2. 4589327 0101010 ABFC... Home page Introduction Binary number system Hexadecimal number system Binary coded decimal Objectives Octal number system Click Next to continue

3. Objectives Upon completion of this Topic, you will be able to :  Differentiate the 4 commonly used number systems.  Convert numbers from one system to another.  Express decimal numbers using BCD code. Number systems Objectives Introduction Binary number system Hexadecimal number system Binary coded decimal Home page Octal number system Click Next to continue

4. The decimal number system is familiar to everyone. This system uses the symbols 0,1,2,3,4,5,6,7,8,9. The decimal system is a positional system where each digit carries a certain weight based on its position. Introduction Click TO Continue Introduction Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system

5. The decimal number system is familiar to everyone. This system uses the symbols 0,1,2,3,4,5,6,7,8,9. The decimal system is a positional system where each digit carries a certain weight based on its position. Introduction 10 0 >> 1s position 10 1 >> 10s position 10 2 >> 100s position E.g. 3 4 5 Introduction Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

6. e.g : 101 10 Beside decimal number; binary (base 2), octal (base 8), and Hexadecimal (base 16) are commonly used in the digital electronics system. e.g : 101 2 e.g : 101 8 e.g : 101 16 Click to evaluate the decimal value Introduction Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system The decimal number system is also known as the base 10 system. It is referred to as base 10 because it has 10 different symbols. The base 10 system is also said to have radix of 10. Radix and base are terms that means exactly the same thing.

7. e.g : 101 10 Beside decimal number; binary (base 2), octal (base 8), and Hexadecimal (base 16) are commonly used in the digital electronics system. e.g : 101 2 e.g : 101 8 e.g : 101 16 5 10 64 10 257 10 Introduction Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click on underlined words on the left to select lesson to learn… The decimal number system is also known as the base 10 system. It is referred to as base 10 because it has 10 different symbols. The base 10 system is also said to have radix of 10. Radix and base are terms that means exactly the same thing.

8. Binary number system Objectives Upon completion of this lesson, you will be able to: Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue 1. Explain why the binary number system is ideal for digital logic applications. 2. Convert decimal whole numbers and fractional numbers into binary numbers and vice versa. 3. Explain the following terms:- Bit, Nibble, Byte, and Word.

9. Binary number system In digital electronics, we only deal with 2 possible states (High and Low, On and OFF, or Open and Closed), for this reason, almost all digital systems use the binary (base 2) number system as the system of its operations. The two states of the binary number are:- 0 0  Low  OFF  Open Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

10. In digital electronics, we only deal with 2 possible states (High and Low, On and OFF, or Open and Closed), for this reason, almost all digital systems use the binary (base 2) number system as the system of its operations. The two states of the binary number are:- 0 and 1 1  High  ON  Closed Binary number system Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Objectives Octal number system Click Next to continue Home page

11. Binary number system Binary number are strings of two (hence ‘bi’), symbols 0’s and 1’s, that represent numbers. They may be expanded in the usual way with a base of 2. E.g.1101 2 2020 2121 2 2323 Click TO Continue Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system

12. Binary number system Binary number are strings of two (hence ‘bi’), symbols 0’s and 1’s, that represent numbers. They may be expanded in the usual way with a base of 2. E.g.1101 2 2020 2121 2 2323 10481048 + 13 10 Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

13.  Each digit in the binary number system is called a bit.  A group of four bits binary number is known nibble.  A group of eight bits binary number is known as byte.  Two bytes number form a word. Binary number system Terminology There are also terms for multiple bits using the standard range of prefixes, eg., and. megabit (Mb) gigabit (Gb) kilobit (Kb) Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

14. Convert 11101 2 to decimal number Whole number Click TO Continue Binary – to – decimal conversion Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system

15. Convert 11101 2 to decimal number Whole number 11101 2 = (1)*2 4 + (1)*2 3 + (1)*2 2 + (0)*2 1 + (1)*2 0 = 1*16 + 1*8 + 1*4 + 0*2 + 1*1 = 16 + 8 + 4 + 0 +1 = 29 10 Binary – to – decimal conversion Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

16. Convert 10110 2 to decimal number Try the following Click For Solution Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system

17. Convert 10110 2 to decimal number Try the following 10110 2 = (1)*2 4 + (0)*2 3 + (1)*2 2 + (1)*2 1 + (0)*2 0 = 1*16 + 0*8 + 1*4 + 1*2 + 0*1 = 16 + 0 + 4 + 2 + 0 = 22 10 Solution Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

18. Convert 101.101 2 to decimal number Fraction number Click TO Continue Binary – to – decimal conversion Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system

