Number Systems Calculations - Essay Example

Module 2 background of the of the Answer The vase of a number system basically tells one about the no. of different symbols that is used in a number system. A system which uses 10 different symbols is said to be a decimal no. system (ITL Education Solutions Limited, 2011). Similarly a system which uses 2 different symbols is called a binary no. system. In order to represent the base of a no. system first of all the no. is written in the parenthesis and then the base is written down as subscript. For example to write 46 which is a no. in the decimal no. system the correct way to write it is (46)10. The use of different symbols by different bases means that a no. system uses 10 different symbols and from the 11th no. the cycle of symbol repeats. The concept bases have long existed. Earlier for each of the counted nos. a different symbol was used. However the problem with this system was as the list of the no. counted increased so did the no. of symbols that were used. In fact there would be infinite no. of symbols if this system was followed. This problem was recognized early on by the human civilization and the concept of bases was introduced. Egyptians used a base of 2, Babylonians base of 60, for Mayans the base was 18 and 20. Currently however the number system that is mostly used is the decimal no. system with base 10. The different no. systems that are generally used by computers are: Decimal- This is the no. system which has a base of 10. The lists of symbols of decimal no. system are 0,1,2,3,4,5,6,7,8,9 and after 9 the counting again starts from 1. The symbols in a decimal system range from 0 to 9. So after 9 are reached the system should start from 1. But if the system starts from 1 then it will be difficult to understand. So an empty placeholder should be placed in the one’s place. A symbol which is part of the list of symbols, doesn’t have a value in itself but signifies great value when places along with another no. This job is done by the symbol 0. Hence after reaching 9 the system starts from 10. Binary- The binary no. system is the no. system that is most commonly used and understood by the computers. Binary no. system has only 2 symbols 0 and 1 (Merlot, 2002). 0 and 1 are the two states that are represented by on and off. On state represents 1 and off state represents 0. For writing a no. in the binary no. system the way is to write the no. in the parenthesis and then attach a base 2 as subscript after closing parenthesis is written. For example to write (25)10 in binary no. system, it is written as (11001)2. Octal- The octal no. system has a base of 8. The no. of symbols in an octal no. system is 8. The symbols are 0, 1,2,3,4,5,6,7. After 7 the cycle repeats. For example writing 25 in octal no. system is as follows (31)8. Hexadecimal- The hexadecimal no. system has a base of 16. That is uses 16 different symbols. The symbols are written as 0,1,2,3,4,5,6,7,8,9,A,B,C,D,E,F. For example writing 25 in hexadecimal no. system is as follows (19)16. Answer 2 Conversion of decimal to binary The conversion of decimal number to binary is similar to that of conversion into any other number case. It encompasses successive division by radix until it reaches zero. At each division, the remainder highlights converted digit. Example: convert (68)10 to binary 68/2 = 34 remainder is 0 34/2 = 17 remainder is 0 17/2 = 8 remainder is 1 8/2 = 4 remainder is 0 4/2 = 2 remainder is 0 2/2 = 1 remainder is 0 1/2 = 0 remainder is 1. Answer is read from bottom that is (1000100)2 Conversion of binary number system to decimal In this conversion decimal value of respective binary digit is determined, comprising of a 1 and then adding up the rest. Example: convert (1000100)2 to decimal = 1X26 + 0X25 + 0X24 + 0X23 + 1X22 + 0X21 + 0X20 = 64 + 0 + 0 + 0 + 4 + 0 + 0 = (68)10 Conversion of decimal to octal Octal number system has radix of 8. 3 bits is equivalent to 1 octal digit. Octal numbers range from 0 to 7. The numbers are basically expressed as powers of 8 (Khanna, 2008). Example: convert (177)10 to octal number system In this case firstly each digit is divided by 8 in order to get a remainder. This remainder adding it up states the converted octal number. 177/8 = 22 remainder is 1 22/8 = 2 remainder is 6 2/8 = 0 remainder is 2 The answer is (261)8 and is read from bottom to top. Conversion of octal to decimal This particular conversion classifies each number in octal system as powers of 8. Then individual digits are summed up to highlight converted decimal number. Example: convert (632)8 to decimal number system = (6 X 82) + (3 X 81) + (2 X 80) = (6 X 64) + (3 X 8) + (2 X 1) = 384 + 24 + 2 = (410)10 Conversion of decimal to hexadecimal Hexadecimal number system belongs to radix or base 16. 4 bits is equivalent to 1 hex digit as per this number system. The numbers can be classified as 0,1,2,3,4,...8,9,A,B,C,D,E,F. Where B is 11 and E is 14. These numbers are generally expressed in the form of power 16. It can be classified as 160 = 1, 161 = 16, 162 = 256, 163 = 4096, 164 = 65536. Example: convert (4768)10 to hexadecimal = 4768/16 = 298 remainder is 0 = 298/16 = 18 remainder is 10 (A) = 18/16 = 1 remainder is 2 = 1/16 = 0 remainder is 1 The answer for this conversion is 12A0. Conversion of hexadecimal to decimal In this form of conversion each of the hexadecimal digits is converted in power of 16. Then respective converted digits are summed up for the decimal number. Example: convert (F4C)16 to decimal = (FX162) + (4X161) + (CX160) = (15X256) + (4X16) + (12X1) = 3840 + 64 + 12 = (3916)10 References Khanna, R. (2008). Basics of computer science. USA: New Age International. ITL Education Solutions Limited. (2011). Introduction to computer science. New Delhi: Pearson Education India. Merlot. (2002). Practical guide to binary, decimal and hexadecimal numbers. 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