LESSON TITLE. Language English Local Language Introduction to Computer Science. Mr. VAR Sovannrath Submission Date October 30th, 2014 Version 1.

Transcription

1 LESSON TITLE Country Cambodia Language English Local Language Course Title Introduction to Computer Science Lesson Title 06. Number Systems SME Mr. VAR Sovannrath Submission Date October 30th, 2014 Version 1.0

2 0. Orientation > 0.2 Outline Please provide the outline of course which will A : Text-based + Audio Introduction to computer science I. General knowledge in comput er science 1. Introduction to computer 2. Computer Hardware and its function 3. Operating system & Application 4. Common application 5. Introduction to Computer Network 6. Number Systems 7. Data representation and encoding II. Basic programming knowledg e with Python 8. Introduction Algorithm and Programming 9. Data Type, Operator, and Variable 10. Control Structure 11. Structure Data Type 12. Sub-Program 13. Files

3 1. Introduction > 1.1 Introduction / Overview Please provide the introduction / overview on this lesson A : Text-based + Audio Overview In this chapter, you are going to learn about How to represent number in binary number system, octal number system and hexadecimal number system. Working with arithmetic operation of different number systems. How to convert from a number system to the other number systems.

4 1. Introduction > 1.2 Content Please make sure the hierarch of the content is well formed. Please organize the lesson in 3-5 main topics and use 3-level headings. Level 1 Level 2 Level 3 1. Number Systems 1.1. Decimal System 1.2. Binary Numbering System 1.3. Octal System 1.4. Hexadecimal System 2. Arithmetic Operations 2.1. Binary Calculation Addition 2.2. Octal Calculation 2.3. Hexadecimal Calculation 3. Radix Conversion 3.1. Conversion of Decimal System to Other System 3.2. Conversion by Binary System to Other Systems 3.3. Conversion of Octal System to Other Systems 3.4. Conversation of Hexadecimal System to other Systems Subtraction Multiplication Division

5 1. Introduction > 1.3 Content ID Will do it by looking at 1.1 Lesson overview Introduction to computer science I. General knowledge in computer science 1. Introduction to computer 2. Computer Hardware 3. Operating system & Software 4. Common application 5. Introduction to Computer Network 6. Number Systems 7. Data representation and encoding II. Basic programming knowledge with Python 8. Introduction Algorithm and Programming 9. Data Type, Operator, and Variable 10. Control Structure 11. Structure Data Type 12. Sub-Program 13. Files

6 1. Introduction > 1.4 Objectives Please provide objective of the lesson by high light keyword and follow (Audience, Behavior, Condition, Degree) to write the objective Objective Upon completion of this chapter, you will be able to Obtain knowledge on different number systems. Capable of doing arithmetic operations with different number systems. Able to convert from a number system to another number system.

7 1. Introduction > 1.5 Keywords () Please provide keywords of the lesson with explanation Keywords Description

8 1. Introduction > 1.5 Pre-Test A : Fill in the blank B : Short answer question C : Multiple Choice Feedback type A : Text-based short answer B : Text-based short answer and more information C : Video based feedback Pre-Test Question Possible answers Correct Answer Feedback of the question

9 2. Learn> Topic 1. : Number Systems The smallest peace of data recognized and used by computer is the bit, a binary digit. A bit is a single binary value 1 or 0. A grouping of eight bits is a byte. Byte is the basic unit for measuring the size of memory. 1 Kilobyte = 1024 bytes 1 Megabyte = 1024 x 1024 bytes = bytes A number of adjacent bits that can be stored and manipulated as a unit is called a computer word. The longer the lengths of the computer word that register can hold, the faster th e computer can process data. 9

10 2. Learn> Topic 1. : Number Systems In a numbering system, there are only a few symbols which represent different v alues depending on the position they occupy in the number. The total number of digits applicable to any system is called its radix/base. The actual number of symbols used in a potential system depends on its base. In any numbering system, the highest numerical symbol always has a value of o ne less than the base. Ex, base of 10, 0 to 9. 10

11 2. Learn> Topic 1.1. : Decimal System Decimal system is a base 10 system which means there are 10 distinct digits 0 to 9. The value that the digits represent depends on the weights or positions they hol d. 11

12 2. Learn> Topic 1.2. : Binary Numbering System The numbering system in which these two digits are found is binary numbering system. Binary numbering system uses a base of 2. The 0s and 1s can be arranged in various combinations to represent all the numb ers, letters, and symbols that can be entered into the computer. As binary system is base 2 system, the position weights are used on the powers of 2. 12

13 2. Learn> Topic 1.2. : Binary Numbering System Computers have been designed to use binary numbers because of the followin g reasons: 1. Computer circuits have to handle only 2 binary digits or bits rather than 10 use d in decimal numbering system. 2. Only identifies signals in the form of digital pulses which represent either high voltage or a low voltage (0). 3. Everything that can be done with a base of 10 can also be done in binary. 13

14 2. Learn> Topic 1.3. : Octal System Octal system was issued to provide a shorthand way to deal with long strings of 1s and 0 s created in binary. It is a base 8 system using the digits 0 through 7. 14

