Numbering Systems

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.

24. 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.

26. Hexadecimal System

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

28. HexadecimalDecimal

29. 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

30. 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

31. HexadecimalBinary

32. 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

33. Binary Arithmetic

34. The binary addition facts

35. 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