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1. 1. CHAPTER 1: NUMBER AND CODE SYSTEMS EE202 DIGITAL ELECTRONICS BY: SITI SABARIAH SALIHIN sabariah@psa.edu.my www.sabariahsalihin.com
2. 2. CHAPTER 1: NUMBER AND CODE SYSTEMS Programme Learning Outcomes, PLO Upon completion of the programme, graduates should be able to: � PLO 1 : Apply knowledge of mathematics, scince and engineering fundamentals to well defined electrical and electronic engineering procedures and practices Course Learning Outcomes, CLO � CLO 1 : Illustrate the knowledge of Digital Number Systems, codes and logic operations correctly. EE202 DIGITAL ELECTRONICS
3. 3. CHAPTER 1: NUMBER AND CODE SYSTEMS Upon completion of this Chapter, students should be able to: � Know the following number system:decimal, binary, octal and Hexadecimal and convert the numbers from ne system to another. � Understand Octal Number System. � Understande Hexadecimal Number System. � Understand how binary codes are used in computers representing decimal digits, alphanumeric characters and symbols. EE202 DIGITAL ELECTRONICS
4. 4. 1.1INTRODUCTION � Many number system are in use in digital technology. The Most common are the decimal, binary, octal and hexadecimal system. � The Binary number system is the most important one in digital systems. � The Decimal System is the most familiar because it is universally used to represent quantities outside a digital system and it is a tool that we use every day. EE202 DIGITAL ELECTRONICS
5. 5. 1.1 INTRODUCTION � This means that there will be situations where decimal values have to be converted to binary values before they are entered into the digital system. � For example, when you punch a decimal number into your hand calculator(or computer), the circuitry inside the device converts the decimal number to a binary value. EE202 DIGITAL ELECTRONICS
6. 6. 1.1 INTRODUCTION � Likewise, there will be situations where the binary values at the outputs of a digital circuit have to be converted to decimal values for presentation to the outside world. � For example, your calculator (or computer) uses binary numbers to calculate answers to a prolem, then converts the answers to a decimal value before displaying them. � The octal (base-8) and hexadecimal (based-16) number systems are both used for the same purpose-to provide an efficient means for representing large binary numbers. EE202 DIGITAL ELECTRONICS
7. 7. 1.1 INTRODUCTION � An understanding of the system operation requires the ability to convert from one number system to another. � This chapter will show you how to perform these conversions . � This chapter will also introduce some of the binary codes that are used to represent various kinds of information. � These binary codes will used 1’s and 0’s . EE202 DIGITAL ELECTRONICS
8. 8. 1.1.1 Decimal Number System � The Decimal System is composed of 10 numerals or symbols. � These 10 symbols are 0,1,2,3,4,5,6,7,8,9 : using these symbols as digits of a number, we can express any quantity. � The decimal system, also called the base-10 system because it has 10 digits, has evolved naturally as a result of the fact that man has 10 fingers. � In fact, the word “digit” is derived from the Latin word for “Finger”. EE202 DIGITAL ELECTRONICS
