Number system....

1. University Computer Center Topic:- Number System Presented by:- Mohammed Shoyeb Submitted to:- Mrs. Priyanka ma’am

2. Objective Understand the concept of number systems. Distinguish between non-positional and positional number systems. Describe the decimal, binary, OCTAL and HEXADECIMAL system. Convert a number in binary, octal or hexadecimal to a number in the decimal system. Convert a number in the decimal system to a number in binary, octal and hexadecimal. Convert a number in binary to octal and vice versa. Convert a number in binary to hexadecimal and vice versa.

3. Introduction A Number System (or system of numeration) is a writing system for expressing numbers, that is a mathematical notation for representing numbers of a given set, using digits or other symbols in a consistent manner. Ideally, a numeral system will Represent a useful set of numbers (e.g. all integers, or rational numbers) Give every number represented a unique representation (or at least a standard representation) Reflect the algebraic and arithmetic structure of the numbers.

4. Types of Number System Positional Number Non-Positional Number Positional Number - In a Positional Number System there are only a fe w symbols called represent different values, depending on the position they occupy in a number. The value of each digit in such a number is determined by three considerations The digit itself The position of the digit in the number The base of the number system(where base is defined as the total number of digits available in the number system) Non-positional Number - In This system we have symbols such as I for 1,II for 2,III for 3 etc. Each symbols represents the same value regardless of its position in a number and to find the value of a number.

5. Types of positonal Number System

6. The Binary System(base 2) The word binary is derived from the Latin root bini (or two by two). In this system the base b = 2 and we use only two symbols S ={1,0} The symbols in this system are often referred to as binary digits or bits (binary digit).

7. The Decimal System(base 10) The word decimal is derived from the Latin root decem (ten). In this system the base b = 10 and we use ten symbols s={0,1,2,3,4,5,6,7,8,9} Example:-

8. The Octal System(base 8) The word octal is derived from the Latin root octo (eight). In this system the base b = 8 and we use eight symbols to represent a number. The set of symbols is S={0,1,2,3,4,5,6,7} The base of the octal number system is eight , so each position of the octal number represents a successive power of eight. From right to left

9. The Hexadecimal System(base 16) The word hexadecimal is derived from the Greek root hex (six) and the Latin root decem (ten). In this system the base b = 16 and we use sixteen symbols to represent a number. The set of symbols is S={0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F} Note that the symbols A, B, C, D, E, F are equivalent to 10, 11, 12, 13, 14, and 15 respectively. The symbols in this system are often referred to as hexadecimal digits.

10. 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 No No

11. Quantities/Counting (1 of 2) Decimal Binary Octal Hexa - decimal 1 1 1 1 2 10 2 2 3 11 3 3 4 100 4 4 5 101 5 5 6 110 6 6 7 111 7 7

12. Quantities/Counting (2 of 2) 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 Etc

13. Conversion Among Bases The possibilities: Decimal Octal Binary Hexadecimal

14. Quick Example 25 10 = 11001 2 = 31 8 = 19 16 Base

15. Binary to Decimal Technique Multiply each bit by 2 n , 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 101011 2 => 1 x 2 = 1 1 x 2 1 = 2 0 x 2 2 = 0 1 x 2 3 = 8 0 x 2 4 = 0 1 x 2 5 = 32 43 10

17. Octal to Decimal Technique Multiply each bit by 8 n , 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

18. Example 724 8 => 4 x 8 = 4 2 x 8 1 = 16 7 x 8 2 = 448 468 10

19. Hexadecimal to Decimal Technique Multiply each bit by 16 n , 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

20. Example ABC 16 => C x 16 = 12x1 = 12 B x 16 1 = 11x16 = 176 A x 16 2 = 10x256= 2560 2748 10

21. 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.

22. Example 125 10 = ? 2 2 125 62 1 2 31 0 2 15 1 2 7 1 2 3 1 2 1 1 2 1 1 125 10 = 1111101 2

23. Octal to Binary Technique Convert each octal digit to a 3-bit equivalent binary representation

24. Example 705 8 = ? 2 7 5 111 000 101 705 8 = 111000101 2

25. Hexadecimal to Binary Technique Convert each hexadecimal digit to a 4-bit equivalent binary representation

26. Example 10AF 16 = ? 2 1 0 A F 0001 0000 1010 1111 10AF 16 = 0001000010101111 2

27. Decimal to Octal Technique Divide by 8 Keep track of the remainder

28. Example 1234 10 = ? 8 8 1234 154 2 8 19 2 8 2 3 8 0 2 1234 10 = 2322 8

29. Decimal to Hexadecimal Technique Divide by 16 Keep track of the remainder

30. Example 16 1234 2 16 77 13=D 16 4 1234 10 = ? 16 1234 10 = 4D2 16

31. Binary to Octal Technique Group bits in threes, starting on right Convert to octal digits

32. Example 1011010111 2 = ? 8 1 011 010 111 1 3 2 7 1011010111 2 = 1327 8

33. Binary to Hexadecimal Technique Group bits in fours, starting on right Convert to hexadecimal digits

34. Example 1010111011 2 = ? 16 10 1011 1011 2 B B 1010111011 2 = 2BB 16

35. Octal to Hexadecimal Technique Use binary as an intermediary

36. Example 1076 8 = ? 16 1 0 7 6 001 000 111 110 2 3 E 1076 8 = 23E 16

37. Hexadecimal to Octal Technique Use binary as an intermediary

38. Example 1F0C 16 = ? 8 1 F 0 C 0001 1111 0000 1100 0 1 7 4 1 4 1F0C 16 = 17414 8

39. Fractions Binary to decimal 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 2 = 0.0 1 x 2 1 = 2.0 2.6875

40. Fractions Decimal to binary 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 . 11.001001...

41. Thank you…………