15 2. Learn> Topic 1.4. : Hexadecimal System It contains the digits 0 through 9 and the letters A through F. The letters are used because 16 placeholders are needed and there are only 10 distinct digits in the decimal system. 15

16 2. Learn> Topic 2. : Arithmetic Operations Binary Calculations Addition Subtraction Multiplication Division Octal Calculation Hexadecimal Calculation 16

17 2. Learn> Topic 2.1. : Binary Calculation Addition Rules for carrying out addition of binary numbers are as follows: = = = = 0 with 1 carry over Example 1: For adding (22 10 ) and (13 10 ): Example 2: For adding (53 10 ) and (47 10 ): 17

18 2. Learn> Topic 2.1. : Binary Calculation Subtraction Rules for subtraction of binary numbers are as follow: 0 0 = = = = 1 with one borrow Example 1: Subtraction (13 10 ) from (22 10 ): Example 2: Subtraction (53 10 ) from (47 10 ): 18

19 2. Learn> Topic 2.1. : Binary Calculation Multiplication Rules for multiplication of binary numbers are as follows: 0 x 0 = 0 0 x 1 = 0 1 x 1 = 1 Example 1: Multiplication (7 10 ) with (5 10 ): Example 2: Multiplication (22 10 ) with (13 10 ) : 19

20 2. Learn> Topic 2.1. : Binary Calculation Division Division for binary numbers can be carried out by following same rules as those ap plicable to decimal number. Example 1: Dividing by : Example 2: Dividing by : 20

21 2. Learn> Topic 2.2. : Octal Calculation To add two octal numbers, we proceed as we do in the decimal system. If any addition produces an octal number in excess of 7, we must utilize the next position: 8 10 = 10 8, 9 10 = 11 8 and so on. Addition: Example 1: (337) 8 + (228) 8 = (?) 8 Ans. (567) 8 Example 2: (72) 8 + (25) 8 = (?) 8 Ans. (117) 8 We can check the result by comparing to the decimal system: (72) 8 is equivalent to (58) 10 (25) 8 is equivalent to (21) 10 Total (117) 8 is equivalent to (79) 10. Hence the addition is correct. 21

22 2. Learn> Topic 2.2. : Octal Calculation To subtract in the octal numbering system, we may use the complementation and end-around-carry method. Subtraction: Example 1: (72) 8 -(25) 8 = (?) 8 Step 1: Complement the subtrahend: 52 ( 25 + [its complement] = 77) (any number + [its complement] = 77) Step 2: Proceed as in addition: Step 3: End-around carry Ans. (45) 8 22

23 2. Learn> Topic 2.2. : Octal Calculation Subtraction: Example 2: (72) 8 -(25) 8 = (?) 8 Step 1: Complement the subtrahend: 173 ( [its complement] = 777) Step 2: Proceed as in addition: Step 3: End-around carry Ans. (112) 8 23

24 2. Learn> Topic 2.3. : Hexadecimal Calculation Perform the operation on each column decimally, convert the decimal number to hexadecimal, and proceed. Example 1: (BAD) 16 + (627) 16 = (?) 16 Ans. (FDF) 16 Example 2: (74E) 16 + (F7E) 16 = (?) 16 (E + E) 16 = ( ) 10 = (28) 10 = (1C) 16 (Carry 1) (4 + 7) 16 + (Carry 1) = ( ) 10 = (12) 10 = C (7 + F) 16 = (7 + 15) 10 = (22) 10 = (16) 16 Ans. (16CC) 16 24

25 2. Learn> Topic 2.3. : Hexadecimal Calculation We can subtract hexadecimal numbers by again converting every digit to decimal for each position and then converting the difference obtained back to hexadecimal. Note that the system of borrowing from or exchanging with the next position results in an exchanges of 16 rather than 10. Example 3: (26) 16 - (7) 16 = (?) 16 Ans. (1F) 16 25

26 2. Learn> Topic 3. : Radix Conversion Conversion of Decimal System to Other System Conversion Methods Remainder Method Power Method Decimal to Binary Using the Remainder Method Using Power Method Decimal to Octal Decimal to Hexadecimal Conversion by Binary System to Other System Binary to Decimal Binary to Octal Binary to Hexadecimal Conversion of Octal System to Other Systems Octal to Decimal Octal to Binary Conversation of Hexadecimal System to other Systems Hexadecimal to Decimal Hexadecimal to Binary 26

27 2. Learn> Topic 3.1. : Conversion of Decimal System to Other Systems Conversion Methods Remainder Method 1) Dividing the given number of decimal system by the radix R of the proposed system. From this, we ll get a quotient Q1 and a remainder R1. 2) Dividing the quotient Q1 by the radix R again to get quotient Q2 and a remainder R2. 3) Dividing the quotient Q2 by the radix R again to get quotient Q3 and a remainder R3. This process of division of the successive quotients by the Radix R of the proposed system should be repeated until the quotients become less than the radix R. RnRn-1Rn-2 R2R1 Decimal to Binary (300) 10 = ( ) 2 27