9. 9. 1.1.1 Decimal Number System � The decimal system is a positional-value system in � � � � which the value of a digit depends on its position. For example: consider the decimal number 453. Digit 4 actually represents 4 hundreds, the 5 represent 5 tens, and the 3 represent 3 units. In essence, the 4 carries the most weight of the three digits; it is referred to as the most significant digit (MSD). The 3 carries the least weight and is called the least significant digit (LSD). EE202 DIGITAL ELECTRONICS
10. 10. 1.1.1 Decimal Number System � Consider another example, 27. 35 � This number is actually equal to 2 tens plus 7 units plus 3 tenth plus 5 hndredths or 2 x 10 + 7 x 1 + 3 x 0.1 + 5 x 0.01 � The decimal point is used to separate the integer and fractional parts of the number. EE202 DIGITAL ELECTRONICS
11. 11. 1.1.1 Decimal Number System � Moreover, the various positions relative to the decimal point carry weights that can be expressed as powers of 10. Figure 1 : Decimal position values as power of 10 � Example : 2745.214 10 10 10 10 10 10 10 3 2 2 7 1 4 MSD 0 5 -1 . -2 2 1 2 1 0 4 LSD Decimal Point 3 -3 -1 -2 -3 = (2x 10 )+(7x 10 )+(4x 10 )+(5x 10 )+(2x 10 )+(1x 10 )+(4x 10 ) EE202 DIGITAL ELECTRONICS
12. 12. 1.1.1 Decimal Number System � In general any number is simply the sum of products of each digit value and its positional value. � Unfortunately, Decimal number system does not lend itself to convinient implementation in digital system. � It is very difficult to design electronics equipment so that it can work with 10 different voltage levels (0-9). � For this reason, almost every digital system uses the binary number system base-2 as the basic number to design electronics circuits that operate with only two voltage levels. EE202 DIGITAL ELECTRONICS
13. 13. 1.1.2 Binary Number System � In binary system, there are only two symbols or possible digit values, 0 and 1. � Even so, this base-2 system can be used to represent any quantity that can be represented in decimal or other number systems. � All the statements made ealier concering the decimal system are equally applicable to the binary system. � The binary system is also a positional-value system, where in each binary digit has its own value or weight expressed as a power of 2. EE202 DIGITAL ELECTRONICS
14. 14. 1.1.2 Binary Number System � Example: Figure 2 : Binary position values as power of 2 1 2 2 2 1 0 1 3 MSB 2 -1 2 1 -2 -3 2 0 . 2 2 1 0 1 Binary Point MSB • Here,places to the left of the binary point (counterpart of the decimal point) are positive power of 2 and places to the right are negative power of 2. � Exercise : Find the equivalent in the decimal system for the number 1011.1012 Answer : 11.62510 (How?) EE202 DIGITAL ELECTRONICS
15. 15. 1.1.2 Binary Number System � In the binary system, the term binary digit is often abbreviated to the term bit, which we will use henceforth. EE202 DIGITAL ELECTRONICS
16. 16. 1.1.2 Binary Number System Figure 3 : Binary Counting Sequence Weights 3 2 =8 2 =4 LSB 2 2 =2 1 2 =1 Decimal equivalent 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 1 0 1 2 3 0 0 0 0 1 1 1 1 0 0 1 1 0 1 0 1 4 5 6 7 1 1 1 1 0 0 0 0 0 0 1 1 0 1 0 1 8 9 10 11 1 1 1 1 1 1 1 EE202 1 0 1 0 1 12 13 14 15 0 0 1 SEMICONDUCTOR DEVICES 1
17. 17. 1.1.2 Binary Number System � Example: What is the largest number that can be represented using 8 bits? � Solution : 2 N 8 - 1 =2 = 255 10 = 111111112 EE202 DIGITAL ELECTRONICS