19. Convert 101.101 2 to decimal number Fraction number 101.101 2 = (1)*2 2 + (0)*2 1 + (1)*2 0 + (1)*2 -1 + (0)*2 -2 + (1)*2 -3 = 1*4 + 0*2 + 1*2 + 1*0.5 + 0*0.25 +1*0.125 = 4 + 0 + 1 + 0.5 + 0 + 0.125 = 5.625 10 Binary – to – decimal conversion Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

20. Try the following Convert 110.11 2 to decimal number Click For Solution Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system

21. Try the following Convert 110.11 2 to decimal number 110.11 2 = (1)*2 2 + (1)*2 1 + (0)*2 0 + (1)*2 -1 + (1)*2 -2 = 4 + 2 + 0 + 0.5 + 0.25 = 6.75 10 Solution Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

22. Convert 25 10 to decimal number by repeated division Whole number Click TO Continue Decimal – to – binary conversion 12 2 62 32 12 02 Remainder 1 LSB MSB 2 12 25 0 1 0 1 Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system

23. Convert 25 10 to decimal number by repeated division Whole number 2 5 10 = 1 1 0 01 2 Decimal – to – binary conversion 12 2 62 32 12 02 Remainder 1 LSB MSB 2 12 25 0 1 0 1 Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

24. Try the following Convert 30 10 to binary number Click For Solution 15 2 72 32 12 02 Remainder 0 1 1 2 30 Ans = 11110 2 Solution 1 1 Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

25. Convert 0.375 10 to decimal number by repeated multiplication Fraction number 0.375 10 = 0.011 2 0.375 x 2 = 0.75 Carry 0 1 0.75 x 2 = 0.5 1 0.5 x 2 = 0 Click TO Continue Decimal – to – binary conversion Introduction Binary number system Binary number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

26. Try the following Convert 13.125 10 to binary number Click For Solution 6 2 32 12 02 Remainder 1 1 2 13 Ans = 1101.001 2 0.125 x 2 = 0.25 Carry 0 0 0.25 x 2 = 0.5 10.5 x 2 = 0 Whole number Fraction number Solution 0 1 Introduction Binary number system Binary number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives Click Next to continue

27. Summary Upon completion of this lesson, you should to answer the following questions :  What is the weight of the 1s in the binary number 000 ?  What is the binary equivalent of decimal number ?  What is the decimal equivalent of binary number ? 1 1 37 101.01 Introduction Binary number system Binary number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives http://courses.cs.vt.edu/~cs1104/Bingo/bingo.kevin.html Click Next to continue

28. Objectives Upon completion of this lesson, you will be able to :  Convert decimal whole numbers into octal numbers and vice versa.  Convert binary numbers into octal numbers and vice versa. Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

29. Octal number are base 8 number. It uses symbols 0, 1, 2, 3, 4, 5, 6, 7. E.g. 3721 8 Click TO Continue Octal number system Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives

30. Octal number are base 8 number. It uses symbols 0, 1, 2, 3, 4, 5, 6, 7. E.g. 3721 8 8080 8181 8282 8383 1 2001 10 Octal number system + 16 448 1536 Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives Click Next to continue

31. Convert 326 8 to decimal number Whole number Click TO Continue Octal – to – decimal conversion Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives

32. Convert 326 8 to decimal number Whole number 326 8 = (3)*8 2 + (2)*8 1 + (6)*8 0 = 3*64 + 2*8 + 6*1 = 192 + 16 + 6 = 214 10 Octal – to – decimal conversion Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives Click Next to continue

33. Convert 277 8 to decimal number Try the following Click For Solution Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives

34. Convert 277 8 to decimal number Try the following 277 8 = (2)*8 2 + (7)*8 1 + (7)*8 0 = 128 + 56 + 7 = 191 10 Solution Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives Click Next to continue

35. Convert 330 10 to decimal number by repeated division 41 8 58 08 Remainder 2121 5 LSD MSD 8 330 Click TO Continue Decimal – to – octal conversion Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives

36. Convert 330 10 to decimal number by repeated division 330 10 = 512 8 41 8 58 08 Remainder 2121 5 LSD MSD 8 330 Decimal – to – octal conversion Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives Click Next to continue

37. Try the following Convert 230 10 to octal number Click For Solution Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives

38. Try the following Convert 230 10 to octal number 28 8 38 0 Remainder 6464 3 8 230 Ans = 346 8 Solution Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives Click Next to continue

