Number system
- 1. NUMBER SYSTEM Dr. (Mrs.) Gargi Khanna Associate Professor Electronics & Communication Engg. Deptt.. National Institute of Technology Hamirpur (HP) Chapter-I
- 2. INTRODUCTION Decimal number system (Base 10). Some other number systems : Number System Base/Radix No of possible Digits Decimal 10 10 Binary 2 2 Octal 8 8 Hexadecimal 16 16 The number system with weights on position is called weighted number system. e.g. Binary, Octal, Decimal, etc. Non-weighted number system e.g. gray code excess-3 code G.Khanna, NITH
- 3. Characteristics of Numbering Systems The number of digits is equal to the size of the base. Zero is always the first digit and digits are consecutive. The base number is never a digit. When 1 is added to the largest digit, a sum of zero and a carry of one results. Numeric values determined by sum of the each digit multiplied by positional values of the digits. G.Khanna, NITH
- 4. Decimal Number System Possible digits 0,1,2,3,4,5,6,7,8,9 Number d3d2 d1 d0. d-1d-2 (Integer) (fractional) D = d3×103+d2×102 + d1×101 +d0×100 + d -1×10-1 +d -2×10-2 The value of the number is the sum of each digit multiplied by the corresponding power of the radix G.Khanna, NITH
- 5. Significant Digits Binary: 1101101 Most significant digit Least significant digit Decimal :4566 Hexadecimal: 196CA7A Most significant digit Least significant digit G.Khanna, NITH
- 6. Binary Number System “Base 2 system” 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 G.Khanna, NITH
- 7. Computer perform all of their operations using the binary (base 2). – Program code and data are stored and manipulated in binary. – Each digit in a binary number is known as a bit (value 0 or 1). – Bits are commonly stored and manipulated in groups of: • 8 bit: Byte. • 16 bit : Halfword. • 32 bit: Word. • 64 bit: Doubleword G.Khanna, NITH
- 8. Binary Numbering Scale Base 2 Number Base 10 Equivalent Power Positional Value 000 0 20 1 001 1 21 2 010 2 22 4 011 3 23 8 100 4 24 16 101 5 25 32 110 6 26 64 111 7 27 128 G.Khanna, NITH
- 9. Decimal to Binary Conversion 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 G.Khanna, NITH
- 10. Division Algorithm Convert 67 to its binary equivalent: 6710 = x2 Step 1: 67 / 2 = 33 R 1 Divide 64 by 2. Record quotient in next row Step 2: 33 / 2 = 16 R 1 Again divide by 2; record quotient in next row Step 3: 16 / 2 = 8 R 0 Repeat again Step 4: 8 / 2 = 4 R 0 Repeat again Step 5: 4 / 2 = 2 R 0 Repeat again Step 6: 2 / 2 = 1 R 0 Repeat again Step 7: 1 / 2 = 0 R 1 STOP when quotient equals 0 1 0 0 0 0 1 12 G.Khanna, NITH
- 11. 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 G.Khanna, NITH
- 12. Multiplication Algorithm Convert (10101111)2 to its decimal equivalent: Binary 1 0 1 0 1 1 1 1 Positional Values xxxxxxxx 2021222324252627 128 + 32 + 8 + 4 +2+1Products 17510 G.Khanna, NITH
- 13. Octal Number System Base 8 System Uses symbols 0 - 7 Ease of convertion to binary Groups of three binary bits can be used to represent each octal symbol Multiplication and division algorithms for conversion to and from base 10 G.Khanna, NITH
- 14. Decimal to Octal Conversion Convert 42910 to its octal equivalent: 429 / 8 = 53 R 5 Divide by 8; R is LSD 53 / 8 = 6 R 5 Divide Q by 8; R is next digit 6 / 8 = 0 R 6 Repeat until Q = 0 6558 G.Khanna, NITH
- 15. Octal to Decimal Conversion Convert 6538 to its decimal equivalent: 6 5 3 xxx 82 81 80 384 + 40 + 3 42710 Positional Values Products Octal Digits G.Khanna, NITH
- 16. Octal to Binary Conversion Each octal number converts to 3 binary digits 475.038 =(100111101.000011) 2 To convert 6538 to binary, just substitute code: 6 5 3 110 101 011 G.Khanna, NITH
- 17. 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 G.Khanna, NITH
- 18. Decimal to Hexadecimal Conversion Convert 83110 to its hexadecimal equivalent: 831 / 16 = 51 R 15 51 / 16 = 3 R 3 3 / 16 = 0 R 3 33F16 = F in Hex G.Khanna, NITH
- 19. Hexadecimal to Decimal Conversion Convert (3B4A)16 to its decimal equivalent: Hex Digits 3 B 4 F xxx 163 162 161 160 12288 +2816 + 64 +10 15,17810 Positional Values Products x G.Khanna, NITH
- 20. 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 G.Khanna, NITH
- 21. Convert 0111001010101111011010112 to hex using the 4-bit substitution code : 0111 0010 1010 1111 0110 1011 Substitution Code 7 2 A F 6 B 76AF6B16 G.Khanna, NITH
- 22. Substitution code can also be used to convert binary to octal by using 3-bit groupings: 010 101 101 010 111 001 101 010 Substitution Code 2 5 5 2 7 1 5 2 255271528 G.Khanna, NITH
- 23. G.Khanna, NITH Number Decimal Binary Octal Hexadecimal ------ ------- ------- ----- ----------- Zero 0 0 0 0 One 1 1 1 1 Two 2 10 2 2 Three 3 11 3 3 Four 4 100 4 4 Five 5 101 5 5 Six 6 110 6 6 Seven 7 111 7 7 Eight 8 1000 10 8 Nine 9 1001 11 9 Ten 10 1010 12 A Eleven 11 1011 13 B Twelve 12 1100 14 C Thirteen 13 1101 15 D Fourteen 14 1110 16 E Fifteen 15 1111 17 F Sixteen 16 10000 20 10 Seventeen 17 10001 21 11 Eighteen 18 10010 22 12 Nineteen 19 10011 23 13 Twenty 20 10100 24 14
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