18. 18. 1.1.2 Binary Number System � Review Questions: 1. What is the decimal equivalent of 11010112? 2. What is the next binary number following 101112 in the counting sequence? 3. What is the largest decimal value that can be represented using 12 bits? EE202 DIGITAL ELECTRONICS
19. 19. 1.1.3 Binary to Decimal Conversion � A Binary number can be converted to decimal by multiplying the weight of each position with the binary digit and adding together. � Example : Convert the Binary number 101102 to its Decimal equivalent. � Solution: Binary number 1 0 1 1 02 4 3 2 1 0 2 +2 +2 +2 +2 4 3 2 1 0 (2 x 1)+(2 x 0)+(2 x 1)+(2 x 1)+(2 x 0 ) = 16 + 0 + 4 + 2 + 0 = 2210 EE202 DIGITAL ELECTRONICS
20. 20. 1.1.3 Binary to Decimal Conversion � Example : Convert the Fractional Binary Number 101.102 to its Decimal equivalent. � Solution: Binary Number = 1 0 1 . 1 0 2 1 0 -1 -2 Power of 2 position = 2 2 2 . 2 2 = (22 x 1)+(21 x 0)+(20 x 1) . (2-1 x 1)+(2-2 x 0 ) Decimal Value = 4 + 0 + 1 = 5.5 10 EE202 DIGITAL ELECTRONICS . 0.5 + 0
21. 21. 1.1.3 Binary to Decimal Conversion EE202 DIGITAL ELECTRONICS
22. 22. 1.1.3 Decimal To Binary conversion � The most convenient method is called division by 2 method. � In which first decimal number will be divided by 2. � The quotient will be dividend for the next step. In each step the remainder part will be recorded separately. � The 1st reminder of the 1st division will be the LSB in the Binary Number. � The quotient should repeatedly divide by 2 until the quotient becomes 0. � The final remainder will be the MSB in Binary number. EE202 DIGITAL ELECTRONICS
23. 23. 1.1.3 Decimal To Binary conversion � Example : Convert Decimal 2010 to its Binary equivalent. Solution: 2 20 remainder of 0 2 10 remainder of 0 2 5 remainder of 1 2 2 remainder of 0 2 1 remainder 0f 1 0 1 0 1 0 02 EE202 DIGITAL ELECTRONICS
24. 24. 1.1.3 Decimal To Binary conversion � When converting a decimal fractional number to its binary, the decimal fractional part will be multiply by 2 till the fractional part gets 0 or till the number of decimal places reached. EE202 DIGITAL ELECTRONICS
25. 25. 1.1.3 Decimal To Binary conversion � Below example show the steps to convert decimal fraction 0.625 to its binary equivalent. Step 1 : 0.625 will be multiply by 2 ( 0.625 x 2 = 1.25) Step 2 : The integer part will be the MSB in the binary result Step 3 : The fractional part of the earlier result will be multiply again ( 0.25 x 2 = 0.5 ) Step 4 : Each time after the multiplication the integer part of the result will be written as the Binary number. Step 5 : The procedure should continue till the fractional part gets 0. EE202 DIGITAL ELECTRONICS
26. 26. 1.1.3 Decimal To Binary conversion � Example : Convert Decimal 0.62510 to its binary equivalent. Solution: 0.25 x 2 = 0.50 0. 5 x 2 = 1. 00 carried MSB = 0.62510 = . 1012 EE202 DIGITAL ELECTRONICS LSB 1 0 1
27. 27. 1.1.4 Binary Addition � Adding of two binary numbers follows same as addition of two decimal numbers. � Some times binary addition is very much easier then Decimal or any other number system addition, because in binary you deal with only 2 numbers. � There are mainly 4 rules should be followed in the process of addition in binary numbers: sum carryout Rule 1 : 0 + 0 = 0 0 Rule 2 : 0 + 1 = 1 1 Rule 3 : 1 + 0 = 1 1 Rule 4 : 1 + 1 = 0 1 EE202 DIGITAL ELECTRONICS