39. Convert 10111010 2 to octal (base 8) number 8 = 2 3 || where n =3 Binary can easily be converted into other based that are powers of 2, and each group of n bits converts directly into a symbol for another representation base -- 2 n Click TO Continue Binary – to – Octal conversion Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives

40. Convert 10111010 2 to octal (base 8) number 8 = 2 3 || where n =3 Binary can easily be converted into other based that are powers of 2, and each group of n bits converts directly into a symbol for another representation base -- 2 n 10 111 010 2 2 7 2 Binary – to – Octal conversion Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives Click Next to continue

41. Convert 111101011 2 to octal number Try the following Click For Solution Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives

42. Convert 111101011 2 to octal number Try the following 111 101 011 2 7 5 3 Solution Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives Click Next to continue

43. Summary Upon completion of this lesson, you should to answer the following questions :  What is the weight of the 3 in the octal number 2 70 ?  What is the octal equivalent of decimal number ?  What is the octal equivalent of binary number ? 3 17 10111 Introduction Binary number system Octal number system Octal number system Hexadecimal number system Binary coded decimal Home page Objectives Click on underlined words on the left to select lesson to learn…

44. Objectives Upon completion of this lesson, you will be able to : Introduction Binary number system Octal number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Hexadecimal number system Click Next to continue 1.Convert decimal whole numbers into hexadecimal numbers and vice versa. 2. Convert binary numbers into hexadecimal numbers and vice versa.

45. Hexadecimal number system Hexadecimal number are base 16 number. It uses symbols 0, 1, 2, 3, 4, 5, 6, 7,8.9,A,B,C,D,E,F. The Hex digit A to F are equivalent to decimal values 10 to 15 Click TO Continue Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system

46. Hexadecimal number system Hexadecimal number are base 16 number. It uses symbols 0, 1, 2, 3, 4, 5, 6, 7,8.9,A,B,C,D,E,F. A 16  10 10 B 16  11 10 C 16  12 10 D 16  13 10 E 16  14 10 F 16  15 10 The Hex digit A to F are equivalent to decimal values 10 to 15 Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

47. Hexadecimal number system Hexadecimal number are base 16 number. It uses symbols 0, 1, 2, 3, 4, 5, 6, 7,8.9,A,B,C,D,E,F. E.g.4AF2 16 16 0 16 1 16 2 16 3 Click TO Continue Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system

48. Hexadecimal number system Hexadecimal number are base 16 number. It uses symbols 0, 1, 2, 3, 4, 5, 6, 7,8.9,A,B,C,D,E,F. E.g.4AF2 16 16 0 16 1 16 2 16 3 2 240 2560 16384 + 19186 10 Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

49. Hexadecimal – to – decimal conversion Convert 2FE 16 to decimal number Whole number Click TO Continue Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system

50. Hexadecimal – to – decimal conversion Convert 2FE 16 to decimal number Whole number 2FE 16 = (2)*16 2 + (F)*16 1 + (E)*16 0 = 2*256 + 15*16 + 14*1 = 512 + 240 + 14 = 766 10 Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

51. Convert 2A6 16 to decimal number Try the following Click For Solution Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system

52. Convert 2A6 16 to decimal number Try the following 2A6 16 = (2)*16 2 + (10)*16 1 + (6)*16 0 = 512 + 160 + 6 = 678 10 Solution Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

53. Decimal – to – Hexadecimal conversion Convert 498 10 to decimal number by repeated division 31 16 1 0 Remainder 2F2F 1 LSD MSD 16 498 Click TO Continue Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system

54. Decimal – to – Hexadecimal conversion Convert 498 10 to decimal number by repeated division 498 10 = 1F2 16 31 16 1 0 Remainder 2F2F 1 LSD MSD 16 498 Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

55. Try the following Convert 255 10 to Hex number Click For Solution Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system

56. Try the following Convert 255 10 to Hex number 15 16 0 FFFF 255 Ans = FF 16 Solution Remainder Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

57. Binary – to – Hexadecimal conversion Convert 101110101010 2 to hexadecimal number Binary can easily converts into Hexadecimal symbol by forming the binary number into group of 4 bits, and each group is then directly into a Hexadecimal number. Click TO Continue Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system

58. Binary – to – Hexadecimal conversion Convert 101110101010 2 to hexadecimal number Binary can easily converts into Hexadecimal symbol by forming the binary number into group of 4 bits, and each group is then directly into a Hexadecimal number. 1011 1010 1010 2 B A A Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

59. Convert 111101011 2 to Hexadecimal number Try the following Click For Solution Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system

60. Convert 111101011 2 to Hexadecimal number Try the following 0001 1110 1011 2 1 E B Solution Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Octal number system Click Next to continue

