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Number System

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Published in: Basic Computer
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Ashish S / Indore

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  1. Number Systems Ashish Soni
  2. Thought of the Day Individual commitment to group effort --- that is what makes a team work, a company work, a civilization work. -- Vince Lombardi Number Systems
  3. To be Discussed Base or Radix. Binary Numbers. Octal Numbers. ' Hexadecimal Numbers. ' Inter-system Conversions. Number Systems
  4. Digital Circuits Operates with binary numbers—only two states : — High (1) or Low (O) - True or False ' Normally, in a — O represents O volts, and — 1 represents 5 volts. ' In reverse situation, it is 29 2017 Number Systems
  5. Number System We express a number in any or the base is , For A number with radix X, MSD + b (X) 4 + b. (-0-2 + + x. 29 2017 LSD Number Systems
  6. Binary System •Only two digits (or bits) - 0 and 1. •The position of 0 or 1 in a number indicates its " •For example, in decimal s stem, 29 2017 1 x 102+9 x 101+8 x 100 Units Hundreds Tens Positional Weights Number Systems 6
  7. Similarly, in binary system, (1 98)10 (11000110)2 = Ix27 +1x26 +0x25 +0x24 +0x23 +1x21 +0x20 = 128 + 64 +4+2+0 = 198 A group of 4 and 8 bits is called a and a •With n bits, you can represents highest number 2n -l numbers. 29 2017 Number Systems
  8. Decimal-Binary Conversion Double-dabble method : Example . Convert (53.625)10 into binary. •Take integer and fraction part separately. 29 2017 Number Systems
  9. step .1 Step 2 The integer is 53, The fraction is 0 • 625, Integer conversion: Division 2) 2) 2) 2) 2) 2) 2) 53 26 13 6 3 O Generated remainder -9 1 -.9 MSB
  10. Generated Multiplication o. 625x2= 0.250x2= o. ooox2= integer 1 MSB 1-25+ 0 0 50-4 1 I 00-—5 o 0 * 00 -.........> Hence, (53.625)10 29 2017 (110101.101)2 Number Systems
  11. decimal equivalent. First, the integer part Ix21- Ix22= Ix23= ox24= Ix25 29 2 4 8 32 47
  12. Similarly, the fraction part 0-1101 I—Ix2 - 0-0625 ox 2-3 0000 - 0-2500 I x 21 - -0-5000 8125 Hence, (110101.101)2 = (47.8125)10 29 2017 Number Systems 12
  13. Octal Number System The base is 8. Symbols : O, 1, 2, 3, 4, 5, 6, and 7. ' Positional weights : 80=1 81=8 83 = 256 82 64 29 2017 8-1 = 1/8 8-2 = 1/64 8-3 = 1/256 Number Systems
  14. Example . Convert (444.456)10 to its octal equivalent. Integer part Division 8) 55 Generated remainder 4 7 6 29 2017 Number Systems 14
  15. Fraction Part : Multiplication 0. 456 x 8— 0-47'2x8= 0 776x8= 3 • 648 5 184 472 3 776 6 208 Generated integer 3 5 3 6 The process is terminated when significant digits are obtained. Thus, the octal equivalent of (444-456)10 is (674.35136)8 29 2017 Number Systems
  16. Example . Convert (237)8 and (120)8 to their decimal numbers. = +3x81 +7 x8 (237)g =2x64+3x8+7xl 128+24+7 = (159)10 (120)8 = Ix82 +2x81 +0x8 =lx64+2x8+Oxl = 64+16+0 = (80)10 29
  17. What is the conversion of (293)8 into decimal number ? The Question is wrong. Ans. . 29 2017 Number Systems 17
  18. Octal-Binary Conversion 'Replace each digit by its binary 3-bit equivalent. For example, 29 2017 2-011 rut exarnple, (376)8 7 6 -011 111 110 (011 Ill 110b . For converting a binary nt Number Systems 18
  19. 61 101 010 L LOC ????? &6 ????? SW91SÅS 19qwnN- 0 ()1 = 1010101)) (1001 ()11 UO!Se.löAUOD 8!EUÅ-IQO0
  20. Hexadecimal Numbers ' Radix of 16. Sixteen symbols: ' Positional weights : 163 162 161 160 29 2017 Number Systems 20
  21. Convert (115)10 to hexadecimal. Example . Division 16) 115 16) 7 16) Hence, (115)10 = (73)16 29 2017 Remainder 3 7 Number Systems 21
  22. Exercise : Convert (235)10 to hexadecimal. Ans : (235) 29 2017 = (EB)16 Number Systems 22
  23. Inverse Conversion -(A3B)16 A3BH = +3x161 +1 =10x256+3x16+11xl = 2560+48+11 = (2619)10 Convert 2F3H into decimal number Exercise . (755)10 Ans. . 29 2017 Number Systems 23
  24. Hexadecimal-Binary Conversion •Each digit replaced by its 4-bit binary equivalent. (2D5)16 D 5 = 0010 1101 0101 16 = (0010 1101 01002 . The reverse procedure is use = 111 7 29 2017 1011 B 0101 5 24
  25. Hexadecimal-Octal Conversion (i) Convert the given hexadecimal number to its binary equivalent. (ii) Form groups of 3 bits, starting from the LSB (least significant digit). (iii) Write the equivalent octal number for each group of 3 bits. (0100 0111b (47)16 29 2017 = (01 000 111)2 = (107)8 Number Systems 25
  26. Reverse conversiOn Convert (32)8 to hexadecimal Exercise . Solution . 29 2017 (32)8 = (011 010)2 = (01 1010)2 Number Systems 26
  27. Review Base or Radix. Binary Numbers. Octal Numbers. ' Hexadecimal Numbers. ' Inter-system Conversions. 29 2017 Number Systems 27

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