28. 28. 1.1.4 Binary Addition Example: Perform Binary Addition for 1012 + 0102 Solution: 1 0 12 0 1 02 + 1 1 1 Exercise: Perform Binary Addition for 10112 + 01112 Answer: 10 01 02 EE202 DIGITAL ELECTRONICS
29. 29. 1.1.5 Binary Subtraction � When subtracting one binary number A (subtrahend) from another binary number B (Minuend) where B > A, the answer is called the difference. � There are four basic rules that should be followed in binary subtracting To perform Rule 2 you have to borrow 1 from the next left column. � The weight of the binary you borrow will be 2. EE202 DIGITAL ELECTRONICS
30. 30. 1.1.5 Binary Subtraction � � Rule 1 � Rule 2 � Rule 3 � Rule 4 Minuend (B) Subtrahend (A) Difference 0 0 - 0 1 = = 1 1 - 0 1 = = Example: Subtract binary 1012 from 1102 0 1 1 1 0 0 0 0 1 1 0 0 1 1 = 0012 EE202 DIGITAL ELECTRONICS Borrow out with a borrow of 1
31. 31. 1.1.8 Signed Binary Numbers � A signed number consist both positive and negative � � � � sign with magnitude. The additional bit for representing the sign of the number (+ or -) is known as sign bit. in general, 0 in the sign bit represents a positive number and 1 in the sign bit represents a negative number. The leftmost bit 0 is the sign bit represent + The leftmost bit 1 is the sign bit represent - EE202 DIGITAL ELECTRONICS
32. 32. 1.1.8 Signed Binary Numbers � Therefore the stored number in register A and B is 13 and -13 respectively in Decimal form. � The signed bit is used to indicate the positive or negative nature of the stored binary numbers. � Here the magnitude bits are the binary equivalent of the decimal value being represented. � This is called the sign magnitude system. EE202 DIGITAL ELECTRONICS
33. 33. 1.1.8 Signed Binary Numbers � Sign bit (+) A4 A3 0 A2 A1 1 1 0 A0 1 = +13 B4 1 B3 1 B2 B1 1 B0 0 1 = - 13 Representing Signed Number EE202 DIGITAL ELECTRONICS
34. 34. 1.1.8 Signed Binary Numbers Example: � Express the Decimal number -46 in 8 bit Signed magnitude system Solution: True Binary number for +46 = 00101110 Change the sign bit to 1 and remain unchanged magnitude nits = 10101110 = -46 EE202 DIGITAL ELECTRONICS
35. 35. 1.2 Octal Number System � Octalnumber has eight possible symbols: 0 , 1 , 2 , 3 , 4 , 5 , 6 , 7 and used to express binary numbers, which is called as base of 8 number system or Radix of 8. � Figure: illustrated how it decrease with negative power of 8: 5 4 3 2 1 0 -1 -2 -3 -4 -5 8 8 8 8 8 8 . 8 8 8 8 8 Decrease with negative power of 8 EE202 DIGITAL ELECTRONICS
36. 36. 1.2.2 Octal to Binary Conversion � Any octal number can be represent by 3 bit binary number, such as 0002 to represent 08 and 111 2 to represent 78 Example: Convert 4358 to its Binary equivalent. Solution: 4 3 58 100 011 101 = 1000111012 EE202 DIGITAL ELECTRONICS
37. 37. 1.2.2 Octal to Binary Conversion � Exercise: Convert 54.78 to its Binary equivalent. Solution : 5 4 . 78 101 100 . 111 = 101100.1112 EE202 DIGITAL ELECTRONICS
38. 38. 1.2.1 Binary to Octal Conversion � This is the reverse form of the octal to binary conversion. � First, the Binary number should be divided into group of three from LSB. � Then each three-bit binary number is converted to an Octal form. Example: Convert 1001010112 to its equivalent Octal number. Solution: 100 101 011 4 5 3 = 4 5 38 EE202 DIGITAL ELECTRONICS
