Number System

Number System Q1. What is a number system? Ans: A number system is a method of representing numbers using different symbols or digits. It helps us write and perform calculations with numbers in mathematics and computers. Q2. Define the base or radix of the number system. Ans: The base or radix of a number system is the total number of unique digits or symbols used in that number system. For example: Decimal system has base 10 because it uses 10 digits (0–9). Binary system has base 2 because it uses 2 digits (0 and 1). Q3. List out the different types of number systems. Ans: The main types of number systems are: i) Binary Number System (Base 2) ii) Octal Number System (Base 8) iii) Decimal Number System (Base 10) iv) Hexadecimal Number System (Base 16) Q4. What is a hexadecimal number system? Write down the symbols used in the hexadecimal number system. Ans: The hexadecimal number system is a number system with base 16. It is commonly used in computers and digital electronics. It uses the following symbols: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F Here: A = 10 , B = 11 ,C = 12 , D = 13 , E = 14 ,F = 15

Binary to Decimal

1. (1101101)2 = (?)10 Face Value 1 1 0 1 1 0 1 Place Value 26 25 24 23 22 21 20 =1×26+1×25+0×24+1×23+1×22+0×21+1×20 =64+32+0+8+4+0+1 =(109)10 ∴ (1101101)2 = (109)10

2. (101010)2 = (?)10 Face Value 1 0 1 0 1 0 Place Value 25 24 23 22 21 20 =1×25+0×24+1×23+0×22+1×21+0×20 =32+0+8+0+2+0 =(42)10 ∴ (101010)2 = (42)10

3. (1100110)2 = (?)10 Face Value 1 1 0 0 1 1 0 Place Value 26 25 24 23 22 21 20 =1×26+1×25+0×24+0×23+1×22+1×21+0×20 =64+32+0+0+4+2+0 =(102)10 ∴ (1100110)2 = (102)10

4. (1100011)2 = (?)10 Face Value 1 1 0 0 0 1 1 Place Value 26 25 24 23 22 21 20 =1×26+1×25+0×24+0×23+0×22+1×21+1×20 =64+32+0+0+0+2+1 =(99)10 ∴ (1100011)2 = (99)10

5. (111011)2 = (?)10 Face Value 1 1 1 0 1 1 Place Value 25 24 23 22 21 20 =1×25+1×24+1×23+0×22+1×21+1×20 =32+16+8+0+2+1 =(59)10 ∴ (1101101)2 = (59)10

6. (1010101)2 = (?)10 Face Value 1 0 1 0 1 0 1 Place Value 26 25 24 23 22 21 20 =1×26+0×25+1×24+0×23+1×22+0×21+1×20 =64+0+16+0+4+0+1 =(85)10 ∴ (1010101)2 = (85)10

Octal to Decimal

7. (740)8 = (?)10 Face Value 7 4 0 Place Value 82 81 80 =7×82+4×81+0×80 =7x64+4x8+0x1 =448+32+0 =(480)10 ∴ (740)8 = (480)10

8. (570)8 = (?)10 Face Value 5 7 0 Place Value 82 81 80 =5×82+7×81+0×80 =5x64+7x8+0x1 =320+56+0 =(376)10 ∴ (570)8 = (376)10

9. (250)8 = (?)10 Face Value 2 5 0 Place Value 82 81 80 =2×82+5×81+0×80 =2x64+5x8+0x1 =128+40+0 =(168)10 ∴ (250)8 = (168)10

10. (1230)8 = (?)10 Face Value 1 2 3 0 Place Value 83 82 81 80 =1×83+2×82+3×81+0×80 =1x512+2x64+3x8+0x1 =512+128+24+0 =(664)10 ∴ (1230)8 = (664)10

11. (330)8 = (?)10 Face Value 3 3 0 Place Value 82 81 80 =3×82+3×81+0×80 =3x64+3x8+0x1 =192+24+0 =(216)10 ∴ (330)8 = (216)10

12. (2020)8 = (?)10 Face Value 2 0 2 0 Place Value 83 82 81 80 =2×83+0×82+2×81+0×80 =2x512+0x64+2x8+0x1 =1024+0+16+0 =(1040)10 ∴ (2020)8 = (1040)10

Hexadecimal to Decimal

13. (130)16 = (?)10 Face Value 1 3 0 Place Value 162 161 160 =1×162+3×161+0×160 =1x256+3x16+0x1 =256+48+0 =(304)10 ∴ (130)16 = (304)10

