1. Writer:-Rashedul Hasan. Editor:- Jasim Uddin
Number system
Writer:-Rashedul Hasan.
Editor:- Jasim Uddin

2. Commonly used Number System
Base
Number system
Digits/symbol used 2
Binary
0,1 8
Octal
0,1,2,3,4,5,6,7 10
Decimal
0,1,2,3,4,5,6,7,8,9 16
Hexadecimal
0,1,2,3,4,5,6,7,8,9,
A,B,C,D,E,F

3. Decimal Number System The decimal numeral system has ten as its base.
It is the most widely used numeral system, perhaps because humans have ten digits over both hands.
uses various symbols (called digits) for no more than ten distinct values (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9) to represent any numbers,

4. Decimal Number System
The decimal system is a positional numeral system; it has positions for units, tens, hundreds, etc.
The position of each digit conveys the multiplier (a power of ten) to be used with that digit—each position has a value ten times that of the position to its right.

5. For example 848 8 Hundreds or 8*100 or 8*102 4 tens or 4*10 or 4*101
8 units or 8*1 or 8*100

6. Form the example, Both the 8 are not equal.
Left most 8 occupies the hundred or 102 position is called MSD that is Most Significant digit.
Right most 8 occupies the units or 100 position is called LSD that is Least Significant digit.
The total value=8* *101 +8*100 =
848.

7. Another example 1492.76 1 thousand or 1*1000 or 1* 103
4 Hundreds or 4*100 or 4*102
9 tens or 9*10 or 9*101
2 units or 2*1 or 2*100
7 tenths or 7*0.1 or 7*10-1
6 hundredths or 6*0.01 or 6*10-2

8. Total value = 1* 103 + 4*102 + 9*101 + 2*100 + 7*10-1 + 6*10-2= =

9. Binary Number System
The binary numeral system, or base-2 number system, is a numeral system that represents numeric values using two symbols, 0 and 1.
The digits in binary system are called bits.
In binary number system, the value of each digit is based on 2, and powers of 2.

10. Since binary is a base-2 system, each digit represents an increasing power of 2, with the rightmost digit representing 20, the next representing 21, then 22, and so on.
In binary system, if the bit is zero (0), its value is zero.

11. Bit: O or 1
Byte: a group of 8 bits is called Byte.
Word: a group of 16 bits is called Word.
Kilobytes KB 210
Megabyte MB 220
Gigabyte GB 230
Terabyte TB 240

12. Binary to Decimal
If the digit is one (1), its value is determined by its position from the right.
For example, the binary number is converted to decimal form by,
[(1) × 25] + [(0) × 24] + [(0) × 23] + [(1) × 22] + [(0) × 21] + [(1) × 20] =
[1 × 32] + [0 × 16] + [0 × 8] + [1 × 4] + [0 × 2] + [1 × 1] = 37

13. Decimal to Binary
Divide the number by 2, the remainder is either 0 or 1.
Place the remainder to the right of partial quotient obtained in step 1.
Divide the partial quotient of step 1 by 2, placing the remainder to the right of new partial quotient
Repeat the step 1,2,3 until a quotient of zero is obtained.
The binary number is equal to the remainders arranged so that first remainder is the LSB and the last remainder is MSB of binary number.

14. Operation Remainder
118 ÷ 2 =
59 ÷ 2 =
29 ÷ 2 =
14 ÷ 2 = 7 0
7 ÷ 2 = 3 1
3 ÷ 2 = 1 1
1 ÷ 2 = 0 1
Reading the sequence of remainders from the bottom up gives the binary numeral

15. Decimal to Binary 47 Dividers number remainders 2 47 2 23 1 (LSB)
(LSB)
(MSB)
4710 =

16. Decimal to Binary 76 Dividers number remainders 2 76 2 38 0 (LSD)
(LSD)
0 1(MSD)
7610 =

17. Incase of Fraction 0.625 Multiplier decimal fraction 2 * 0.625 1.25 1
2 *
2 *
2 *

18. Incase of Fraction 0.86 Multiplier decimal fraction 2 * 0.86 1.72 1
2 *
2 *
2 *
2 *

19. 87.125 87 [Integral part] Dividers number remainders 2 87 2 43 1 (LSD)
(LSD)
0 1(MSD)
8710 =