39. 39. 1.2.1 Binary to Octal Conversion � Sometimes the Binary numberwill not have even groups of 3 bits. � For those cases, we can add one or two 0s to the left of the MSB of the binary number to fill out the last group. Example: Convert 110101102 to its equivalent Octal number. Solution: 011 010 110 3 2 68 =3268 NOTE that a 0 was placed to the left of the MSB to produce even groups of 3 Bits. EE202 DIGITAL ELECTRONICS
40. 40. 1.2.2 Octal Number to Decimal Conversion � Octal number can be converted to decimal by multiplying the weight of each position with the octal number and adding together. � Example: Convert the Octal number 2578 to its decimal equivalent � Solution: 2 1 0 2578 = (2 x 8 )+ (5 x 8 )+ (7 x 8 ) = (2 x 64) + (5 x 8) + (7 x 1) = 17510 EE202 DIGITAL ELECTRONICS
41. 41. 1.2.2 Octal Number to Decimal Conversion � Exercise : Convert the Octal number 17.78 to its Decimal equivalent Solution: 1 0) + (7 x 8-1) 17.78 = (1 x 8 ) + (7 x 8 = (1 x 8) + (7 x 1) + (7 x .125) = 64.87510 EE202 DIGITAL ELECTRONICS
42. 42. 1.2.2 Decimal to Octal Conversion � Here we can apply the same method done in decimal to binary conversion. Dividing the decimal number by 8 can do conversion to octal. Example : Convert 9710 to its Octal equivalent. Solution : 8 97 + remainder of 1 8 12 + remainder of 4 8 1 + remainder of 1 0 1 4 EE202 DIGITAL ELECTRONICS 18
43. 43. Review Question: Convert 6148 to decimal. 2. Convert 14610 to Octal, then from Octal to Binary. 1. Convert 100111012 to Octal. 4. Convert 97510 to Binary by First Coverting to Octal. 5. Convert Binary 1010111011 2 to Decimal by first converting to Octal. 3. Answer: 1. 396 2. 222, 010010010 4. 1111001111 5. 699 3. 235 EE202 DIGITAL ELECTRONICS
44. 44. 1.3 Hexadecimal Number System � Hexadecimal number system is called as base 16 number system. � It uses 10 decimal numbers and 6 alphabetic characters to represent all 16 possible symbols. � Table below, shows Hexadecimal numbers with its equivalent in decimal and Binary. EE202 DIGITAL ELECTRONICS
45. 45. 1.3 Hexadecimal Number System Hexadecimal Number Binary Number Decimal Nmber 0 0000 0 1 0001 1 2 0010 2 3 0011 3 4 0100 4 5 0101 5 6 0110 6 7 0111 7 8 1000 8 9 1001 9 A 1010 10 B 1011 11 C 1100 12 D 1101 13 E 1110 14 F 1111 15 EE202 DIGITAL ELECTRONICS
46. 46. 1.3.1 Hexadecimal to Binary Conversion � Hexadecimal number can be represent in Binary form by using 4 bits for each hexadecimal number. 016 can be written in binary = 0000 716 in binary can be written = 0111 A16 in binary can be written = 1010 Example: Convert the Hexadecimal A516 to its Binary equivalent. Solution : A 516 1010 01012 = 101001012 EE202 DIGITAL ELECTRONICS
47. 47. 1.3.1 Hexadecimal to Binary Conversion Exercise: Convert the Hexadecimal 9F216 to its Binary equivalent. Solution: 9 F 2 1011 1111 0010 = 100111110010 2 EE202 DIGITAL ELECTRONICS
48. 48. 1.3.1 Binary to Hexadecimal Conversion � This is the reverse form of the Hexadecimal to Binary Conversion. � First the Binary number should be Divided into group of Four bits from LSB. � Then each four-bit binary number is converted to a Hexadecimal form. EE202 DIGITAL ELECTRONICS