14. (1A7)16 = (?)10 Face Value 1 A 7 Place Value 162 161 160 =1×162+A×161+7×160 =1x256+10x16+7x1 =256+160+7 =(423)10 ∴ (1A7)16 = (423)10

15. (240)16 = (?)10 Face Value 2 4 0 Place Value 162 161 160 =2×162+4×161+0×160 =2x256+4x16+0x1 =512+64+0 =(576)10 ∴ (240)16 = (576)10

16. (2AB)16 = (?)10 Face Value 2 A B Place Value 162 161 160 =2×162+A×161+B×160 =2x256+10x16+11x1 =512+160+11 =(683)10 ∴ (2AB)16 = (683)10

17. (4FD)16 = (?)10 Face Value 4 F D Place Value 162 161 160 =4×162+F×161+D×160 =4x256+15x16+13x1 =1024+240+13 =(1277)10 ∴ (4FD)16 = (1277)10

18. (530F)16 = (?)10 Face Value 5 3 0 F Place Value 163 162 161 160 =5x163+3×162+0×161+F×160 =5x4096+3x256+0x16+15x1 =20480+768+0+15 =(21263)10 ∴ (530F)16 = (21263)10

Homework

19. (10011)2 = (?)10 Face Value 1 0 0 1 1 Place Value 24 23 22 21 20 =1×24+0×23+0×22+1×21+1×20 =1×16+0×8+0×4+1×2+1×1 =16+0+0+2+1 =(19)10

20. (10101)2 = (?)10 Face Value 1 0 1 0 1 Place Value 24 23 22 21 20 =1×24+0×23+1×22+0×21+1×20 =1×16+0×8+1×4+0×2+1×1 =16+0+4+0+1 =(21)10

21. (1010)2 = (?)10 =1x23+0x22+1x21+0x20 =1x8+0x4+1x2+0x1 =8+0+2+0 =(10)10

22. (1111)2 = (?)10 =1x23+1x22+1x21+1x20 =1x8+1x4+1x2+1x1 =8+4+2+1 =(15)10

23.(1001101)2 = (?)10 =1x26+0x25+0x24+1x23+1x22+0x21+1x20 =1x64+0x32+0x16+1x8+1x4+0x2+1x1 =64+0+0+8+4+0+1 =(77)10

24. (110010)2 = (?)10 =1x25+1x24+0x23+0x22+1x21+0x20 =1x32+1x16+0x8+0x4+1x2+0x1 =32+16+0+0+2+0 =(50)10

25. (10000001)2 = (?)10 =1x27+0x26+0x25+0x24+0x23+0x22+0x21+1x20 =1x128+0x64+0x32+0x16+0x8+0x4+0x2+1x1 =128+0+0+0+0+0+0+1 =(129)10

26. (10101010)2 = (?)10 =1x27+0x26+1x25+0x24+1x23+0x22+1x21+0x20 =1x128+0x64+1x32+0x16+1x8+0x4+1x2+0x1 =128+0+32+0+8+0+2+0 =(170)10

27. (11010111)2 = (?)10 =1x27+1x26+0x25+1x24+0x23+1x22+1x21+1x20 =1x128+1x64+0x32+1x16+0x8+1x4+1x2+1x1 =128+64+0+16+0+4+2+1 =(215)10

28. (11111100)2 = (?)10 =1x27+1x26+1x25+1x24+1x23+1x22+0x21+1x20 =1x128+1x64+1x32+1x16+1x8+1x4+0x2+0x1 =128+64+32+16+8+4+0+0 =(252)10

29. (1001101101)2 = (?)10 =1x29+0x28+0x27+1x26+1x25+0x24+1x23+1x22+0x21+1x20 =1x512+0x256+0x128+1x64+1x32+0x16+1x8+1x4+0x2+1x1 =512+0+0+64+32+0+8+4+0+1 =(621)10

30. (1111111111)2 = (?)10 =1x29+1x28+1x27+1x26+1x25+1x24+1x23+1x22+1x21+1x20 =1x512+1x256+1x128+1x64+1x32+1x16+1x8+1x4+1x2+1x1 =512+256+128+64+32+16+8+4+2+1 =(1023)10

31. (2074)8 = (?)10 =2×83+0×82+7×81+4×80 =2×512+0×64+7×8+4×1 =1024+0+56+4 =(1084)10

32. (2026)8 = (?)10 =2×83+0×82+2×81+6×80 =2×512+0×64+2×8+6×1 =1024+0+16+6 =(1046)10

