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The document provides an overview of basic digital systems, binary numbers, and their conversions, including octal and hexadecimal systems. It covers essential topics such as binary logic, binary codes, and different number systems, focusing on their applications in digital electronics. The document also explains methods for performing arithmetic operations and conversions between these number systems.

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Basic Digital Systems and Binary Numbers Basic Digital Electronics (EP) U Tin Maung Lwin Associate Professor EP Dept; GTI, Insein. 2nd July 2020 By
Outline of Chapter 1 & 2  Digital Systems  Binary Numbers  Number-base Conversions  Octal and Hexadecimal Numbers  Complements  Signed Binary Numbers  Binary Codes  Binary Storage and Registers  Binary Logic ဒစ်ဂျစ်တယ်စနစ်များ Binary နံပါတ်များ နံပါတ်အခြေခံ ပြောင်းလဲမှု များ Octal နှင့် Hexadecimal နံပါတ်များ ဖြည့်စွက်ချက်များ Binary နံပါတ်များ သတ်မှတ်ခြင်း Binary ကုဒ်များ Binary သိုလှောင်မှုနှင့် မှတ်ပုံတင်မှုများ Binary Logic
ဒစ်ဂျစ်တယ်စနစ်များနှင့် Binary နံပါတ်များ  ဒစ်ဂျစ်တယ်ခေတ်နှင့် သတင်းအချက်အလက်ခေတ်  Digital computers  General purposes  Many scientific, industrial and commercial applications  Digital systems /ဒစ်ဂျစ်တယ်စနစ်များ  Telephone switching exchanges  Digital camera  Electronic calculators, Digital Meter  Digital TV  Discrete information-processing systems သီးသန ့ ် သတင်းအချက်အလက် စီမံဆောင်ရွက်ပေးသော စနစ်များ  Manipulate discrete elements of information /သတင်းအချက်အလက်၏ သီးခြားအစိတ်အပိုင်းများကို ကိုင်တွယ်ဖြေရှင်းပါ။  For example, {1, 2, 3, …} and {A, B, C, …}…
Analog နှင့် Digital Signal များ  Analog system  The physical quantities or signals may vary continuously over a specified range. ရုပ်ပိုင်းဆိုင်ရာ ပမာဏ သို့ မဟုတ် အချက်ပြမှုများသည် သတ်မှတ် ထားသည့် ပမာဏထက် အဆက်မပြတ် ကွဲပြားနိုင်သည်။  Digital system  The physical quantities or signals can assume only discrete values. ရုပ်ပိုင်းဆိုင်ရာပမာဏ သို့ မဟုတ် အချက်ပြမှုများသည် သီးခြားတန်ဖိုးများ ကိုသာ ယူဆနိုင်သည်။  Greater accuracy (ပိုတိကျတယ်) t X(t) t X(t) Analog signal Digital signal
Binary Digital Signal  An information variable represented by physical quantity. ရုပ်ပိုင်းဆိုင်ရာ ပမာဏဖြင့် ကိုယ်စားပြုသည့် အချက်အလက် ကိန်းရှင်တစ်ခု။  For digital systems, the variable takes on discrete values.  Two level, or binary values are the most prevalent values.  Binary တန်ဖိုးများကို အနှစ်ချုပ်အားဖြင့် ကိုယ်စားပြုသည်။  Digits 0 and 1  Words (symbols) False (F) and True (T)  Words (symbols) Low (L) and High (H)  And words On and Off  Binary တန်ဖိုးများကို ရုပ်ပိုင်းဆိုင်ရာ ပမာဏ တန်ဖိုးများ ၏ အပိုင်းအခြား တန်ဖိုးများ ဖြင့် ကိုယ်စားပြုသည်။ t V(t) Binary digital signal Logic 1 Logic 0 undefine
Decimal Number System /ဆယ်လီစိပ်စနစ်  Base (also called radix) = 10  10 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }  Digit Position  Integer & fraction /ကိန်းပြည့်နှင့် အပိုင်းကိန်း  Digit Weight  Weight = (Base) Position  Magnitude  Sum of “Digit x Weight”  Formal Notation 1 0 -1 2 -2 5 1 2 7 4 10 1 0.1 100 0.01 500 10 2 0.7 0.04 d2*B2 +d1*B1 +d0*B0 +d-1*B-1 +d-2*B-2 (512.74)10
Octal(base 8) Number System  Base = 8  8 digits { 0, 1, 2, 3, 4, 5, 6, 7 }  Weights  Weight = (Base) Position  Magnitude  Sum of “Digit x Weight”  Formal Notation 1 0 -1 2 -2 8 1 1/8 64 1/64 5 1 2 7 4 5 *82 +1 *81 +2 *80 +7 *8-1 +4 *8-2 =(330.9375)10 (512.74)8
Hexadecimal (base 16) Number System  Base = 16  16 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F }  Weights  Weight = (Base) Position  Magnitude  Sum of “Digit x Weight”  Formal Notation 1 0 -1 2 -2 16 1 1/16 256 1/256 1 E 5 7 A 1 *162 +14 *161 +5 *160 +7 *16-1 +10 *16-2 =(485.4765625)10 (1E5.7A)16
Binary (base 2) Number System  Base = 2  2 digits { 0, 1 }, called binary digits or “bits”  Weights  Weight = (Base) Position  Magnitude  Sum of “Bit x Weight”  Formal Notation  Groups of bits 4 bits = Nibble 8 bits = Byte 1 0 -1 2 -2 2 1 1/2 4 1/4 1 0 1 0 1 1 *22 +0 *21 +1 *20 +0 *2-1 +1 *2-2 =(5.25)10 (101.01)2 1 0 1 1 1 1 0 0 0 1 0 1
The Power of 2 n 2n 0 20 =1 1 21 =2 2 22 =4 3 23 =8 4 24 =16 5 25 =32 6 26 =64 7 27 =128 n 2n 8 28 =256 9 29 =512 10 210 =1024 11 211 =2048 12 212 =4096 20 220 =1M 30 230 =1G 40 240 =1T Mega Giga Tera Kilo