20. 87.125 .125 [Fraction part] Multiplier decimal fraction
2 *
2 *
2 *

21. Binary to Decimal 10101 1 0 1 0 1 20*1 = 1 21*0 = 0 22*1 = 4 23*0 = 0
20*1 = 1
21*0 = 0
22*1 = 4
23*0 = 0
24*1 = 16 21
= 21

22. In case of fraction 0.1011 0. 1 0 1 1 2-1*1 = 0.5 2-2*0 = 0
2-1*1 = 0.5
2-2*0 = 0
2-3*1 =0.125
2-4*1 =
0.6875 =

23. Convert binary to Decimal

24. Octal Number system
The octal number system has a base of eight. And they are,
0,1,2,3,4,5,6 and 7.
The digit position of an octal number can have only value for 0 to 7. the digit positions in an octal number have weights as follows,

25. Decimal to Octal conversion
Divide the Decimal number by 8
Place reminder to the right of partial quotient obtained in step 1.
Divide the partial quotient of step 1 by 8, placing the remainder to the right of new partial quotient
Repeat the step 1,2,3 until a quotient of zero is obtained.
The binary number is equal to the remainders arranged so that first remainder is the LSD and the last remainder is MSD of octal number.

26. Decimal to Octal conversion
573
Dividers number remainders
(LSD)
(MSD)
= 1075

27. Decimal to Octal conversion
2536
Dividers number remainders
(LSD)
(MSD)
= 4750

28. Octal to Decimal
The extreme right hand digit is multiplied by 80 the second from the right by 81 and So on.
Then add all this products to get decimal equivalent of the octal number.
In case of octal fraction, multiply the first digit after octal point by 8-1, second digit from octal point by 8-2

29. Octal to Decimal 1075 1 0 7 5 80 * 5 = 5 81 * 7 = 56 82 * 0 = 0
80 * 5 = 5
81 * 7 = 56
82 * 0 = 0
83* 1 = 512
573

30. Octal to Decimal
0.44
8-1 * 4 = 0.5
8-2 * 4 =
0.5625

31. Octal to Decimal Assignment
4750
0.6256

32. Octal to Binary
576
111
101
110
576 =

33. Octal to Binary In case of Fraction 0.216 0. 2 1 6 010 001 110
0.216 =

34. Octal to Binary
27.12

35. Binary to Octal Group the binary bits in three.
For grouping the bits in three, move towards left from binary point.
In case of even number, add zero or zeros at appropriate place.
Replace each group of threes bits by equivalent octal numbers.

36. 6 5 7

37. 3 6

38. Assignment
Ans. 170

39. Hexadecimal The base is 16, it has 16 possible digit symbol.
0,1,2,3,4,5,6,7,8,9,A,B,C,D,E and F
The digit positions in an octal numbers have weights as follows,
Each hexadecimal digit represent group of four (4) binary digits.
hexadecimal digit A-F are equivalent to decimal values 10 – 15.

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

41. Decimal to Hexadecimal
Divide the number by 16
Place reminder to the right of partial quotient obtained in step 1.
Divide the partial quotient of step 1 by 16, placing the remainder to the right of new partial quotient
Repeat the step 1,2,3 until a quotient of zero is obtained.
The binary number is equal to the remainders arranged so that first remainder is the LSD and the last remainder is MSD of octal number.

42. Decimal to Hexadecimal
741
Dividers number remainders
(LSD)
i.e E
0 2(MSD)
= 2E5

43. Decimal to Hexadecimal
2536

44. Decimal to Hexadecimal
In case of Fraction
0.256
16 *
16 *
16 *
16 *

45. Decimal to Hexadecimal
0.3942

46. Decimal to Hexadecimal
97.236
Integral Part
Dividers number remainders
(LSD)
(MSD)

47. Decimal to Hexadecimal
Fraction Part
Multiplier fraction & Partial
product
16 *
16 * /C
16 *
16 * /A

48. Hexadecimal to Decimal
1F95
1 F 9 5
160 * 5 = 5
161 * 9 = 144
162 * F = 3840
163* 1 = 4096
8085

49. Hexadecimal to Decimal

50. Hexadecimal to Decimal
0.48
16-1 * 4 = 0.25
16-2 * 8 =

51. Assignment
0.D2F
61.3C6A

52. Hexadecimal to Binary
59C C

53. Hexadecimal to Binary
0.2D6
D 6

54. Hexadecimal to Binary
ABCD
A B C D

55. Hexadecimal to Binary
D E

56. Hexadecimal to Binary