Number Systems Basic Concepts

Introduction to Computing lecture presentation to analyze the number systems handled by digital computing devices to process data, convert decimal to binary, solve Binary Arithmetic, and extend understanding of other number systems (Octal and Hexadecimal).

1. NUMBER SYSTEMS FOR-IAN V. SANDOVAL

2. Analyze the number systems handled by digital computing devices to process data Convert decimal to binary Solve Binary Arithmetic Extend understanding of other number systems (Octal and Hexadecimal) Learning Objectives

3. Decimal Number System Data Representation in Digital Computing Binary Number System Contents

4. NUMBER SYSTEMS digital devices deals with numbers decimal number system for numerical calculations number system used to represents numerical data when using the computer

5. DECIMAL NUMBER SYSTEM Base 10 Number System The word “Decimal” comes or derived from the Latin word “Ten” The numerals run from 0 to 9 {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}; these numerals are called Arabic Numerals Radix is the other term for the base of the number system

6. DECIMAL NUMBER SYSTEM Power of 10 may be expressed as 10 or 1, 10 1 or 10, 10 2 or 100, etc. and this is called place value . Each digit in decimal number system is called face value Example: The digit 3 in the decimal integer 321 has a face value of 3 and place value of 10 2 .

7. DECIMAL INTEGER Decimal Integer is a string of decimal digits. Example: 1234, 2509, etc.

8. DECIMAL FRACTION Decimal Fraction is a string of decimal digits with an embedded decimal point. Example: 1234.56, 2509.325 etc. In a decimal fraction, the place values to the right of the decimal are expressed to the negative powers of 10 such as 10 -1 or 1/10 or 0.1, 10 -2 or 1/100 or 0.01, etc.

9. EXPANDED NOTATION FOR DECIMAL INTEGER Any decimal integer can be expressed as the sum of each digit times the power of ten. For example, 2509 can be expressed as

10. EXPANDED NOTATION FOR DECIMAL FRACTION Any decimal fraction may also be expressed in expanded notation. For example, 2509.325 can be expressed as

11. DATA REPRESENTATION IN DIGITAL COMPUTING Data Data Representation Digitization Digital Revolution

12. DATA REPRESENTATION IN DIGITAL COMPUTING

13. DATA REPRESENTATION IN DIGITAL COMPUTING Representing Decimal Data by Binary Components

14. DATA REPRESENTATION IN DIGITAL COMPUTING

15. DATA REPRESENTATION IN DIGITAL COMPUTING

16. REPRESENTING NUMBERS

17. REPRESENTING TEXT

18. REPRESENTING TEXT

19. REPRESENTING TEXT

20. REPRESENTING TEXT – ASCII TABLE

21. REPRESENTING TEXT – EXTENDED ASCII CODES

22. REPRESENTING TEXT

23. BINARY NUMBER SYSTEM Binary is derived from the Latin word for “Two” Two or 2 is the base for the binary number system It uses only two numerals (0 & 1); these are called as BITS. A bit is a short term for binary digits. Zero or 0 represents the absence of an assigned value One or 1 represents the presence of the assigned value

24. BINARY NUMBER SYSTEM

25. BINARY INTEGERS binary numbers that do not have fractional part or without an embedded binary point. Example: 101 2 , 1110 2 , etc.

26. BINARY FRACTIONS binary numbers with an embedded binary point Example: 110.01 2 , 10110.010 2 , etc.

27. DECIMAL TO BINARY CONVERSIONS Convert 63 10 number system to binary number system.

28. DECIMAL TO BINARY CONVERSIONS Convert 63 10 number system to binary number system.

29. BINARY TO DECIMAL CONVERSIONS OF INTEGERS Convert 1001 2 to decimal number system

30. SEAT WORK ACTIVITY

31. DECIMAL TO BINARY CONVERSIONS OF FRACTIONS Convert the decimal fraction 0.375 10 to binary fraction.

32. DECIMAL TO BINARY CONVERSIONS OF FRACTIONS Convert the decimal fraction 0.375 10 to binary fraction.

33. NON-TERMINATION CONVERSIONS OF FRACTIONS The decimal fraction 0.8 10 is to be converted to its binary equivalent.

34. DECIMAL TO BINARY CONVERSIONS WITH INTEGRAL & FRACTIONAL PARTS Convert the decimal number 24.625 10 to its binary equivalent.

35. DECIMAL TO BINARY CONVERSIONS WITH INTEGRAL & FRACTIONAL PARTS Convert the decimal number 24.625 10 to its binary equivalent.