Addition  Decimal Addition 5 5 5 5 + 0 1 1 = Ten ≥ Base  Subtract a Base 1 1 Carry
Binary Addition  Column Addition 1 0 1 1 1 1 1 1 1 1 0 + 0 0 0 0 1 1 1 ≥ (2)10 1 1 1 1 1 1 = 61 = 23 = 84
Binary Subtraction  Borrow a “Base” when needed 0 0 1 1 1 0 1 1 1 1 0 − 0 1 0 1 1 1 0 = (10)2 2 2 2 2 1 0 0 0 1 = 77 = 23 = 54
Binary Multiplication  Bit by bit 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 x
Number Base Conversions Decimal (Base 10) Octal (Base 8) Binary (Base 2) Hexadecimal (Base 16) Evaluate Magnitude Evaluate Magnitude Evaluate Magnitude
Decimal (Integer) to Binary Conversion  Divide the number by the ‘Base’ (=2)  Take the remainder (either 0 or 1) as a coefficient  Take the quotient and repeat the division Example: (13)10 Quotient Remainder Coefficient Answer: (13)10 = (a3 a2 a1 a0)2 = (1101)2 MSB LSB MSB LSB 13/ 2 = 6 1 a0 = 1 6 / 2 = 3 0 a1 = 0 3 / 2 = 1 1 a2 = 1 1 / 2 = 0 1 a3 = 1
Decimal (Fraction) to Binary Conversion  Multiply the number by the ‘Base’ (=2)  Take the integer (either 0 or 1) as a coefficient  Take the resultant fraction and repeat the division Example: (0.625)10 Integer Fraction Coefficient Answer: (0.625)10 = (0.a-1 a-2 a-3)2 = (0.101)2 MSB LSB MSB LSB 0.625 * 2 = 1 . 25 0.25 * 2 = 0 . 5 a-2 = 0 0.5 * 2 = 1 . 0 a-3 = 1 a-1 = 1
Decimal to Octal Conversion Example: (175)10 Quotient Remainder Coefficient Answer: (175)10 = (a2 a1 a0)8 = (257)8 175 / 8 = 21 7 a0 = 7 21 / 8 = 2 5 a1 = 5 2 / 8 = 0 2 a2 = 2 Example: (0.3125)10 Integer Fraction Coefficient Answer: (0.3125)10 = (0.a-1 a-2 a-3)8 = (0.24)8 0.3125 * 8 = 2 . 5 0.5 * 8 = 4 . 0 a-2 = 4 a-1 = 2
Binary − Octal Conversion  8 = 23  Each group of 3 bits represents an octal digit Octal Binary 0 0 0 0 1 0 0 1 2 0 1 0 3 0 1 1 4 1 0 0 5 1 0 1 6 1 1 0 7 1 1 1 Example: ( 1 0 1 1 0 . 0 1 )2 ( 2 6 . 2 )8 Assume Zeros Works both ways (Binary to Octal & Octal to Binary)
Binary − Hexadecimal Conversion  16 = 24  Each group of 4 bits represents a hexadecimal digit Hex Binary 0 0 0 0 0 1 0 0 0 1 2 0 0 1 0 3 0 0 1 1 4 0 1 0 0 5 0 1 0 1 6 0 1 1 0 7 0 1 1 1 8 1 0 0 0 9 1 0 0 1 A 1 0 1 0 B 1 0 1 1 C 1 1 0 0 D 1 1 0 1 E 1 1 1 0 F 1 1 1 1 Example: ( 1 0 1 1 0 . 0 1 )2 ( 1 6 . 4 )16 Assume Zeros Works both ways (Binary to Hex & Hex to Binary)
Octal − Hexadecimal Conversion  Convert to Binary as an intermediate step Example: ( 0 1 0 1 1 0 . 0 1 0 )2 ( 1 6 . 4 )16 Assume Zeros Works both ways (Octal to Hex & Hex to Octal) ( 2 6 . 2 )8 Assume Zeros
Decimal, Binary, Octal and Hexadecimal Decimal Binary Octal Hex 00 0000 00 0 01 0001 01 1 02 0010 02 2 03 0011 03 3 04 0100 04 4 05 0101 05 5 06 0110 06 6 07 0111 07 7 08 1000 10 8 09 1001 11 9 10 1010 12 A 11 1011 13 B 12 1100 14 C 13 1101 15 D 14 1110 16 E 15 1111 17 F
Complements  There are two types of complements for each base-r system: the radix complement and diminished radix complement.  Diminished Radix Complement - (r-1)’s Complement  Given a number N in base r having n digits, the (r–1)’s complement of N is defined as: (rn –1) – N  Example for 6-digit decimal numbers:  9’s complement is (rn – 1)–N = (106 –1)–N = 999999–N  9’s complement of 546700 is 999999–546700 = 453299  Example for 7-digit binary numbers:  1’s complement is (rn – 1) – N = (27 –1)–N = 1111111–N  1’s complement of 1011000 is 1111111–1011000 = 0100111  Observation:  Subtraction from (rn – 1) will never require a borrow  Diminished radix complement can be computed digit-by-digit  For binary: 1 – 0 = 1 and 1 – 1 = 0
Complements  1’s Complement (Diminished Radix Complement)  All ‘0’s become ‘1’s  All ‘1’s become ‘0’s Example (10110000)2  (01001111)2 If you add a number and its 1’s complement … 1 0 1 1 0 0 0 0 + 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1
Complements  Radix Complement  Example: Base-10  Example: Base-2 The r's complement of an n-digit number N in base r is defined as rn – N for N ≠ 0 and as 0 for N = 0. Comparing with the (r  1) 's complement, we note that the r's complement is obtained by adding 1 to the (r  1) 's complement, since rn – N = [(rn  1) – N] + 1. The 10's complement of 012398 is 987602 The 10's complement of 246700 is 753300 The 2's complement of 1101100 is 0010100 The 2's complement of 0110111 is 1001001
