Number systems r002
- 1. Number Systems Prepared By M Prasannakumar Asst.Professor Dept of Electronics
- 2. • In digital electronics, the number system is used for representing the information. • The number system has different bases. • The base or radix of the number system is the total number of the digit used in the number system. • Suppose if the number system representing the digit from 0 – 9 then the base of the system is the 10. Types of Number Systems Some of the important types of number system are Decimal Number System Binary Number System Octal Number System Hexadecimal Number System These number systems are explained below in details.
- 3. 1. Decimal Number Systems The number system is having digit 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 this number system is known as a decimal number system because total ten digits are involved. The base of the decimal number system is 10.
- 4. 2. Binary Number Systems • The modern computers do not process decimal number they work with another number system known as a binary number system. • The Binary Number System uses only two digits 0 and1. • The base of binary number system is 2 because it has only two digit 0 and 1. • The digital electronic equipments are works on the binary number system and hence the decimal number system is converted into binary system.
- 5. 3. Octal Numbers • The base of a number system is equal to the number of digits used, i.e., for decimal number system the base is ten while for the binary system the base is two. • The octal system has the base of eight as it uses eight digits 0, 1, 2, 3, 4, 5, 6, 7. 4. Hexadecimal Numbers • These numbers are used extensively in microprocessor work. • The hexadecimal number system has a base of 16, and hence it consists of the following sixteen number of digits. • 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, A, B, C, D, E, F
- 6. Common Number Systems System Base Symbols Used by humans? Used in computers? Decimal 10 0, 1, … 9 Yes No Binary 2 0, 1 No Yes Octal 8 0, 1, … 7 No No Hexa- decimal 16 0, 1, … 9, A, B, … F Yes No
- 7. Quantities/Counting (1 of 3) Decimal Binary Octal Hexa- decimal 0 000 0 0 1 0001 1 1 2 0010 2 2 3 011 3 3 4 100 4 4 5 101 5 5 6 110 6 6 7 111 7 7 p. 33
- 8. Quantities/Counting (2 of 3) Decimal Binary Octal Hexa- decimal 8 1000 10 8 9 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
- 9. Quantities/Counting (3 of 3) Decimal Binary Octal Hexa- decimal 16 10000 20 10 17 10001 21 11 18 10010 22 12 19 10011 23 13 20 10100 24 14 21 10101 25 15 22 10110 26 16 23 10111 27 17 Etc.
- 10. Conversion Among Bases • The possibilities: Hexadecimal Decimal Octal Binary pp. 40-46
- 11. Quick Example 2510 = 110012 = 318 = 1916 Base
- 12. Decimal to Decimal (just for fun) Hexadecimal Decimal Octal Binary Next slide…
- 13. 125410 => 4 x 100 = 4 5 x 101 = 50 2 x 102 = 200 1 x 103 =1000 1254 Base Weight
- 14. Binary to Decimal Hexadecimal Decimal Octal Binary
- 15. Binary to Decimal • Technique – Multiply each bit by 2n, where n is the “weight” of the bit – The weight is the position of the bit, starting from 0 on the right – Add the results
- 16. Example 1010112 => 1 x 20 = 1 1 x 21 = 2 0 x 22 = 0 1 x 23 = 8 0 x 24 = 0 1 x 25 = 32 4310 Bit “0”
- 17. Decimal to Binary Hexadecimal Decimal Octal Binary
- 18. Decimal to Binary • Technique – Divide by two, keep track of the remainder – First remainder is bit 0 (LSB, least-significant bit) – Second remainder is bit 1 – Etc.