49. 49. 1.3.1 Binary to Hexadecimal Conversion � Example: Convert the Binary 1011011011111010 2 to its equivalent Hexadecimal number. Solution: 1011 0110 1111 1010 B 6 F A EE202 DIGITAL ELECTRONICS = B6FA16
50. 50. 1.3.1 Binary to Hexadecimal Conversion � Exercise: Convert the Binary 1110100110 2 to its equivalent Hexadecimal number. Solution: 0011 3 1010 01102 A 6 = 3A616 EE202 DIGITAL ELECTRONICS
51. 51. 1.3.2 Hexadecimal to Decimal Conversion � Hexadecimal number can be converted to decimal by multiplying the weight of each position of the hexadecimal number (power of 16) and adding togather. Example: Convert the Hexadecimal number 32716 to its Decimal Equivalent. Solution : 2 1 0 32716 = (3 x 16 ) + (3 x 16 ) + (7 x 16 ) = ( 3 x 256) + ( 2 x 16 ) + ( 7 x 1 ) = 807 10 EE202 DIGITAL ELECTRONICS
52. 52. 1.3.2 Hexadecimal to Decimal Conversion Example : Convert the Hexadecimal number 2AF16 to its Decimal Equivalent. Solution: 2AF16 = (2 x 162)+(10 x 161 )+(15 x 160 ) = (512) + (160) + (15) = 68710 EE202 DIGITAL ELECTRONICS
53. 53. 1.3.2 Hexadecimal to Decimal Conversion � Exercise : Convert the Hexadecimal number 1BC216 to its Decimal Equivalent. Solution: 2 2 2 2 = (1 x 16 )+(11 x 16 )+(12 x 16 )+(2 x 16 ) = 710610 EE202 DIGITAL ELECTRONICS
54. 54. 1.3.2 Decimal to Hexadecimal Conversion � Here we can apply the same method done in Decimal to Binary conversion. � Since we need to convert to Hexadecimal, so we have to divide the Decimal number by 16. � Example : Convert the Decimal 38210 to its Hexadecimal equivalent. Solution: 16 382 + remainder of 14 16 23 + remainder of 7 0 7 E16 EE202 DIGITAL ELECTRONICS
55. 55. Counting Hexadecimal � When counting in Hex, each digit position can be incremented (increased by 1) from 0 to F. � Once a digit position reaches, the value F, it is RESET to 0 and the next digit position is incremented. � This illustrated in the following Hex counting sequences. (a). 38, 39, 3A, 3B, 3C, 3D, 3E, 3F, 40, 41, 42 (b). 6F8, 6F9, 6FA, 6FB, 6FC, 6FD, 6FE, 6FF, 700 EE202 DIGITAL ELECTRONICS
56. 56. Review Question: 1. Convert 24CE16 to Decimal. 2. Convert 311710 to Hex, then from Hex to Binary. 3. Convert 10010111101101012 to Hex. 4. Write the next four numbers in this Hex counting sequence. E9A, E9B, E9C, E9D, ___,___,___,___ 5. Convert 35278 to Hex. Answer: 1. 9422 2. C2D ; 110000101101 3. 97B5 4. E9E, E9F, EA0, EA1 5. 757 EE202 DIGITAL ELECTRONICS
57. 57. 1.3.3 1's Complement Form � The 1's complement form of any binary number is simply obtained by taking the complement form of 0 and 1. � 1 to 0 and 0 to 1 * The range is –(2n-1 – 1) to +(2n-1-1). Example: Find the 1's complement 101001 Solution: 1 0 1 0 0 1 Binary Number 0 1 0 1 1 0 1's complement of Binary Number EE202 DIGITAL ELECTRONICS
58. 58. 1.3.3 2's Complement Form � The 2's complement form of binary number is obtained by taking the complement form of 0, 1 and adding 1 to LSB. � The range is –(2n-1) to (2n-1-1). Example: Find the 2's complement of 1110010 Solution: 1 1 1 0 0 1 0 Binary Number 0 0 0 1 1 0 1 1's complement + 1 Add 1 0 0 0 1 1 1 0 2's complement EE202 DIGITAL ELECTRONICS
59. 59. 1.3.3 Signed Number Representing Using 2's Complement Example: Express the Decimal number -25 in the 2's complement system using 8-bits. Solution: Represent the +25 in Binary for 0 0 0 1 1 0 0 1 1 1 1 0 0 1 1 0 ( 1's complement) +1 1 1 1 0 0 1 1 1 ( 2's complement) EE202 DIGITAL ELECTRONICS
