Number Systems Basic Concepts
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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).
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Number Systems Basic Concepts
- 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 100 or 1, 101 or 10, 102 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 102.
- 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
- 20. REPRESENTING TEXT – ASCII TABLE
- 21. REPRESENTING TEXT – EXTENDED ASCII CODES
- 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
- 25. BINARY INTEGERS • binary numbers that do not have fractional part or without an embedded binary point. • Example: 1012 , 11102 , etc.
- 26. BINARY FRACTIONS • binary numbers with an embedded binary point • Example: 110.012 , 10110.0102 , etc.
- 27. DECIMAL TO BINARY CONVERSIONS • Convert 6310 number system to binary number system.
- 28. DECIMAL TO BINARY CONVERSIONS • Convert 6310 number system to binary number system.
- 29. BINARY TO DECIMAL CONVERSIONS OF INTEGERS • Convert 10012 to decimal number system
- 31. DECIMAL TO BINARY CONVERSIONS OF FRACTIONS • Convert the decimal fraction 0.37510 to binary fraction.
- 32. DECIMAL TO BINARY CONVERSIONS OF FRACTIONS • Convert the decimal fraction 0.37510 to binary fraction.
- 33. NON-TERMINATION CONVERSIONS OF FRACTIONS • The decimal fraction 0.810 is to be converted to its binary equivalent.
- 34. DECIMAL TO BINARY CONVERSIONS WITH INTEGRAL & FRACTIONAL PARTS • Convert the decimal number 24.62510 to its binary equivalent.
- 35. DECIMAL TO BINARY CONVERSIONS WITH INTEGRAL & FRACTIONAL PARTS • Convert the decimal number 24.62510 to its binary equivalent.
- 36. DECIMAL TO BINARY CONVERSIONS WITH INTEGRAL & FRACTIONAL PARTS • Convert the decimal number 24.62510 to its binary equivalent.
- 37. BINARY TO DECIMAL CONVERSIONS WITH INTEGRAL & FRACTIONAL PARTS • Convert the binary number 11.0112 to its decimal equivalent.
- 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
- 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
- 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
- 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
- 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 1910 to its equivalent octal number.
- 58. OCTAL TO DECIMAL CONVERSION • Convert the octal number 4858 to its equivalent decimal number.
- 59. OCTAL TO BINARY CONVERSION • Convert the octal number 7328 to its equivalent binary number.
- 60. BINARY TO OCTAL CONVERSION • Convert the binary number 101101112 to its equivalent octal number.
- 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 5910 to its equivalent hexadecimal number.
- 66. HEXADECIMAL TO DECIMAL CONVERSION • Convert the hexadecimal number AD16 to its equivalent decimal number.
- 67. HEXADECIMAL TO BINARY CONVERSION • Convert the hexadecimal number 1AC16 to its equivalent binary number.
- 68. BINARY TO HEXADECIMAL CONVERSION • Convert the binary number 100111012 to its equivalent hexadecimal number.
- 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
Editor's Notes
- it is important to know what kind of numbers can be handled most easily when using these machines but there are some number systems that are far better suited to the capabilities of digital computer
- Base is a number raised to a power 10 is the base of the decimal number system Radix is the other term for the base of the number system defined as the number of different digits which can occur in each position in the number system
- Base is a number raised to a power 10 is the base of the decimal number system Radix is the other term for the base of the number system defined as the number of different digits which can occur in each position in the number system
- Data refers to the symbols that represent people, events, things, and ideas. Data can be a name, a number, the colors in a photograph, or the notes in a musical composition Data Representation refers to the form in which data is stored, processed, and transmitted. Devices such as smartphones, iPods, and computers store data in digital formats that can be handled by electronic circuitry. Digitization is the process of converting information, such as text, numbers, photo, or music, into digital data that can be manipulated by electronic devices. The Digital Revolution has evolved through four phases, beginning with big, expensive, standalone computers, and progressing to today’s digital world in which small, inexpensive digital devices are everywhere
- Data is recorded as electronic signals or indications. The presence and absence of these signals in specific circuitry represents data in the computer just as the presence or absence of punched holes represents data on a punch card. Representing the data within the computer is accomplished by assigning a specific value to each binary component or groups or components. The values that the designer assigns to individual binary components become the code for representing data in computer.
- Power of Two and its equivalent decimal value
- To convert decimal whole numbers from base 10 to any other base, divide that number repeatedly by the value of the base to which the number is being converted. The division operation is repeated until the quotient is zero. The remainders – written in reverse of the order in which they were obtained from the equivalent numeral.
- Make a table with the power of two. Assign value as presence (1) and absence (0) with the decimal equivalent Write the conversion from left to right.
- Binary numerals can be converted to decimal by the use of Expanded Notation or Tabulation Method. When this approach is used, the position values of the original numeral are written out.
- A decimal fraction may also be converted into an equivalent binary notation. The conversion may be accomplished using several techniques. A much simpler method consists of repeatedly doubling the decimal fraction and noting the integral part of the product.
- The binary equivalent of a terminating decimal fraction does not always terminate or is not exactly converted. It will be noted that the first four steps will continuously be repeated and the same four bits will be obtained again and again. Here, the fractional part of the decimal number does not become zero after a series of multiplications. Therefore, 0.810 = 0.110011001……….2
- In each example we checked our solution by converting the binary numbers to decimal and the determining if the decimal sum was equal to the binary total. If not, then an error was made in the process.
- The solution shows that three (3) repeated subtractions were performed. Since, the equivalent of 310 in binary notation is 112, therefore, 11002 / 1002 = 112.
- Binary numbers are extremely awkward to read or handle. It requires many more positions for data than any other numbering system. To represent decimal numbers, we must use so many binary digits. Thus, in most computers, binary numbers are grouped in order to conserve storage location. The octal system overcome this problem since it is essentially a shorthand method for replacing groups of three binary digits by single octal digit. In this way, the numbers of digits required to represent any number is significantly reduced and still maintain the binary concept. Octal numbers are important in digital computers, although many computer specialists and users are not thoroughly familiar with binary, octal, and other numbering systems used by computers. Knowledge of these concepts can be very helpful in debugging programs, understanding how computer operates, and in selecting computer equipment.
- When converting from decimal to octal, divide the decimal number by the radix of octal number system and note the remainder after each division. This technique is called as Remainder Method also known as the Division – Multiplication Method. When the divide operation produces a quotient or result of zero, then the process is terminated. The remainders in reverse order, as shown by the arrow, for the octal number.
- To convert from octal to decimal, multiply each octal digit by its positional value and add the resulting products.