28 2. Learn> Topic 3.1. : Conversion of Decimal System to Other Systems Conversion Methods Power Method 1) Subtracting the highest number which is obtained by raising the radix to t he power of the proposed system. R1 be the remainder. 2) Subtracting the next highest number from this remainder R1. 3) This procedure should be repeated until the remainder is zero. 4) Write the multiplication factors in the sequential order in such a way that t he powers which have been used should be multiplied by one and the ones missing should be multiplied by zero (0). Decimal to Binary 28

29 2. Learn> Topic 3.1. : Conversion of Decimal System to Other Systems Decimal to Octal ( = ) Decimal to Hexadecimal ( = 1D7F 16 ) 29

30 2. Learn> Topic 3.2. : Conversion by Binary System to Other Systems Binary to Decimal Binary to Decimal Example Converting to decimal form. Ans. ( ) 2 = (218) 10 30

31 2. Learn> Topic 3.2. : Conversion by Binary System to Other Systems Binary to Octal Binary to Octal A binary number can be converted into its octal equivalent by using two me thods: Method-I For converting a binary number to its octal equivalents, first co nvert binary number to decimal form and then convert decimal to octal for m. Example: Convert to octal form. Converting to decimal form by multiplying each binary digit with it s position weights. 31

32 2. Learn> Topic 3.2. : Conversion by Binary System to Other Systems Binary to Octal Now converting the decimal to octal form, using Remainder Method. ( ) 2 = (413) 10 = (635) 8 32

33 2. Learn> Topic 3.2. : Conversion by Binary System to Other Systems Binary to Octal Method-II We can also convert the binary number to octal number by groupin g three binary digits to produce a single octal number. Let us take the same example. The binary number can be repr esented as: ( ) 2 = (635) 8 33

34 2. Learn> Topic 3.3. : Conversion of Octal System to Other Systems Octal to Decimal For converting octal to decimal, each octal digit should be multiplied by its posi tion weights. Example: Convert to decimal form =

35 2. Learn> Topic 3.3. : Conversion of Octal System to Other Systems Octal to Binary For converting octal to binary form, it should first be converted into decimal and the resulting decimal should be converted to binary. Example: Converting to its binary form = =

36 2. Learn> Topic 3.4. : Conversation of Hexadecimal System to other Systems Hexadecimal to Decimal For converting hexadecimal number to its decimal form, each hexadecimal dig it should be multiplied by its position weights. Example: Convert 1D7F into decimal form: 1D7F 16 =

37 2. Learn> Topic 3.4. : Conversation of Hexadecimal System to other Systems Hexadecimal to Binary A hexadecimal number can be converted to its binary equivalent by using two methods. Method-I 1. Hexadecimal number, first, should be converted into its decimal equivalent and that decimal number should then be converted into binary form. Example Converting 2D4AC into decimal form. Multiplying each hexadeci mal digits with its position weights. 2D4AC 10 =

38 2. Learn> Topic 3.4. : Conversation of Hexadecimal System to other Systems Hexadecimal to Binary Now converting this decimal number into binary form by using remainder m ethods, 2D4AC 10 = =

39 2. Learn> Topic 3.4. : Conversation of Hexadecimal System to other Systems Hexadecimal to Binary Method-II This method uses the concept that a hexadecimal digit can be repre sented by four binary digits. 2D4AC16 = or which is sa me as obtained by Method-I. 39

40 4. Test Question Possible answers Correct Answer 1. Conversion of decimal number to it s binary number equivalent is: A : Fill in the blank B : Short answer question C : Multiple Choice a b c d d) Feedback type A : Text-based short answer B : Text-based short answer and more information C : Video based feedback 2. Conversion of Test decimal number to it s octal number equivalent is: a b c d c) Multiplication of by is: a b c d b) Division of by is: a b c d b) 111 2

41 5. Outro > 5.1 Summarize Please give a lesson summary. Each topic can be summarized into a sentence, diagram, or even a word. Summarize Number Systems that are used in computing are Binary Numbers, Octal Numbers and Hexadecimal Numbers. Addition, subtraction, multiplication and division of binary/octal/hexadecimal numbers is carried out in the same way as it is in decimal system. There are two conversion methods, Remainder Method and Power Method.

42 5. Outro > 5.2 References Provide references if you think the students need. Reference A. Goel, Computer Fundamentals, Pearson, 2nd Impression, 2011 D. P. Nagpal, Computer Fundamentals, S. Chand, Revised Edition 2009

43 5. Outro > 5.3 Assignment Please provide the assignment such as exercise, discussion, research topic, Short essay, case studies,. A : Text-based + Audio Assignment

44 5. Outro > 5.4 Next Lesson This is the end of the lesson. Ending message and introduction to next lesson including lesson title and topics should be given. Overview Introduce to Character representation, Integer number representation and Real number representation. How computing represent information data. Next Lesson Title Data Representation and Encoding Character representation Integer number representation Real number representation