Complements  2’s Complement (Radix Complement)  Take 1’s complement then add 1  Toggle all bits to the left of the first ‘1’ from the right Example: Number: 1’s Comp.: 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 1 1 1 + 1 OR 1 0 1 1 0 0 0 0 0 0 0 0 1 0 1 0
Complements  Subtraction with Complements  The subtraction of two n-digit unsigned numbers M – N in base r can be done as follows:
Complements  Example 1.5  Using 10's complement, subtract 72532 – 3250.  Example 1.6  Using 10's complement, subtract 3250 – 72532. There is no end carry. Therefore, the answer is – (10's complement of 30718) =  69282.
Complements  Example 1.7  Given the two binary numbers X = 1010100 and Y = 1000011, perform the subtraction (a) X – Y ; and (b) Y  X, by using 2's complement. There is no end carry. Therefore, the answer is Y – X =  (2's complement of 1101111) =  0010001.
Complements  Subtraction of unsigned numbers can also be done by means of the (r  1)'s complement. Remember that the (r  1) 's complement is one less then the r's complement.  Example 1.8  Repeat Example 1.7, but this time using 1's complement. There is no end carry, Therefore, the answer is Y – X =  (1's complement of 1101110) =  0010001.
1.6 Signed Binary Numbers To represent negative integers, we need a notation for negative values. It is customary to represent the sign with a bit placed in the leftmost position of the number since binary digits. The convention is to make the sign bit 0 for positive and 1 for negative. Example: Table 1.3 lists all possible four-bit signed binary numbers in the three representations.
Signed Binary Numbers
Signed Binary Numbers  Arithmetic addition  The addition of two numbers in the signed-magnitude system follows the rules of ordinary arithmetic. If the signs are the same, we add the two magnitudes and give the sum the common sign. If the signs are different, we subtract the smaller magnitude from the larger and give the difference the sign if the larger magnitude.  The addition of two signed binary numbers with negative numbers represented in signed-2's-complement form is obtained from the addition of the two numbers, including their sign bits.  A carry out of the sign-bit position is discarded.  Example:
Signed Binary Numbers  Arithmetic Subtraction  In 2’s-complement form:  Example: 1. Take the 2’s complement of the subtrahend (including the sign bit) and add it to the minuend (including sign bit). 2. A carry out of sign-bit position is discarded. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A B A B A B A B               ( 6)  ( 13) (11111010  11110011) (11111010 + 00001101) 00000111 (+ 7)
Binary Codes  BCD Code  A number with k decimal digits will require 4k bits in BCD.  Decimal 396 is represented in BCD with 12bits as 0011 1001 0110, with each group of 4 bits representing one decimal digit.  A decimal number in BCD is the same as its equivalent binary number only when the number is between 0 and 9.  The binary combinations 1010 through 1111 are not used and have no meaning in BCD.
Binary Code  Example:  Consider decimal 185 and its corresponding value in BCD and binary:  BCD addition
Binary Code  Example:  Consider the addition of 184 + 576 = 760 in BCD:  Decimal Arithmetic: (+375) + (-240) = +135 Hint 6: using 10’s of BCD
Binary Codes  Other Decimal Codes
Binary Codes  Gray Code  The advantage is that only bit in the code group changes in going from one number to the next. » Error detection. » Representation of analog data. » Low power design. 000 001 010 100 110 111 101 011 1-1 and onto!!
Binary Codes  American Standard Code for Information Interchange (ASCII) Character Code
Binary Codes  ASCII Character Code
ASCII Character Codes  American Standard Code for Information Interchange (Refer to Table 1.7)  A popular code used to represent information sent as character- based data.  It uses 7-bits to represent:  94 Graphic printing characters.  34 Non-printing characters.  Some non-printing characters are used for text format (e.g. BS = Backspace, CR = carriage return).  Other non-printing characters are used for record marking and flow control (e.g. STX and ETX start and end text areas).
