1 / 35

Numbering Systems

Numbering Systems. Introduction to Numbering Systems. Decimal System We are all familiar with the decimal number system (Base 10). Some other number systems that we will work with are: Binary  Base 2 Octal  Base 8 Hexadecimal  Base 16.

leonburton
Download Presentation

Numbering Systems

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  1. Numbering Systems

  2. Introduction to Numbering Systems • Decimal System • We are all familiar with the decimal number system (Base 10). Some other number systems that we will work with are: • Binary  Base 2 • Octal  Base 8 • Hexadecimal  Base 16

  3. Characteristics of Numbering Systems • The digits are consecutive. • The number of digits is equal to the size of the base. • Zero is always the first digit. • 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 are determined by the implicit positional values of the digits.

  4. Significant Digits Binary: 11101101 Most significant digit Least significant digit Hexadecimal: 1D63A7A Most significant digit Least significant digit

  5. Binary System

  6. Binary Number System • Also called the “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

  7. Binary Decimal

  8. Decimal to Binary Conversion • The easiest way to convert a decimal number to its binary equivalent is to use the 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

  9. An algorithm for finding the binary representation of a positive integer

  10. Division Algorithm Convert 67 to its binary equivalent: 6710 = x2 Step 1: 67 / 2 = 33 R 1Divide 67 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

  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

  12. Multiplication Algorithm Convert (10101101)2 to its decimal equivalent: Binary 1 0 1 0 1 1 0 1 Positional Values x x x x x x x x 27 26 25 24 23 22 21 20 128 + 32 + 8 + 4 + 1 Products 17310

  13. BINARY TO DECIMAL CONVERTION • Any binary number can be converted to its decimal equivalent simply by summing together the weights of the various positions in the binary number which contain 1. • Example 1: convert 110112 to decimal value Solve: 1 1 0 1 1 = 16+8+2+1 =

  14. Example 2 : Convert 101101012 to decimal value Solve: 1 0 1 1 0 1 1 0 = 128 + 32 + 16 + 4 + 1 = You should noticed the method is find the weights (i.e., powers of 2) for each bit position that contains 1, and then to add them up.

  15. 3 25 1 12 6 2 2 2 2 2 DECIMAL TO BINARY CONVERTION Example : convert 2510 to binary 2510 = ?2 Solve = = 12 balance 1 LSB = 6 balance 0 = 3 balance 0 = 1 balance 1 = 0 balance 1 MSB . . . Answer = 110012

  16. Octal System

  17. Octal Number System • Also known as the Base 8 System • Uses digits 0 - 7 • Readily converts to binary • Groups of three (binary) digits can be used to represent each octal digit • Also uses multiplication and division algorithms for conversion to and from base 10

  18. Octal Decimal

  19. OCTAL TO DECIMAL CONVERTION • Convert from octal to decimal by multiplying each octal digit by its positional weight. Example 1: Convert 1638 to decimal value Solve = = 1 x 64 + 6 x 8 + 3 x 1 = 11510 Example 2: Convert 3338 to decimal value Solve = = 3 x 64 + 3 x 8 + 3 x 1 = 21910

  20. 5 359 44 8 8 8 DECIMAL TO OCTAL CONVERTION • Convert from decimal to octal by using the repeated division method used for decimal to binary conversion. • Divide the decimal number by 8 • The first remainder is the LSB and the last is the MSB. Example : convert 35910 to Decimal Value 35910 = ?8 Solve = = 44 balance 7 LSB = 5 balance 4 = 0 balance 5 MSB . . . Answer = 5478

  21. Octal Binary

  22. OCTAL TO BINARY CONVERTION • Convert from octal to binary by converting each octal digit to a three bit binary equivalent • Convert from binary to octal by grouping bits in threes starting with the LSB. • Each group is then converted to the octal equivalent • Leading zeros can be added to the left of the MSB to fill out the last group.

  23. BINARY TO OCTAL CONVERSION • Can be converted by grouping the binary bit in group of three starting from LSB • Octal is a base-8 system and equal to two the power of three, so a digit in Octal is equal to three digit in binary system.

  24. Hexadecimal System

  25. Hexadecimal Number System • Base 16 system • Uses digits 0-9 & letters A,B,C,D,E,F • Groups of four bitsrepresent eachbase 16 digit

  26. HexadecimalDecimal

  27. Decimal to Hexadecimal Conversion Convert 83010 to its hexadecimal equivalent: 830 / 16 = 51 R14 51 / 16 = 3 R3 3 / 16 = 0 R3 = E in Hex 33E16

  28. Hexadecimal to Decimal Conversion Convert 3B4F16 to its decimal equivalent: Hex Digits 3 B 4 F x x x x Positional Values 163 162 161 160 12288 +2816 +64 +15 Products 15,18310

  29. HexadecimalBinary

  30. 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

  31. Binary Arithmetic

  32. The binary addition facts

  33. Binary Arithmetic • The individual digits of a binary number are referred to as bits • Each bit represents a power of two 01011 = 0 • 24 + 1 • 23 + 0 • 22 + 1 • 21 + 1 • 20 = 11 00010 = 0 • 24 + 0 • 23 + 0 • 22 + 1 • 21 + 0 • 20 = 2 00010 + 01011 01101 2 + 11 13 Equivalent decimaladdition Binary addition

More Related