- 19. Example 12510 = ?2 2 125 62 1 2 31 0 2 15 1 2 7 1 2 3 1 2 1 1 12510 = 11111012
- 20. Hexadecimal to Decimal Hexadecimal Decimal Octal Binary
- 21. Hexadecimal to Decimal • Technique – Multiply each bit by 16n, where n is the “weight” of the bit – The weight is the position of the bit, starting from 0 on the right – Add the results
- 22. Example ABC16 => C x 160 = 12 x 1 = 12 B x 161 = 11 x 16 = 176 A x 162 = 10 x 256 = 2560 274810
- 23. Decimal to Hexadecimal Hexadecimal Decimal Octal Binary
- 24. Decimal to Hexadecimal • Technique – Divide by 16 – Keep track of the remainder
- 25. Example 123410 = ?16 123410 = 4D216 16 1234 77 2 16 4 13 = D
- 26. Octal to Decimal Hexadecimal Decimal Octal Binary
- 27. Octal to Decimal • Technique – Multiply each bit by 8n, where n is the “weight” of the bit – The weight is the position of the bit, starting from 0 on the right – Add the results
- 28. Example 7248 => 4 x 80 = 4 2 x 81 = 16 7 x 82 = 448 46810
- 29. Decimal to Octal Hexadecimal Decimal Octal Binary
- 30. Decimal to Octal • Technique – Divide by 8 – Keep track of the remainder
- 31. Example 123410 = ?8 8 1234 154 2 8 19 2 8 2 3 123410 = 23228
- 32. Binary to Hexadecimal Hexadecimal Decimal Octal Binary
- 33. Binary to Hexadecimal • Technique – Group bits in fours, starting on right – Convert to hexadecimal digits
- 34. Example 10101110112 = ?16 0010 1011 1011 2 B B 10101110112 = 2BB16
- 35. Hexadecimal to Binary Hexadecimal Decimal Octal Binary
- 36. Hexadecimal to Binary • Technique – Convert each hexadecimal digit to a 4-bit equivalent binary representation
- 37. Example 10AF16 = ?2 1 0 A F 0001 0000 1010 1111 10AF16 = 00010000101011112
- 38. Binary to Octal Hexadecimal Decimal Octal Binary
- 39. Binary to Octal • Technique – Group bits in threes, starting on right – Convert to octal digits
- 40. Example 10110101112 = ?8 1 011 010 111 1 3 2 7 10110101112 = 13278
- 41. Octal to Binary Hexadecimal Decimal Octal Binary
- 42. Octal to Binary • Technique – Convert each octal digit to a 3-bit equivalent binary representation
- 43. Example 7058 = ?2 7 0 5 111 000 101 7058 = 1110001012
- 44. Octal to Hexadecimal Hexadecimal Decimal Octal Binary
- 45. Octal to Hexadecimal • Technique – Use binary as an intermediary
- 46. Example 10768 = ?16 1 0 7 6 001 000 111 110 2 3 E 10768 = 23E16
- 47. Hexadecimal to Octal Hexadecimal Decimal Octal Binary
- 48. Hexadecimal to Octal • Technique – Use binary as an intermediary
- 49. Example 1F0C16 = ?8 1 F 0 C 0001 1111 0000 1100 1 7 4 1 4 1F0C16 = 174148
- 50. Exercise – Convert ... Don’t use a calculator! Decimal Binary Octal Hexa- decimal 33 1110101 703 1AF Skip answer Answer
- 52. Exercise – Convert … Decimal Binary Octal Hexa- decimal 33 100001 41 21 117 1110101 165 75 451 111000011 703 1C3 431 110101111 657 1AF Answer
- 53. Common Powers (1 of 2) • Base 10 Power Preface Symbol 10-12 pico p 10-9 nano n 10-6 micro 10-3 milli m 103 kilo k 106 mega M 109 giga G 1012 tera T Value .000000000001 .000000001 .000001 .001 1000 1000000 1000000000 1000000000000