ASCII Properties  ASCII has some interesting properties:  Digits 0 to 9 span Hexadecimal values 3016 to 3916  Upper case A-Z span 4116 to 5A16  Lower case a-z span 6116 to 7A16 » Lower to upper case translation (and vice versa) occurs by flipping bit 6.
Binary Codes  Error-Detecting Code  To detect errors in data communication and processing, an eighth bit is sometimes added to the ASCII character to indicate its parity.  A parity bit is an extra bit included with a message to make the total number of 1's either even or odd.  Example:  Consider the following two characters and their even and odd parity:
Binary Codes  Error-Detecting Code  Redundancy (e.g. extra information), in the form of extra bits, can be incorporated into binary code words to detect and correct errors.  A simple form of redundancy is parity, an extra bit appended onto the code word to make the number of 1’s odd or even. Parity can detect all single-bit errors and some multiple-bit errors.  A code word has even parity if the number of 1’s in the code word is even.  A code word has odd parity if the number of 1’s in the code word is odd.  Example: 10001001 10001001 1 0 (odd parity) Message B: Message A: (even parity)
Binary Storage and Registers  Registers  A binary cell is a device that possesses two stable states and is capable of storing one of the two states.  A register is a group of binary cells. A register with n cells can store any discrete quantity of information that contains n bits.  A binary cell  Two stable state  Store one bit of information  Examples: flip-flop circuits, ferrite cores, capacitor  A register  A group of binary cells  AX in x86 CPU  Register Transfer  A transfer of the information stored in one register to another.  One of the major operations in digital system.  An example in next slides. n cells 2n possible states
A Digital Computer Example Synchronous or Asynchronous? Inputs: Keyboard, mouse, modem, microphone Outputs: CRT, LCD, modem, speakers Memory Control unit Datapath Input/Output CPU
Transfer of information Figure 1.1 Transfer of information among register
Transfer of information  The other major component of a digital system  Circuit elements to manipulate individual bits of information  Load-store machine LD R1; LD R2; ADD R3, R2, R1; SD R3; Figure 1.2 Example of binary information processing
Binary Logic  Definition of Binary Logic  Binary logic consists of binary variables and a set of logical operations.  The variables are designated by letters of the alphabet, such as A, B, C, x, y, z, etc, with each variable having two and only two distinct possible values: 1 and 0,  Three basic logical operations: AND, OR, and NOT.
Binary Logic  Truth Tables, Boolean Expressions, and Logic Gates x y z 0 0 0 0 1 0 1 0 0 1 1 1 x y z 0 0 0 0 1 1 1 0 1 1 1 1 x z 0 1 1 0 AND OR NOT x y z x y z z = x • y = x y z = x + y z = x = x’ x z
Examples for Switching Circuits AND OR
Binary Logic  Logic gates  Example of binary signals 0 1 2 3 Logic 1 Logic 0 Un-define Figure 1.3 Example of binary signals
Binary Logic  Logic gates  Graphic Symbols and Input-Output Signals for Logic gates: Fig. 1.4 Symbols for digital logic circuits Fig. 1.5 Input-Output signals for gates
Binary Logic  Logic gates  Graphic Symbols and Input-Output Signals for Logic gates: Fig. 1.6 Gates with multiple inputs
THANK YOU For Your Participation!
အားလုံးကို ကျေးဇူး အထူးတင် ပါတယ်! 3G EP ကျောင်းသူ၊ သားများအားလုံး Covid 19 ကင်းဝေး၍ စိတ်၏ ချမ်းသာခြင်း၊ ကိုယ်၏ကျန်းမာခြင်းများ ဖြင့် ပြည့်စုံကြပါစေဗျာ!
Assessment Questions!  (1) အောက်ပါပေးထားသော Logic Gates များအား Truth Tables နှင့် LD = Ladder Diagram ဖြင့် ရေးသားဖော်ပြပါ။  (a) AND (b) OR (c) NOT (d) NAND (e) NOR  (f) XOR (g) XNOR  (2) Binary Digits များကို ဖော်ပြပါ။  (3) PLC နှင့်ပတ်သတ်သော Number System (5) မျ ိုးအား ဖော်ပြပါ။  (4) 1 Nibble တွင် မည်မျှ Bits ရှိသနည်း။  (5) အောက်တွင်ဖော်ပြထားမှုများအား Nibble, Byte, Word ခွဲခြားပြပါ။ 1 0 1 1 1 0 0 1 1 0 1 0 1 0 1 1 MSB LSB Bit (a) ? (b) ? (c) ?

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Basic Digital_Systems_and_Binary_Numbers Sample.ppt

  • 1.
    Basic Digital Systemsand Binary Numbers Basic Digital Electronics (EP) U Tin Maung Lwin Associate Professor EP Dept; GTI, Insein. 2nd July 2020 By
  • 2.
