LECTURE 1-2. Introduction and Number Systems

Transcription

1 LECTURE 1-2 Introduction and Number Systems 1

2 BASIC INFORMATION Course Code: CSE 115 Course Title: Computing Concepts Course Teacher: Dr. Muhammad Asif H. Khan (Mfs), Associate Professor, Dept. of Computer Sci. & Eng., Dhaka University. Text Book 1. Programming in ANSI C E. Balagurusamy 2. Problem Solving and Program Design in C (7 th Edition) -- Jeri R. Hanly & Elliot B. Koffman

3 REFERENCE BOOKS Other Books You can Look into: 1. C The Complete Reference Herbert Schildt

4 COURSE WEBSITE How to use? Go to If you do not have an account with Engrade, then Join as a student and provide the Access Code. Collect the Access code from me during the next lab session. If you already have an account, just add a new course by providing the Access Code. Why to Use? All lecture notes and other deliverables will be provided through Engrade.

5 COURSE EVALUATION Topic Marks Attendance (Randomly checked) 5 Class Performance and HW 10 Quizzes (4 in total) 20 Midterm-1 15 Midterm-2 25 Final 25

6 WHAT IS A COMPUTER Computer Device capable of performing computations and making logical decisions Computer programs A set of instructions that control computer s processing of data to do a particular work Hardware Various devices comprising computer Software Application programs developed to do a specific set of tasks for the users

7 INTRODUCTION TO NUMBER SYSTEMS We are all familiar with the decimal number system (Base 10). Some other number systems that we will work with are: Binary Base 2 Octal Base 8 Hexadecimal Base 16 7

8 CHARACTERISTICS OF NUMBER SYSTEMS 1) The digits are consecutive. 2) The number of digits is equal to the size of the base. 3) Zero is always the first digit. 4) The base number is never a digit. 5) When 1 is added to the largest digit, a sum of zero and a carry of one results. 6) Numeric values determined by the implicit positional values of the digits. 8

9 SIGNIFICANT DIGITS Binary: Most significant digit Least significant digit Hexadecimal: 1D63A7A Most significant digit Least significant digit 9

10 BINARY NUMBER SYSTEM Also called the Base 2 system Two digits: 0 and 1 The binary number system is used to model the series of electrical signals computers use to represent information 0 represents the no voltage or an off state 1 represents the presence of voltage or an on state 10

11 BINARY NUMBERING SCALE Base 2 Number Base 10 Equivalent Power Positional Value

12 DECIMAL TO BINARY CONVERSION The easiest way to convert a decimal number to its binary equivalent is to use the Division Algorithm This method repeatedly divides a decimal number by 2 and records the quotient and remainder The remainder digits (a sequence of zeros and ones) form the binary equivalent in least significant to most significant digit sequence 13

13 DIVISION ALGORITHM Convert 67 to its binary equivalent: = x 2 Step 1: 67 / 2 = 33 R 1 Step 2: 33 / 2 = 16 R 1 Step 3: 16 / 2 = 8 R 0 Step 4: 8 / 2 = 4 R 0 Step 5: 4 / 2 = 2 R 0 Step 6: 2 / 2 = 1 R 0 Divide 67 by 2. Record quotient in next row Again divide by 2; record quotient in next row Repeat again Repeat again Repeat again Repeat again Step 7: 1 / 2 = 0 R 1 STOP when quotient equals

14 BINARY TO DECIMAL CONVERSION The easiest method for converting a binary number to its decimal equivalent is to use the Multiplication Algorithm Multiply the binary digits by increasing powers of two, starting from the right Then, to find the decimal number equivalent, sum those products 15

15 MULTIPLICATION ALGORITHM Convert ( ) 2 to its decimal equivalent: Binary x x x x x x x x Positional Values Products

16 OCTAL NUMBER SYSTEM Also known as the Base 8 System Uses digits 0-7 Readily converts to binary Groups of three (binary) digits can be used to represent each octal digit Also uses multiplication and division algorithms for conversion to and from base 10 17

17 DECIMAL TO OCTAL CONVERSION Convert to its octal equivalent: 427 / 8 = 53 R3 Divide by 8; R is LSD 53 / 8 = 6 R5 Divide Q by 8; R is next digit 6 / 8 = 0 R6 Repeat until Q =

18 OCTAL TO DECIMAL CONVERSION Convert to its decimal equivalent: Octal Digits Positional Values Products x x x

19 OCTAL TO BINARY CONVERSION Each octal number converts to 3 binary digits To convert to binary, just substitute code:

20 HEXADECIMAL NUMBER SYSTEM Base 16 system Uses digits 0-9 & letters A,B,C,D,E,F Groups of four bits represent each base 16 digit 21