60. 60. 1.3.3 2's complement Addition � The two number in addition are addend and augend which result in the sum. � The following five cases can be occur when two binary numbers are added; CASE 1: Both number are positive Straight Foward addition. Example: +8 and +4 in 5 bits 0 1000 ( + 8, augend) 0 0100 ( + 4, addend) 0 1 1 0 0 ( sum = + 12) signed bit EE202 DIGITAL ELECTRONICS
61. 61. 1.3.3 2's complement Addition � CASE 2: Positive number larger than negative number Example: Add two numbers +17 and -6 in six bits Solution: 0 1 0 0 0 1 (+17) 1 1 1 0 1 0 (-6) 1 0 0 1 0 1 1 (+11) The Final carry bit is disregarded EE202 DIGITAL ELECTRONICS
62. 62. 1.3.3 2's complement Addition � CASE 3: Positive number smaller than negative number � Example: � Add two numbers -8 and +4 Solution: 1 1 0 0 0 ( -8, augend) 0 0 1 0 0 ( +4, addend) 1 1 1 0 0 ( sum = -4) sign bit Since the SUM is negative, it is in 2's complement form. EE202 DIGITAL ELECTRONICS
63. 63. 1.3.3 2's complement Addition � Case 4: Both numbers are negative Example: Add two numbers -5 and -9 in 8 bits. Solution: 1 1 1 1 1 0 1 1 (-5) + 1 1 1 1 0 1 1 1 (-9) 1 1 1 1 1 0 0 1 0 (-14) Since the SUM is negative, it is in 2's complement form. The final carry bit is disregarded EE202 DIGITAL ELECTRONICS
64. 64. 1.3.3 2's complement Subtraction � when subtracting the binary number ( the subtrahend) from another binary number ( the minuend) then change the sign of the subtrahend and adds it to the minuend. Case 1: Both numbers are positive : Example: Subtract +41 from +75 in byte. Solution: Minuend ( +75) = 01001011 Subtrahend (+41)=00101001 Take the 2's complement form of subtrahend (+41) and add with miuend. EE202 DIGITAL ELECTRONICS
65. 65. 1.3.3 2's complement Subtraction 0 1 0 0 1 0 1 1 1 1 0 1 0 1 1 1 100 1 0 0 0 1 0 (+75) (-41) (+34) discard EE202 DIGITAL ELECTRONICS
66. 66. 1.3.3 2's complement Subtraction � CASE 2: Both numbers are negative Example: Subtract -30 from -80 in 8bit. Solution: In this case (-80) - (-30) = ( -80) + (30) 1 0 1 1 0 0 0 0 (-80) - 0 0 0 1 1 1 1 0 (+30) 1 1 0 0 1 1 1 0 (-50) EE202 DIGITAL ELECTRONICS
67. 67. 1.3.3 2's complement Subtraction � CASE 3 : Both numbers are opposite sign Example: Subtract -20 from + 24 in byte. Solution: In this case (+24) - (-20) = (+24)+(20) 0 0 0 1 1 0 0 0(+24) 0 0 0 1 0 1 0 0 (+20) 0 0 1 0 1 1 0 0 (+44) EE202 DIGITAL ELECTRONICS
68. 68. 1.4 How Binary CODES are Used in Computers � Binary codes are used in computers representing Decimal digits, alphanumeric characters and symbols. � When Numbers, letters or words are represented by a speacial group of symbols, we say that they are being encoded, and the group of symbols is called a code. � When a decimal number is represented by its equivalent binary number, we call it Straight Binary Coding. EE202 DIGITAL ELECTRONICS
69. 69. 1.4.1 BCD 8421 codes � BCD- Binary Coded Decimal Code � If each digit of a decimal number is represented by its binary equivalent, the result is a code called Binary Coded Decimal , BCD. � To illustrate the BCD code, take a Decimal number such as 874. � Each digit is changed to its Binary Equivalent as follows : 8 7 4 (Decimal) (BCD) 1000 0111 0100 EE202 DIGITAL ELECTRONICS