    Outline of Chapter1 & 2  Digital Systems  Binary Numbers  Number-base Conversions  Octal and Hexadecimal Numbers  Complements  Signed Binary Numbers  Binary Codes  Binary Storage and Registers  Binary Logic ဒစ်ဂျစ်တယ်စနစ်များ Binary နံပါတ်များ နံပါတ်အခြေခံ ပြောင်းလဲမှု များ Octal နှင့် Hexadecimal နံပါတ်များ ဖြည့်စွက်ချက်များ Binary နံပါတ်များ သတ်မှတ်ခြင်း Binary ကုဒ်များ Binary သိုလှောင်မှုနှင့် မှတ်ပုံတင်မှုများ Binary Logic
  • 3.
    ဒစ်ဂျစ်တယ်စနစ်များနှင့် Binary နံပါတ်များ ဒစ်ဂျစ်တယ်ခေတ်နှင့် သတင်းအချက်အလက်ခေတ်  Digital computers  General purposes  Many scientific, industrial and commercial applications  Digital systems /ဒစ်ဂျစ်တယ်စနစ်များ  Telephone switching exchanges  Digital camera  Electronic calculators, Digital Meter  Digital TV  Discrete information-processing systems သီးသန ့ ် သတင်းအချက်အလက် စီမံဆောင်ရွက်ပေးသော စနစ်များ  Manipulate discrete elements of information /သတင်းအချက်အလက်၏ သီးခြားအစိတ်အပိုင်းများကို ကိုင်တွယ်ဖြေရှင်းပါ။  For example, {1, 2, 3, …} and {A, B, C, …}…
  • 4.
    Analog နှင့် DigitalSignal များ  Analog system  The physical quantities or signals may vary continuously over a specified range. ရုပ်ပိုင်းဆိုင်ရာ ပမာဏ သို့ မဟုတ် အချက်ပြမှုများသည် သတ်မှတ် ထားသည့် ပမာဏထက် အဆက်မပြတ် ကွဲပြားနိုင်သည်။  Digital system  The physical quantities or signals can assume only discrete values. ရုပ်ပိုင်းဆိုင်ရာပမာဏ သို့ မဟုတ် အချက်ပြမှုများသည် သီးခြားတန်ဖိုးများ ကိုသာ ယူဆနိုင်သည်။  Greater accuracy (ပိုတိကျတယ်) t X(t) t X(t) Analog signal Digital signal
  • 5.
    Binary Digital Signal An information variable represented by physical quantity. ရုပ်ပိုင်းဆိုင်ရာ ပမာဏဖြင့် ကိုယ်စားပြုသည့် အချက်အလက် ကိန်းရှင်တစ်ခု။  For digital systems, the variable takes on discrete values.  Two level, or binary values are the most prevalent values.  Binary တန်ဖိုးများကို အနှစ်ချုပ်အားဖြင့် ကိုယ်စားပြုသည်။  Digits 0 and 1  Words (symbols) False (F) and True (T)  Words (symbols) Low (L) and High (H)  And words On and Off  Binary တန်ဖိုးများကို ရုပ်ပိုင်းဆိုင်ရာ ပမာဏ တန်ဖိုးများ ၏ အပိုင်းအခြား တန်ဖိုးများ ဖြင့် ကိုယ်စားပြုသည်။ t V(t) Binary digital signal Logic 1 Logic 0 undefine
  • 6.
    Decimal Number System/ဆယ်လီစိပ်စနစ်  Base (also called radix) = 10  10 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 }  Digit Position  Integer & fraction /ကိန်းပြည့်နှင့် အပိုင်းကိန်း  Digit Weight  Weight = (Base) Position  Magnitude  Sum of “Digit x Weight”  Formal Notation 1 0 -1 2 -2 5 1 2 7 4 10 1 0.1 100 0.01 500 10 2 0.7 0.04 d2*B2 +d1*B1 +d0*B0 +d-1*B-1 +d-2*B-2 (512.74)10
  • 7.
    Octal(base 8) NumberSystem  Base = 8  8 digits { 0, 1, 2, 3, 4, 5, 6, 7 }  Weights  Weight = (Base) Position  Magnitude  Sum of “Digit x Weight”  Formal Notation 1 0 -1 2 -2 8 1 1/8 64 1/64 5 1 2 7 4 5 *82 +1 *81 +2 *80 +7 *8-1 +4 *8-2 =(330.9375)10 (512.74)8
  • 8.
    Hexadecimal (base 16)Number System  Base = 16  16 digits { 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F }  Weights  Weight = (Base) Position  Magnitude  Sum of “Digit x Weight”  Formal Notation 1 0 -1 2 -2 16 1 1/16 256 1/256 1 E 5 7 A 1 *162 +14 *161 +5 *160 +7 *16-1 +10 *16-2 =(485.4765625)10 (1E5.7A)16
  • 9.
    Binary (base 2)Number System  Base = 2  2 digits { 0, 1 }, called binary digits or “bits”  Weights  Weight = (Base) Position  Magnitude  Sum of “Bit x Weight”  Formal Notation  Groups of bits 4 bits = Nibble 8 bits = Byte 1 0 -1 2 -2 2 1 1/2 4 1/4 1 0 1 0 1 1 *22 +0 *21 +1 *20 +0 *2-1 +1 *2-2 =(5.25)10 (101.01)2 1 0 1 1 1 1 0 0 0 1 0 1
  • 10.