36. DECIMAL TO BINARY CONVERSIONS WITH INTEGRAL & FRACTIONAL PARTS Convert the decimal number 24.625 10 to its binary equivalent.

37. BINARY TO DECIMAL CONVERSIONS WITH INTEGRAL & FRACTIONAL PARTS Convert the binary number 11.011 2 to its decimal equivalent.

38. SEAT WORK ACTIVITY

39. BINARY ADDITION Four possible combinations when adding these two binary numbers: 0 + 0 = 0 0 + 1 = 1 1 + 0 = 1 1 + 1 = 0 plus a carry-over of 1

40. BINARY ADDITION C

41. BINARY ADDITION C C

42. BINARY SUBTRACTION The table for binary subtraction is as follows: 0 – 0 = 0 1 – 1 = 0 1 – 0 = 1 0 – 1 = 0 with a barrow of 1

43. BINARY SUBTRACTION

44. BINARY SUBTRACTION

45. BINARY SUBTRACTION

46. BINARY MULTIPLICATION The table for binary multiplication is as follows: 0 x 0 = 0 0 x 1 = 0 1 x 0 = 0 1 x 1 = 1

47. BINARY MULTIPLICATION

48. BINARY DIVISION The table for binary division is as follows: 0 / 0 = 0 0 / 1 = 0 1 / 1 = 1 1 / 0 = cannot be

49. BINARY DIVISION

50. BINARY DIVISION

51. BINARY DIVISION

52. SEAT WORK ACTIVITY

53. BINARY DIVISION The table for binary division is as follows: 0 / 0 = 0 0 / 1 = 0 1 / 1 = 1 1 / 0 = cannot be

54. OCTAL NUMBER SYSTEM Octal is derived from the Greek word meaning “eight”. The octal number system was adapted because of the difficulty of dealing with long strings of binary 0s and 1s in converting them into decimals. The radix for the number system is 8. It uses 8 basic digits {0, 1, 2, 3, 4, 5, 6, and 7}.

55. OCTAL NUMBER SYSTEM Power of Eight and its equivalent decimal value

56. OCTAL NUMBER SYSTEM Octal Number and its equivalent Decimal number

57. DECIMAL TO OCTAL CONVERSION Convert the decimal number 19 10 to its equivalent octal number.

58. OCTAL TO DECIMAL CONVERSION Convert the octal number 485 8 to its equivalent decimal number.

59. OCTAL TO BINARY CONVERSION Convert the octal number 732 8 to its equivalent binary number.

60. BINARY TO OCTAL CONVERSION Convert the binary number 10110111 2 to its equivalent octal number.

61. SEAT WORK ACTIVITY

62. HEXADECIMAL NUMBER SYSTEM The term “hexadecimal” is derived from the combining Greek word “six” with the Latin word “ten”. It uses 10 numerals {0,1,2,3,4,5,6,7,8 & 9} and letter {A, B, C, D, E & F}. The radix of the number system is 16.

63. HEXADECIMAL NUMBER SYSTEM Hexadecimal Number and its equivalent Decimal number

64. HEXADECIMAL NUMBER SYSTEM Power of sixteen and its equivalent decimal value

65. DECIMAL TO HEXADECIMAL CONVERSION Convert the decimal number 59 10 to its equivalent hexadecimal number.

66. HEXADECIMAL TO DECIMAL CONVERSION Convert the hexadecimal number AD 16 to its equivalent decimal number.

67. HEXADECIMAL TO BINARY CONVERSION Convert the hexadecimal number 1AC 16 to its equivalent binary number.

68. BINARY TO HEXADECIMAL CONVERSION Convert the binary number 10011101 2 to its equivalent hexadecimal number.

69. SEAT WORK ACTIVITY

70. REFERENCES Byte-Notes (n.d.). Number System in Computer. Retrieved from https://byte-notes.com/number-system-computer/. Cook, D. (n.d.). Number Systems. Retrieved from https://www.robotroom.com/NumberSystems.html. GeeksforGeeks (n.d.). Number System and Base Conversion. Retrieved from https://www.geeksforgeeks.org/number-system-and-base-conversions/. Mendelson, E. (2008). Number Systems and the Foundation of Analysis. New York: Dover Publications, Inc. TutorialPoints (n.d.). Number System Conversion. Retrieved from https://www.tutorialspoint.com/computer_logical_organization/number_system_conversion.htm