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# 1Sem-Basic Electronics Notes-Unit7-Number System

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1st Semester Basic Electronics UNIT 7 Notes - Number System

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### 1Sem-Basic Electronics Notes-Unit7-Number System

1. 1. ODD   SEMESTER   12  BASIC  ELECTRONICS-­‐1-­‐CLASS  NOTES  –  UNIT7  Shivoo  Koteshwar  Professor,  E&C  Department,  PESIT  SC      Number  Systems     • Introduction   • Decimal  System   • Binary,  Octal  and  Hexadecimal  Number  systems   • Additions  and  Subtraction   • Fractional  Number   • Binary  Coded  Decimal  Numbers    Reference  Books:   • Basic  Electronics,  RD  Sudhaker  Samuel,  U  B  Mahadevaswamy,  V.  Nattarsu,  Saguine-­‐Pearson,   2007    UNIT  7:  NUMBER  SYSTEMS:  Introduction,  decimal  system,  Binary,  Octal  and  Hexadecimal  number  systems,  addition  and  subtraction,  fractional  number,  Binary  Coded  Decimal  Numbers          7  Hours      P e o p l e s   E d u c a t i o n   S o c i e t y   S o u t h   C a m p u s   ( w w w . p e s . e d u )
2. 2. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0  Decimal System:When we write decimal (base 10) numbers, we use a positionalnotation system. Each digit is multiplied by an appropriate powerof 10 depending on its position in the number.For example: 843 = 8 x 102 + 4 x 101 + 3 x 100 = 8 x 100 + 4 x 10 + 3 x 1 = 800 + 40 + 3In a positional notation system, the number base is called theradix. Thus, the base ten system that we normally use has a radixof 10. The term radix and base can be used interchangeably. 843 = 843(10) = 84310Other Number Systems: • Binary  System   • Octal  System   • Hexadecimal  System     Shivoo  Koteshwar’s  Notes                                          2                                                                                          shivoo@pes.edu
3. 3. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0   Conversion of Binary to Decimal: Conversion of Decimal to Binary:   Shivoo  Koteshwar’s  Notes                                          3                                                                                          shivoo@pes.edu
4. 4. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0   Conversion of Octal to Decimal: Conversion of Decimal to Octal   Shivoo  Koteshwar’s  Notes                                          4                                                                                          shivoo@pes.edu
5. 5. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0   Conversion of Hexadecimal to Decimal: Conversion of Decimal to Hexadecimal:   Shivoo  Koteshwar’s  Notes                                          5                                                                                          shivoo@pes.edu
6. 6. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0   Conversion of Binary to Hexadecimal:Converting binary to hexadecimal can be done in two steps: 1. Convert binary to decimal 2. Convert decimal to hexadecimalOr you can use this short cut! • Divide into groups for 4 digits • Write the equivalent hexadecimal numberExamples: • 101101012 = 1011 0101 = B 5 101101012 = B516 • 0110101110001100 = 0110 1011 1000 1100 = 6 B 8 C 01101011100011002 = 6B8C16 • 11101101012 = 11 1011 0101 = 3 B 5 11101101012 = 3B516 Conversion of Hexadecimal to Binary:Converting hexadecimal to binary can be done in two steps: 1. Convert hexadecimal to decimal 2. Convert decimal to binaryOr you can use this short cut! • Convert each hexadecimal digit into a group of 4 binary digits • Concatenate allExamples: • 374F16 = 3 7 4 F = 0011 0111 0100 1111 374F 16 = 00110111010011112 • 3B516 = 3 B 5 = 0011 1011 0101 3B5 16 = 0011101101012   Shivoo  Koteshwar’s  Notes                                          6                                                                                          shivoo@pes.edu
7. 7. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0   Conversion of Binary to Octal:Converting binary to octal can be done in two steps: 1. Convert binary to decimal 2. Convert decimal to octalOr you can use this short cut! • Divide into groups for 3 digits • Write the equivalent hexadecimal numberExamples: • 1.1101101012 = 110 110 101 = 6 6 5 1101101012 = 6658 • 101101012 = 10 110 101 = 2 6 5 101101012 = 2658 Conversion of Octal to Binary:Converting octal to binary can be done in two steps: 1. Convert octal to decimal 2. Convert decimal to binaryOr you can use this short cut! • Convert each octal digit into a group of 3 binary digits • Concatenate allExamples: • 6658 = 6 6 5 = 110 110 101 6658 = 1101101012 • 2658 = 2 6 5 = 010 110 101 2658 = 0101101012   Shivoo  Koteshwar’s  Notes                                          7                                                                                          shivoo@pes.edu
8. 8. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0   Conversion of Octal/Hexa to Hexa/Octal:Converting octal/hexa to hexa/octal can be done in two steps: 1. Convert octal/hexa to decimal 2. Convert decimal to hexa/octalOr you can use this short cut! • Conversion from Octal to Hexa o Convert to Binary, regroup 4 bits and write the equivalent Hexa o 2658 = 010 110 101 = 0101101012 = 0 1011 0101 = B516 • Conversion from Hexa to Octal o Convert to Binary, regroup 3 bits, write the equivalent Octal o B516 = 1011 0101 = 101101012 = 10 110 101 = 2658 SUMMARY   Shivoo  Koteshwar’s  Notes                                          8                                                                                          shivoo@pes.edu
9. 9. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0   Binary Addition: Binary Subtraction:   Shivoo  Koteshwar’s  Notes                                          9                                                                                          shivoo@pes.edu