70. 70. 1.4.1 BCD 8421 codes � As another example, let us change 943 to its BCD- code representation: 9 4 3 (Decimal) 1001 0100 0011 ( BCD) � Once again, each decimal digit is changed to its straight Binary equivalent. � Note that 4 Bits are always used for each digit. � The BCD code, then represents each digit of the decimal number by a 4 Bit binary number. EE202 DIGITAL ELECTRONICS
71. 71. 1.4.1 BCD 8421 codes � Clearly only the 4-bit binary numbers from 0000 through 1001 are used. � The BCD code does not use the numbers 1010, 1011, 1100, 1101, 1110 and 1111. � In other words, only 10 of the 16 possible 4-bit binary code groups are used. � If any of the "forbidden" 4-bit numbers ever occurs in a machine using the BCD code, it is usually an indication that an error has occurred. EE202 DIGITAL ELECTRONICS
72. 72. 1.4.1 BCD 8421 codes Example: Convert 0110100000111001 (BCD) to its decimal equivalent. Solution: Divide the BCD number into 4-bit groups and convert each to decimal. 0110 1000 0011 1001 6 8 3 9 EE202 DIGITAL ELECTRONICS
73. 73. 1.4.1 BCD 8421 codes Exercise: Convert the BCD number 011111000001 to its Decimal equivalent. Solution: 0111 1100 0001 7 1 Forbidden code group indicates error in BCD number EE202 DIGITAL ELECTRONICS
74. 74. 1.4.1 Comparison of BCD and Binary � It is important to realize that BCD is not another number � � � � system like binary, octal, decimal and Hexadecimal. It is also important to understand that a BCD number is not the same as a straight binary number. A straight binary code takes the complete decimal number and represents it in binary. BCD code converts each decimal digit to binary individually. Example: 13710 = 100010012 (binary) 13710 = 0001 0011 0111 (BCD) EE202 DIGITAL ELECTRONICS
75. 75. 1.4.2 Alphanumeric Codes � A computer should recognize codes that represent letters of the alphabet, puntuation marks, and other special characters as well as numbers. � A complete alphanumeric code would include the 26 lowercase latters, 26 uppercase latter, 10 numeric digits, 7 punctuation marks, and anywhere from 20 to 40 other characters such as + / * # and so on. � we can say that an alphanumeric code represents all of the various characters and functions that are found on a standard typewriter or computer keyboard. EE202 DIGITAL ELECTRONICS
76. 76. 1.4.2 Alphanumeric Codes - ASCII Code � The most widely used alphanumeric code. � American Standard Code for Information interchange (ASCII). � Pronounced "askee" Refer table next slide to see : Partial Listing of ASCII code. EE202 DIGITAL ELECTRONICS
77. 77. ASCII Reference Table EE202 DIGITAL ELECTRONICS
78. 78. Extended ASCII Codes EE202 DIGITAL ELECTRONICS
79. 79. 1.4.2 Alphanumeric Codes - ASCII Code Example: The following is a message encoded in ASCII code. What is the message? 1001000 1000101 1001100 1010000 Solution: Convert each 7-bit code to its Hexadecimal equivalent. The results are : 48 45 4C 50 Now, locate these Hexadecimal values in table ASCII and determine the character represented by each. The results are: H E L P EE202 DIGITAL ELECTRONICS
80. 80. References � "Digital Systems Principles And Application" Sixth Editon, Ronald J. Tocci. � "Digital Systems Fundamentals" P.W Chandana Prasad, Lau Siong Hoe, Dr. Ashutosh Kumar Singh, Muhammad Suryanata.
81. 81. The End Of Chapter 1... � Do Review Questions � Quiz 1 for Chapter 1- be prepared!! EE201 DIGITAL ELECTRONICS