61. Summary Upon completion of this lesson, you should to answer the following questions :  What is the weight of the F in the Hexadecimal number 2 1?  What is the Hexadecimal equivalent of decimal number ?  What is the hexadecimal equivalent of binary number ? F 63 11110 Introduction Binary number system Hexadecimal number system Hexadecimal number system Binary coded decimal Home page Objectives Click on underlined words on the left to select lesson to learn… Octal number system

62. Objectives Upon completion of this lesson, you will be able to :  Convert decimal numbers into BCD code and vice versa. Introduction Binary number system Hexadecimal number system Binary coded decimal Binary coded decimal Home page Objectives Binary-Coded- Decimal ( BCD ) Octal number system Click Next to continue

63. Binary-Coded-Decimal ( BCD )  BCD is a way to each of the decimal digits is with a binary code ( code).  The BCD code uses only 4 bit binary numbers from 0000 to 1001.  Since a decimal digit can only be as large as 9, binary numbers from 1010 to 1111 are not used in the BCD code. 8421 Introduction Binary number system Hexadecimal number system Binary coded decimal Binary coded decimal Home page Objectives Octal number system Click Next to continue

64. Decimal – to – BCD conversion Convert 25 10 to BCD Click TO Continue Introduction Binary number system Hexadecimal number system Binary coded decimal Binary coded decimal Home page Objectives Octal number system

65. Decimal – to – BCD conversion Convert 25 10 to BCD 2 0010 5 0101 25 10 = 00100101 BCD Each decimal digit is represented by its binary equivalent Introduction Binary number system Hexadecimal number system Binary coded decimal Binary coded decimal Home page Objectives Octal number system Click Next to continue

66. Try the following Convert 91 10 to BCD Click For Solution Introduction Binary number system Hexadecimal number system Binary coded decimal Binary coded decimal Home page Objectives Octal number system

67. Try the following Convert 91 10 to BCD 1 = 0001 9 = 1001 Solution 91 10 = 10010001 BCD Introduction Binary number system Hexadecimal number system Binary coded decimal Binary coded decimal Home page Objectives Octal number system Click Next to continue

68. Convert 11010101111001 BCD to decimal number Try the following Click For Solution Introduction Binary number system Hexadecimal number system Binary coded decimal Binary coded decimal Home page Objectives Octal number system

69. Convert 11010101111001 BCD to decimal number Try the following 11 0101 0111 1001 BCD 3 5 7 9 Solution 11010101111001 BCD = 3579 10 Introduction Binary number system Hexadecimal number system Binary coded decimal Binary coded decimal Home page Objectives Octal number system Click Next to continue

70. Summary Upon completion of this lesson, you should to answer the following questions :  What decimal number is represented by each BCD code?  Convert the decimal number ? 10000001 BCD 911 10 Introduction Binary number system Hexadecimal number system Binary coded decimal Binary coded decimal Home page Objectives Click on underlined words on the left to select lesson to learn… Octal number system

71. The designation of 8421 indicates the binary weights of the four bit ( 2 3 = 8, 2 2 = 4, 2 1 = 2, 2 0 = 1 ) Binary coded decimal means that each decimal digit, 0 through 9, is represented y a binary code of four bits.The remaining six code combinations from 1010 to 1111 are invalided in the BCD code. 8 4 2 1 0000 = 0 8421 Code Decimal 0001 = 1 0010 = 2 0011 = 3 0100 = 4 0101 = 5 0110 = 6 0111 = 7 1000 = 8 1001 = 9 Introduction Binary number system Hexadecimal number system Home page Objectives Octal number system Binary coded decimal Binary coded decimal Click Closed to return to previous page

72. kilobit (Kb) A common convention is to use K for 1000, However, for binary system, K is referring to 10 bits binary number. i.e. 2 10 = 1024 bit. Introduction Hexadecimal number system Home page Objectives Binary coded decimal Octal number system Binary number system Binary number system Click Closed to return to previous page

73. megabit (Mb) A common convention is to use M for 1,000,000 However, for binary system, M is referring to 20 bits binary number. i.e. 2 20 = 1,048,576 bit. Introduction Hexadecimal number system Home page Objectives Binary coded decimal Octal number system Binary number system Binary number system Click Closed to return to previous page

74. gigabit (Gb) A common convention is to use G for 1,000,000,000 However, for binary system, G is referring to 30 bits binary number. i.e. 2 30 = 1,073,741,824 bit. Introduction Hexadecimal number system Home page Objectives Binary coded decimal Octal number system Binary number system Binary number system Click Closed to return to previous page