- 54. Common Powers (2 of 2) • Base 2 Power Preface Symbol 210 kilo k 220 mega M 230 Giga G Value 1024 1048576 1073741824 • What is the value of “k”, “M”, and “G”? • In computing, particularly w.r.t. memory, the base-2 interpretation generally applies
- 55. Fractions • Binary to decimal pp. 46-50 10.1011 => 1 x 2-4 = 0.0625 1 x 2-3 = 0.125 0 x 2-2 = 0.0 1 x 2-1 = 0.5 0 x 20 = 0.0 1 x 21 = 2.0 2.6875
- 56. Fractions • Decimal to binary p. 50 3.14579 .14579 x 2 0.29158 x 2 0.58316 x 2 1.16632 x 2 0.33264 x 2 0.66528 x 2 1.33056 etc. 11.00100
- 57. Fractions • Hex decimal to decimal: B0. A031 => 1 x 16-4 = 0.0001 3 x 16-3 = 0.0007 0 x 16-2 = 0.0000 10 x 16-1 = 0.6250 0 x 160 = 0.0 11 x 161 =176. 176.6258
- 58. Fractions • Decimal to Hex Decimal 32.14579 .14579 x 16 2.33264 x 16 5.32224 x 16 5.15584 x 16 2.49344 x 16 7.89504 x 16 14 .32064 etc. 20.25527E
- 59. Fractions • Octal to decimal: 27.1031 => 1 x 8-4 = 0.0002 3 x 8-3 = 0.005 0 x 8-2 = 0.000 1 x 8-1 = 0.125 7 x 80 = 7. 2 x 81 = 16. 23.1302
- 60. Fractions • Decimal to Octal 32.14579 .14579 x 8 1.16632 x 8 1.33056 x 8 2.64448 x 8 5.15584 x 8 1.24672 x 8 1 .97376 etc. 40.112511
- 61. Fractional Binary to Hexa- decimal (a) 11110.010112 Solution: 11110.010112 = 0001 1110 . 0101 1000 = 1E.5816 Therefore, 11110.010112 = 1E.5816
- 62. Fractional Hexa-decimal to Binary (b) BA2.23C16 Solution: = B A 2 . 2 3 C = 1011 1010 0010 . 0010 0011 11002 = 101110100010.0010001111 Hence the required binary equivalent is 101110100010 . 0010001111.
- 63. Fractional Binary to Octal (a) 111101.011012 Solution: 111101.0110102 = 75.328 Hence the required octal equivalent is 75.32.
- 64. Fractional Octal to Binary (b) 64.1758 Solution: = 6 4 . 1 7 5 = 110 100 . 001 111 101 = 110100.0011111012 Hence the required binary number is 110100.001111101.
- 65. Chapter 1 65 Example: Fractional Octal to Hexadecimal via Binary 1. Convert octal to binary 2. Use groups of four binary bits and express them as hexadecimal digits • Example: Octal Binary Hexadecimal (6 3 5 . 1 7 5)8 110 011 101. 001 111 101 000110011101. 001111101000 = (1 9 D . 3 E 8)16 Represent Octal in binary Group into 4 bit groups for both the integer and fraction parts, starting at the radix point Append leading 0’s to the left of integer part and trailing 0’s to the right of the fraction part as needed Express each group of 4 bits in hex Appended 0’s
- 66. Example: Fractional Hexadecimal to Octal via Binary 1. Convert Hexadecimal to binary 2. Use groups of three binary bits and express them as octal digits • Example: Hexadecimal Binary Octal • (6 3 5 . 1 7 5)16 0110 0011 0101. 0001 0111 0101 =011 000 110 101. 000 101 110 101 = (3065.0565)16
- 67. Exercise – Convert ... Don’t use a calculator! Decimal Binary Octal Hexa- decimal 29.8 101.1101 3.07 C.82 Skip answer Answer
- 68. Exercise – Convert … Decimal Binary Octal Hexa- decimal 29.8 11101.110011… 35.63… 1D.CC… 5.8125 101.1101 5.64 5.D 3.109375 11.000111 3.07 3.1C 12.5078125 1100.10000010 14.404 C.82 Answer