21 DECIMAL TO HEXADECIMAL CONVERSION Convert to its hexadecimal equivalent: 830 / 16 = 51 R14 51 / 16 = 3 R3 3 / 16 = 0 R3 = E in Hex 33E 16 22

22 HEXADECIMAL TO DECIMAL CONVERSION Convert 3B4F 16 to its decimal equivalent: Hex Digits Positional Values Products 3 B 4 F x x x x ,

23 BINARY TO HEXADECIMAL CONVERSION The easiest method for converting binary to hexadecimal is to use a substitution code Each hex number converts to 4 binary digits 24

24 SUBSTITUTION CODE Convert to hex using the 4-bit substitution code : A 5 6 A E AE6A 16 25

25 SUBSTITUTION CODE Substitution code can also be used to convert binary to octal by using 3-bit groupings:

26 SUBTRACTION BY ADDITION Follow these steps: take the "complement" of the number you are subtracting (I will show you how) add it to the number you are subtracting from discard the extra "1" on the left 27

27 COMPLEMENT The "complement" is the number to add to make 10 (or 100, 1000, etc, depending on how many digits you have) Example The complement of 3 is 7, because 3+7=10 (you add 7 to make 10) Example: the complement of 85 is 15, because 85+15=100 Example: the complement of 111 is 889, because =

28 CALCULATING THE COMPLEMENT The basic idea is to find the difference between each digit and 9. That will get you to "999...", so you only need to add 1 to make it " Steps Starting at the right (the "units" position) Skip over any zeros at the start For the first digit that isn't zero: find what would make it to 10 For all other digits: find what would make it to 9 What is the complement of 1700? 29

29 SUBTRACTION BY ADDITION Follow these steps: take the "complement" of the number you are subtracting add it to the number you are subtracting from discard the extra "1" on the left 30

30 SUBTRACTION BY ADDITION Example Find Complement of 372 is 628 (verify yourself) = 1281 Discard the leading 1, and that s your answer! 31

31 WHAT IF THE NUMBER YOU ARE SUBTRACTING HAS LESS DIGITS? How would you, for example, do ? the complement of 56 is 44, but we need to "pad it" out to 4 digits, so we end up with

32 COMPLEMENT Complement is the negative equivalent of a number. If we have a number N then complement of N will give us another number which is equivalent to N So if complement of N is M, then we can say M = -N So complement of M = -M = -(-N) = N So complement of complement gives the original number 33

33 TYPES OF COMPLEMENT For a number of base r, two types of complements can be found 1. r s complement 2. (r-1) s complement Definition: If N is a number of base r having n digits then r s complement of N = r n N and (r-1) s complement of N = r n -N-1 34

34 EXAMPLE Suppose N = (3675) 10 So we can find two complements of this number. The 10 s complement and the 9 s complement. Here n = 4 10 s complement of (3675) = = s complement of (3675) = =

35 SHORT CUT WAY TO FIND (R-1) S COMPLEMENT In the previous example we see that 9 s complement of 3675 is We can get the result by subtracting each digit from 9. Similarly for other base, the (r-1) s complement can be found by subtracting each digit from r-1 (the highest digit in that system). For binary 1 s complement is even more easy. Just change 1 to 0 and 0 to 1. (Because 1-1=0 and 1-0=1) 36

36 EXAMPLE: Find the (r-1) s complement in short cut method. (620143) 8 Ans: (A4D7E) 16 Ans: 5B281 ( ) 2 Ans:

37 SHORT CUT WAY TO FIND R S COMPLEMENT From the definition we can say, r s complement of (N) = (r-1) s complement +1 So, we can first find the (r-1) s complement in short cut way then add 1 to get the r s complement. Example: r s complement of (620143) 8 = = This method is a two step process. But we can find it in one step process also. 38

38 SHORT CUT WAY TO FIND R S COMPLEMENT One step process: Start from rightmost digit to left. Initial zeros will remain unchanged Rightmost non-zero digit will be subtracted from r Rest of the digits will be subtracted from r-1 Example: Find the 10 s complement of (529400) 10 Rightmost 2 zeros will not change, 4 will be subtracted from 10 and rest of the digits 529 will be subtracted from 9 So the result is

39 Example Find the r s complement in short cut method. (8210) 10 Ans: 1790 (61352) 10 Ans: ( ) 8 Ans: (A4D7E0) 16 Ans: 5B

40 EXAMPLE FOR BINARY For binary: start from rightmost bit Up to first 1 no change. For rest of the bits toggle (Change 1 to 0 and 0 to 1) ( ) 2 Ans: ( ) 2 Ans: ( ) 2 Ans:

41 USE OF COMPLEMENT Complement is used to perform subtraction using addition Mathematically A-B = A + (-B) So we can get the result of A-B by adding complement of B with A. So A-B = A + Complement of (B) 42