33. (456)8 = (?)10 =4×82+5×81+6×80 =4×64+5×8+6×1 =256+40+6 =(302)10

34. (1234)8 = (?)10 =1×83+2×82+3×81+4×80 =2×512+2×64+3×8+4×1 =512+128+24+4 =(668)10

35. (21)8 = (?)10 =2×81+1×80 =2×8+1×1 =16+1 =(17)10

36. (3456)8 = (?)10 =3×83+4×82+5×81+6×80 =3×512+4×64+5×8+6×1 =1536+256+40+6 =(1838)10

37. (2017)8 = (?)10 =2×83+0×82+1×81+7×80 =2×512+0×64+1×8+7×1 =1024+0+8+7 =(1039)10

38. (145)8 = (?)10 =1×82+4×81+5×80 =1×64+4×8+5×1 =64+32+5 =(101)10

39. (421)8 = (?)10 =4×82+2×81+1×80 =4×64+2×8+1×1 =256+16+1 =(273)10

40. (1330)8 = (?)10 =1×83+3×82+3×81+0×80 =1×512+3×64+3×8+0×1 =512+192+24+0 =(728)10

41. (2045)8 = (?)10 =2×83+0×82+4×81+5×80 =2×512+0×64+4×8+5×1 =1024+0+32+5 =(1061)10

42. (2024)8 = (?)10 =2×83+0×82+2×81+4×80 =2×512+0×64+2×8+4×1 =1024+0+16+4 =(1044)10

43. (A27)16 = (?)10 =A×162+2×161+7×160 =10×256+2×16+7×1 =2560+32+7 =(2599)10

44. (CAFE)16 = (?)10 =C×163+A×162+F×161+E×160 =12×4096+10×256+15×16+14×1 =49152+2560+240+14 =(51966)10

45. (DAD)16 = (?)10 =D×162+A×161+D×160 =13×256+10×16+13×1 =3328+160+13 =(3501)10

46. (BED)16 = (?)10 =B×162+E×161+D×160 =11×256+14×16+13×1 =2816+224+13 =(3053)10

47. (ABC)16 = (?)10 =A×162+B×161+C×160 =10×256+11×16+12×1 =2560+176+12 =(2748)10

48. (BBA)16 = (?)10 =B×162+B×161+A×160 =11×256+11×16+10×1 =2816+176+10 =(3002)10

Decimal to Binary

49. (42)10 = (?)2 2 42 Remainder ↑ 2 21 0 2 10 1 2 5 0 2 2 1 2 1 0 0 1 ∴ (42)10 = (101010)2

50. (100)10 = (?)2 2 100 Remainder ↑ 2 50 0 2 25 0 2 12 1 2 6 0 2 3 0 2 1 1 0 1 ∴ (100)10 = (1100100)10

51. (123)10 = (?)2 2 123 Remainder ↑ 2 61 1 2 30 1 2 15 0 2 7 1 2 3 1 2 1 1 0 1 ∴ (123)10 = (1111011)10

52. (456)10 = (?)2 2 456 Remainder ↑ 2 228 0 2 114 0 2 57 0 2 28 1 2 14 0 2 7 0 2 3 1 2 1 1 0 1 ∴ (456)10 = (111001000)2

53. (421)10 = (?)2 2 421 Remainder ↑ 2 210 1 2 105 0 2 52 1 2 26 0 2 13 0 2 6 1 2 3 0 2 1 1 0 1 ∴ (421)10 = (110100101)2

54. (125)10 = (?)2 2 125 Remainder ↑ 2 62 1 2 31 0 2 15 1 2 7 1 2 3 1 2 1 1 0 1 ∴ (125)2 = (1111101)2

Decimal to Octal

55. (45)10 = (?)8 8 45 Remainder ↑ 8 5 5 0 5 ∴ (45)10 = (55)8

56. (98)10 = (?)8 8 98 Remainder ↑ 8 12 2 8 1 4 0 4 ∴ (98)10 = (142)8

57. (156)10 = (?)8 8 156 Remainder ↑ 8 19 4 8 2 3 0 2 ∴ (156)10 = (234)8

58. (512)10 = (?)8 8 512 Remainder ↑ 8 64 0 8 8 0 8 1 0 0 1 ∴ (512)10 = (1000)8

59. (1025)10 = (?)8 8 1025 Remainder ↑ 8 128 1 8 16 0 8 2 0 0 2 ∴ (1025)10 = (2001)8

60. (4096)10 = (?)8 8 4096 Remainder ↑ 8 512 0 8 64 0 8 8 0 8 1 0 0 1 ∴ (4096)2 = (10000)8