    The Power of2 n 2n 0 20 =1 1 21 =2 2 22 =4 3 23 =8 4 24 =16 5 25 =32 6 26 =64 7 27 =128 n 2n 8 28 =256 9 29 =512 10 210 =1024 11 211 =2048 12 212 =4096 20 220 =1M 30 230 =1G 40 240 =1T Mega Giga Tera Kilo
  • 11.
    Addition  Decimal Addition 55 5 5 + 0 1 1 = Ten ≥ Base  Subtract a Base 1 1 Carry
  • 12.
    Binary Addition  ColumnAddition 1 0 1 1 1 1 1 1 1 1 0 + 0 0 0 0 1 1 1 ≥ (2)10 1 1 1 1 1 1 = 61 = 23 = 84
  • 13.
    Binary Subtraction  Borrowa “Base” when needed 0 0 1 1 1 0 1 1 1 1 0 − 0 1 0 1 1 1 0 = (10)2 2 2 2 2 1 0 0 0 1 = 77 = 23 = 54
  • 14.
    Binary Multiplication  Bitby bit 0 1 1 1 1 0 1 1 0 0 0 0 0 0 0 1 1 1 1 0 1 1 1 1 0 0 0 0 0 0 1 1 0 1 1 1 0 x
  • 15.
    Number Base Conversions Decimal (Base10) Octal (Base 8) Binary (Base 2) Hexadecimal (Base 16) Evaluate Magnitude Evaluate Magnitude Evaluate Magnitude
  • 16.
    Decimal (Integer) toBinary Conversion  Divide the number by the ‘Base’ (=2)  Take the remainder (either 0 or 1) as a coefficient  Take the quotient and repeat the division Example: (13)10 Quotient Remainder Coefficient Answer: (13)10 = (a3 a2 a1 a0)2 = (1101)2 MSB LSB MSB LSB 13/ 2 = 6 1 a0 = 1 6 / 2 = 3 0 a1 = 0 3 / 2 = 1 1 a2 = 1 1 / 2 = 0 1 a3 = 1
  • 17.
    Decimal (Fraction) toBinary Conversion  Multiply the number by the ‘Base’ (=2)  Take the integer (either 0 or 1) as a coefficient  Take the resultant fraction and repeat the division Example: (0.625)10 Integer Fraction Coefficient Answer: (0.625)10 = (0.a-1 a-2 a-3)2 = (0.101)2 MSB LSB MSB LSB 0.625 * 2 = 1 . 25 0.25 * 2 = 0 . 5 a-2 = 0 0.5 * 2 = 1 . 0 a-3 = 1 a-1 = 1
  • 18.
    Decimal to OctalConversion Example: (175)10 Quotient Remainder Coefficient Answer: (175)10 = (a2 a1 a0)8 = (257)8 175 / 8 = 21 7 a0 = 7 21 / 8 = 2 5 a1 = 5 2 / 8 = 0 2 a2 = 2 Example: (0.3125)10 Integer Fraction Coefficient Answer: (0.3125)10 = (0.a-1 a-2 a-3)8 = (0.24)8 0.3125 * 8 = 2 . 5 0.5 * 8 = 4 . 0 a-2 = 4 a-1 = 2
  • 19.
    Binary − OctalConversion  8 = 23  Each group of 3 bits represents an octal digit Octal Binary 0 0 0 0 1 0 0 1 2 0 1 0 3 0 1 1 4 1 0 0 5 1 0 1 6 1 1 0 7 1 1 1 Example: ( 1 0 1 1 0 . 0 1 )2 ( 2 6 . 2 )8 Assume Zeros Works both ways (Binary to Octal & Octal to Binary)
  • 20.
    Binary − HexadecimalConversion  16 = 24  Each group of 4 bits represents a hexadecimal digit Hex Binary 0 0 0 0 0 1 0 0 0 1 2 0 0 1 0 3 0 0 1 1 4 0 1 0 0 5 0 1 0 1 6 0 1 1 0 7 0 1 1 1 8 1 0 0 0 9 1 0 0 1 A 1 0 1 0 B 1 0 1 1 C 1 1 0 0 D 1 1 0 1 E 1 1 1 0 F 1 1 1 1 Example: ( 1 0 1 1 0 . 0 1 )2 ( 1 6 . 4 )16 Assume Zeros Works both ways (Binary to Hex & Hex to Binary)
  • 21.
    Octal − HexadecimalConversion  Convert to Binary as an intermediate step Example: ( 0 1 0 1 1 0 . 0 1 0 )2 ( 1 6 . 4 )16 Assume Zeros Works both ways (Octal to Hex & Hex to Octal) ( 2 6 . 2 )8 Assume Zeros
  • 22.
    Decimal, Binary, Octaland Hexadecimal Decimal Binary Octal Hex 00 0000 00 0 01 0001 01 1 02 0010 02 2 03 0011 03 3 04 0100 04 4 05 0101 05 5 06 0110 06 6 07 0111 07 7 08 1000 10 8 09 1001 11 9 10 1010 12 A 11 1011 13 B 12 1100 14 C 13 1101 15 D 14 1110 16 E 15 1111 17 F
  • 23.
