significant bit (LSB) and the last as the most significant bit (MSB).
(43)10 = (101011)2
Repeated Multiplication-by-2 Method (for fractions)
To convert decimal fractions to binary, repeated multiplication by 2 is used, until the fractional product is 0 (or until the desired number of decimal places). The carried digits, or carries, produce the answer, with the first carry as the MSB, and the last as the LSB.
(0.3125)10 = (.0101)2
Farazul H Bhuiyan
Lecturer
Dept. of CSE
BRAC University
Bangladesh

22. Example
12510 = ?2
12510 =

23. Octal to Binary
Decimal
Octal
Binary
Hexadecimal

24. Octal to Binary Technique
Convert each octal digit to a 3-bit equivalent binary representation

25. Example
7058 = ?2
7058 =

26. Hexadecimal to Binary
Decimal
Octal
Binary
Hexadecimal

27. Hexadecimal to Binary Technique
Convert each hexadecimal digit to a 4-bit equivalent binary representation

28. Example
10AF16 = ?2
A F
10AF16 =

29. Decimal to Octal
Decimal
Octal
Binary
Hexadecimal

30. Decimal to Octal
Technique
Divide by 8
Keep track of the remainder

31. Example
= ?8 8
19 2 8
2 3 8
0 2
= 23228

32. Decimal to Hexadecimal
Octal
Binary
Hexadecimal

33. Decimal to Hexadecimal
Technique
Divide by 16
Keep track of the remainder

34. Example
= ?16
77 2 16
= D
0 4
= 4D216

35. Binary to Octal
Decimal
Octal
Binary
Hexadecimal

36. Binary to Octal Technique Group bits in threes, starting on right
Convert to octal digits

37. Example
= ?8
= 13278

38. Binary to Hexadecimal
Decimal
Octal
Binary
Hexadecimal

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

40. Example
= ?16
B B
= 2BB16

41. Octal to Hexadecimal
Decimal
Octal
Binary
Hexadecimal

42. Octal to Hexadecimal
Technique
Use binary as an intermediary

43. Example
10768 = ?16 E
10768 = 23E16

44. Hexadecimal to Octal
Decimal
Octal
Binary
Hexadecimal

45. Hexadecimal to Octal
Technique
Use binary as an intermediary

46. Example
1F0C16 = ?8
1 F C
1F0C16 =

47. Exercise – Convert ... Decimal Binary Octal Hexa- decimal 33 1110101
703
1AF
Don’t use a calculator!
Skip answer
Answer

48. Exercise – Convert … Decimal Binary Octal Hexa- decimal 33 100001 41
Answer
Decimal
Binary
Octal
Hexa- decimal 33
100001 41 21
117
165 75
451
703
1C3
431
657
1AF

49. Thank you