Number System Conversion In number system conversion we will study to convert a number of one base to a number of another base There are a variety of number systems such as binary numbers decimal numbers hexadecimal numbers octal numbers which can be exercised The general representation of number systems are Decimal Number Base 10 N10 Binary Number Base 2 N2 Octal Number Base 8 N8 Hexadecimal Number Base 16 N16 Number System Conversion Table Binary Numbers Octal Numbers Decimal Numbers Hexadecimal Numbers 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 0 1 2 3 4 5 6 7 10 11 12 13 14 0 1 2 3 4 5 6 7 8 9 10 11 12 0 1 2 3 4 5 6 7 8 9 A B C 1101 1110 1111 15 16 17 13 14 15 D E F Number System Conversion Methods Number system conversions deal with the operations to change the base of the numbers For example to change a decimal number with base 10 to binary number with base 2 We can also perform the arithmetic operations like addition subtraction multiplication on the number system Here we will learn the methods to convert the number of one base to the number of another base starting with the decimal number system The representation of number system base conversion in general form for any base number is Number b dn 1 dn 2 d1 d0 d 1 d 2 d m In the above expression dn 1 dn 2 d1 d0 represents the value of integer part and d 1 d 2 d m represents the fractional part Also dn 1 is the Most significant bit MSB and d m is the Least significant bit LSB Now let us learn conversion from one base to another Other Base System to Decimal Conversion Binary to Decimal In this conversion binary number to a decimal number we use multiplication method in such a way that if a number with base n has to be converted into a number with base 10 then each digit of the given number is multiplied from MSB to LSB with reducing the power of the base Let us understand this conversion with the help of an example Example 1 Convert 1101 2 into a decimal number Solution Given a binary number 1101 2 Now multiplying each digit from MSB to LSB with reducing the power of the base number 2 1 23 1 22 0 21 1 20 8 4 0 1 13 Therefore 1101 2 13 10 Example 2 Covert 1101 101 2 10 1x20 1 0x21 0 1x22 4 1x23 8 1x2 1 0 5 0x2 2 0 1x2 3 0 125 Total 13 62510 To convert octal to decimal we multiply the digits of octal number with decreasing power of the base number 8 starting from MSB to LSB and then add them all together Example 2 Convert 228 to decimal number Octal to Decimal Solution Given 228 2 x 81 2 x 80 16 2 18 Therefore 228 1810 Hexadecimal to Decimal Example 3 Convert 12116 to decimal number Solution 1 x 162 2 x 161 1 x 160 256 32 1 289 289 16 x 16 2 x 16 1 x 1 Therefore 12116 28910 Decimal to Binary Number 25 2 12 2 6 2 3 2 1 2 12 6 3 1 0 Therefore from the above table we can write 25 10 11001 2 Suppose if we have to convert decimal to binary then divide the decimal number by 2 Example 1 Convert 25 10 to binary number Solution Let us create a table based on this question Operation Output Remainder 1 MSB 0 0 1 1 LSB Decimal to Octal Number To convert decimal to octal number we have to divide the given original number by 8 such that base 10 changes to base 8 Let us understand with the help of an example Example 2 Convert 12810 to octal number Solution Let us represent the conversion in tabular form Operation Output Remainder 0 MSB 0 2 LSB 0 MSB 8 LSB Therefore the equivalent octal number 2008 Decimal to Hexadecimal Again in decimal to hex conversion we have to divide the given decimal number by 16 Example 3 Convert 12810 to hex Solution As per the method we can create a table Operation Output Remainder Therefore the equivalent hexadecimal number is 8016 Here MSB stands for a Most significant bit and LSB stands for a least significant bit 16 2 0 8 0 128 8 16 8 2 8 128 16 8 16 1011 2 138 1x20 1 1x21 2 0x22 0 1x23 8 Total 1110 11 8 1 8 1 38 0 1