    Complements  There aretwo types of complements for each base-r system: the radix complement and diminished radix complement.  Diminished Radix Complement - (r-1)’s Complement  Given a number N in base r having n digits, the (r–1)’s complement of N is defined as: (rn –1) – N  Example for 6-digit decimal numbers:  9’s complement is (rn – 1)–N = (106 –1)–N = 999999–N  9’s complement of 546700 is 999999–546700 = 453299  Example for 7-digit binary numbers:  1’s complement is (rn – 1) – N = (27 –1)–N = 1111111–N  1’s complement of 1011000 is 1111111–1011000 = 0100111  Observation:  Subtraction from (rn – 1) will never require a borrow  Diminished radix complement can be computed digit-by-digit  For binary: 1 – 0 = 1 and 1 – 1 = 0
  • 24.
    Complements  1’s Complement(Diminished Radix Complement)  All ‘0’s become ‘1’s  All ‘1’s become ‘0’s Example (10110000)2  (01001111)2 If you add a number and its 1’s complement … 1 0 1 1 0 0 0 0 + 0 1 0 0 1 1 1 1 1 1 1 1 1 1 1 1
  • 25.
    Complements  Radix Complement Example: Base-10  Example: Base-2 The r's complement of an n-digit number N in base r is defined as rn – N for N ≠ 0 and as 0 for N = 0. Comparing with the (r  1) 's complement, we note that the r's complement is obtained by adding 1 to the (r  1) 's complement, since rn – N = [(rn  1) – N] + 1. The 10's complement of 012398 is 987602 The 10's complement of 246700 is 753300 The 2's complement of 1101100 is 0010100 The 2's complement of 0110111 is 1001001
  • 26.
    Complements  2’s Complement(Radix Complement)  Take 1’s complement then add 1  Toggle all bits to the left of the first ‘1’ from the right Example: Number: 1’s Comp.: 0 1 0 1 0 0 0 0 1 0 1 1 0 0 0 0 0 1 0 0 1 1 1 1 + 1 OR 1 0 1 1 0 0 0 0 0 0 0 0 1 0 1 0
  • 27.
    Complements  Subtraction withComplements  The subtraction of two n-digit unsigned numbers M – N in base r can be done as follows:
  • 28.
    Complements  Example 1.5 Using 10's complement, subtract 72532 – 3250.  Example 1.6  Using 10's complement, subtract 3250 – 72532. There is no end carry. Therefore, the answer is – (10's complement of 30718) =  69282.
  • 29.
    Complements  Example 1.7 Given the two binary numbers X = 1010100 and Y = 1000011, perform the subtraction (a) X – Y ; and (b) Y  X, by using 2's complement. There is no end carry. Therefore, the answer is Y – X =  (2's complement of 1101111) =  0010001.
  • 30.
    Complements  Subtraction ofunsigned numbers can also be done by means of the (r  1)'s complement. Remember that the (r  1) 's complement is one less then the r's complement.  Example 1.8  Repeat Example 1.7, but this time using 1's complement. There is no end carry, Therefore, the answer is Y – X =  (1's complement of 1101110) =  0010001.
  • 31.
    1.6 Signed BinaryNumbers To represent negative integers, we need a notation for negative values. It is customary to represent the sign with a bit placed in the leftmost position of the number since binary digits. The convention is to make the sign bit 0 for positive and 1 for negative. Example: Table 1.3 lists all possible four-bit signed binary numbers in the three representations.
  • 32.
  • 33.
    Signed Binary Numbers Arithmetic addition  The addition of two numbers in the signed-magnitude system follows the rules of ordinary arithmetic. If the signs are the same, we add the two magnitudes and give the sum the common sign. If the signs are different, we subtract the smaller magnitude from the larger and give the difference the sign if the larger magnitude.  The addition of two signed binary numbers with negative numbers represented in signed-2's-complement form is obtained from the addition of the two numbers, including their sign bits.  A carry out of the sign-bit position is discarded.  Example:
  • 34.
    Signed Binary Numbers Arithmetic Subtraction  In 2’s-complement form:  Example: 1. Take the 2’s complement of the subtrahend (including the sign bit) and add it to the minuend (including sign bit). 2. A carry out of sign-bit position is discarded. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) A B A B A B A B               ( 6)  ( 13) (11111010  11110011) (11111010 + 00001101) 00000111 (+ 7)
  • 35.
    Binary Codes  BCDCode  A number with k decimal digits will require 4k bits in BCD.  Decimal 396 is represented in BCD with 12bits as 0011 1001 0110, with each group of 4 bits representing one decimal digit.  A decimal number in BCD is the same as its equivalent binary number only when the number is between 0 and 9.  The binary combinations 1010 through 1111 are not used and have no meaning in BCD.
  • 36.
    Binary Code  Example: Consider decimal 185 and its corresponding value in BCD and binary:  BCD addition
  • 37.
    Binary Code  Example: Consider the addition of 184 + 576 = 760 in BCD:  Decimal Arithmetic: (+375) + (-240) = +135 Hint 6: using 10’s of BCD
  • 38.
  • 39.
    Binary Codes  GrayCode  The advantage is that only bit in the code group changes in going from one number to the next. » Error detection. » Representation of analog data. » Low power design. 000 001 010 100 110 111 101 011 1-1 and onto!!
  • 40.
    Binary Codes  AmericanStandard Code for Information Interchange (ASCII) Character Code
  • 41.
    Binary Codes  ASCIICharacter Code
  • 42.