10. 10. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0   Octal Addition: Hexadecimal Addition:   Shivoo  Koteshwar’s  Notes                                          10                                                                                    shivoo@pes.edu
11. 11. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0   Subtraction using Complement: NOTATION • A complement is a number that is used to represent the negative of a given number • When two numbers are to be subtracted, the subtrahend* can either be subtracted directly from the minuend (as we are used to doing in decimal subtraction) or, the complement of the subtrahend can be added to the minuend to obtain the difference. • When the latter method is used, the addition will produce a high-order (leftmost) one in the result (a "carry"), which must be dropped. • This is how the computer performs subtraction: it is very efficient for the computer to use the same "add" circuitry to do both addition and subtraction; thus, when the computer "subtracts", it is really adding the complement of the subtrahend to the minuendExample: Subtract 4589 - 322, using complements   Shivoo  Koteshwar’s  Notes                                          11                                                                                    shivoo@pes.edu
12. 12. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0   Binary Subtraction using compliment: • Match the bit size of the minuend and subtrahend before finding out its complement • Compute the ones complement of the subtrahend by subtracting each digit of the subtrahend by 1. A shortcut for doing this is to simply reverse each digit of the subtrahend - the 1s become 0s and the 0s become 1’s • Add 1 to the ones complement of the subtrahend to get the twos complement of the subtrahend • Add the twos complement of the subtrahend to the minuend and drop the high-order 1. This is your differenceExample: • 1012 -112 (5 – 3 = 2) • Here make sure you add a 0 to subtrahend 11 and make it 011 • 11  011 • Find 1’s complement: 011  100 • Find 2’s complement = 1’s complement +1: 100 101 • 101 +101 = 1010 • Ignore MSB, so answer is 102 = 2   Shivoo  Koteshwar’s  Notes                                          12                                                                                    shivoo@pes.edu
13. 13. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0  Note: • When you are using 2’s complement’s methods note that 2’s complement is a signed representation. So when you get a 1 at the MSB after subtracting, it just means that it is a negative number. o So when you want to find the equivalent number, don’t use MSB as a value bit. Use it as an indicator for a negative number o Equivalent positive number would be a 2’s complement of the same with a negative sign attached to itExample:   Shivoo  Koteshwar’s  Notes                                          13                                                                                    shivoo@pes.edu
14. 14. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0   Octal Subtraction using compliment:   Shivoo  Koteshwar’s  Notes                                          14                                                                                    shivoo@pes.edu
15. 15. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0   Hexadecimal Subtraction using compliment:   Shivoo  Koteshwar’s  Notes                                          15                                                                                    shivoo@pes.edu
16. 16. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0   Fractions: Binary to DecimalDecimal representation: • 953.78 = 9 x 102 + 5 x 101 + 3 x 100 + 7 x 10-1 + 8 x 10-2 = 900 + 50 + 3 + .7 + .08 = 953.78Similarly, binary also can be represented as above with the rightbase considered: • 1011.112 = 1x23 + 0x22 + 1x21 + 1x20 + 1x2-1 + 1x2-2 = 8 + 0 + 2 + 1 + 0.5 + 0.25 = 11.75 Fractions: Decimal to Binary • No change in the integer part of the number conversion • Multiply the fractional part with the base, 2 . Note the integer part of this product and multiply the fractional part with the base, 2 …repeat the steps until you get a repeat of bits or till the required accuracy • Typically 4 bits at least is computed (4 integer part from 4 products)   Shivoo  Koteshwar’s  Notes                                          16                                                                                    shivoo@pes.edu
17. 17. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0   Fractions: Octal to Decimal • 0.4138 = 4 x 8-1 + 1 x 8-2 + 3 x 8-3 = 4(0.125) + 1(0.015625) +3(0.001953125) = 0.5 + 0.015625 + 0.005859375 = 0.521484375 Fractions: Decimal to Octal Fractions: Hexadecimal to Decimal • 0.41316 = 4 x 16-1 + 1 x 16-2 + 3 x 16-3 = 4(0.0625) + 1(0.00390625)+3(0.000244140625) = 0.25 + 0.00390625 + 0.000732421875 = 0.254639 Fractions: Decimal to Hexadecimal   Shivoo  Koteshwar’s  Notes                                          17                                                                                    shivoo@pes.edu
18. 18. Number  Systems  (1st  Semester)                                                                                                            UNIT  7  Notes  v1.0   SUMMARY Binary Coded Decimal Number (BCD)It is possible to represent decimal numbers simply by encodingeach decimal digit in binary for – called binary-coded-decimal(BCD)Because there are 10 digits to represent, it is necessary to use fourbits per digit • From 0=0000 to 9=1001 • (01111000)BCD =(78)10   Shivoo  Koteshwar’s  Notes                                          18                                                                                    shivoo@pes.edu