    ASCII Character Codes American Standard Code for Information Interchange (Refer to Table 1.7)  A popular code used to represent information sent as character- based data.  It uses 7-bits to represent:  94 Graphic printing characters.  34 Non-printing characters.  Some non-printing characters are used for text format (e.g. BS = Backspace, CR = carriage return).  Other non-printing characters are used for record marking and flow control (e.g. STX and ETX start and end text areas).
  • 43.
    ASCII Properties  ASCIIhas some interesting properties:  Digits 0 to 9 span Hexadecimal values 3016 to 3916  Upper case A-Z span 4116 to 5A16  Lower case a-z span 6116 to 7A16 » Lower to upper case translation (and vice versa) occurs by flipping bit 6.
  • 44.
    Binary Codes  Error-DetectingCode  To detect errors in data communication and processing, an eighth bit is sometimes added to the ASCII character to indicate its parity.  A parity bit is an extra bit included with a message to make the total number of 1's either even or odd.  Example:  Consider the following two characters and their even and odd parity:
  • 45.
    Binary Codes  Error-DetectingCode  Redundancy (e.g. extra information), in the form of extra bits, can be incorporated into binary code words to detect and correct errors.  A simple form of redundancy is parity, an extra bit appended onto the code word to make the number of 1’s odd or even. Parity can detect all single-bit errors and some multiple-bit errors.  A code word has even parity if the number of 1’s in the code word is even.  A code word has odd parity if the number of 1’s in the code word is odd.  Example: 10001001 10001001 1 0 (odd parity) Message B: Message A: (even parity)
  • 46.
    Binary Storage andRegisters  Registers  A binary cell is a device that possesses two stable states and is capable of storing one of the two states.  A register is a group of binary cells. A register with n cells can store any discrete quantity of information that contains n bits.  A binary cell  Two stable state  Store one bit of information  Examples: flip-flop circuits, ferrite cores, capacitor  A register  A group of binary cells  AX in x86 CPU  Register Transfer  A transfer of the information stored in one register to another.  One of the major operations in digital system.  An example in next slides. n cells 2n possible states
  • 47.
    A Digital ComputerExample Synchronous or Asynchronous? Inputs: Keyboard, mouse, modem, microphone Outputs: CRT, LCD, modem, speakers Memory Control unit Datapath Input/Output CPU
  • 48.
    Transfer of information Figure1.1 Transfer of information among register
  • 49.
    Transfer of information The other major component of a digital system  Circuit elements to manipulate individual bits of information  Load-store machine LD R1; LD R2; ADD R3, R2, R1; SD R3; Figure 1.2 Example of binary information processing
  • 50.
    Binary Logic  Definitionof Binary Logic  Binary logic consists of binary variables and a set of logical operations.  The variables are designated by letters of the alphabet, such as A, B, C, x, y, z, etc, with each variable having two and only two distinct possible values: 1 and 0,  Three basic logical operations: AND, OR, and NOT.
  • 51.
    Binary Logic  TruthTables, Boolean Expressions, and Logic Gates x y z 0 0 0 0 1 0 1 0 0 1 1 1 x y z 0 0 0 0 1 1 1 0 1 1 1 1 x z 0 1 1 0 AND OR NOT x y z x y z z = x • y = x y z = x + y z = x = x’ x z
  • 52.
    Examples for SwitchingCircuits AND OR
  • 53.
    Binary Logic  Logicgates  Example of binary signals 0 1 2 3 Logic 1 Logic 0 Un-define Figure 1.3 Example of binary signals
  • 54.
    Binary Logic  Logicgates  Graphic Symbols and Input-Output Signals for Logic gates: Fig. 1.4 Symbols for digital logic circuits Fig. 1.5 Input-Output signals for gates
  • 55.
    Binary Logic  Logicgates  Graphic Symbols and Input-Output Signals for Logic gates: Fig. 1.6 Gates with multiple inputs
  • 56.
    THANK YOU ForYour Participation!
  • 57.
    အားလုံးကို ကျေးဇူး အထူးတင် ပါတယ်! 3GEP ကျောင်းသူ၊ သားများအားလုံး Covid 19 ကင်းဝေး၍ စိတ်၏ ချမ်းသာခြင်း၊ ကိုယ်၏ကျန်းမာခြင်းများ ဖြင့် ပြည့်စုံကြပါစေဗျာ!
  • 58.
    Assessment Questions!  (1)အောက်ပါပေးထားသော Logic Gates များအား Truth Tables နှင့် LD = Ladder Diagram ဖြင့် ရေးသားဖော်ပြပါ။  (a) AND (b) OR (c) NOT (d) NAND (e) NOR  (f) XOR (g) XNOR  (2) Binary Digits များကို ဖော်ပြပါ။  (3) PLC နှင့်ပတ်သတ်သော Number System (5) မျ ိုးအား ဖော်ပြပါ။  (4) 1 Nibble တွင် မည်မျှ Bits ရှိသနည်း။  (5) အောက်တွင်ဖော်ပြထားမှုများအား Nibble, Byte, Word ခွဲခြားပြပါ။ 1 0 1 1 1 0 0 1 1 0 1 0 1 0 1 1 MSB LSB Bit (a) ? (b) ? (c) ?