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 0 0 0 0001 1 1 1 0010 2 2 2 0011 3 3 3 0100 4 4 4 0101 5 5 5 0110 6 6 6 0111 7 7 7 1000 10 8 8 1001 11 9 9 1010 12 10 A 1011 13 11 B 1100 14 12 C1101 15 13 D 1110 16 14 E 1111 17 15 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=13.62510Octal to Decimal: 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. 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 = (16 x 16) + (2 x 16) + (1 x 1) = 289 Therefore, 12116 = 28910 Decimal to Binary Number: 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 25 ÷ 2 12 1(MSB) 12 ÷ 2` 6 0 6 ÷ 2 3 0 3 ÷ 2 1 1 1 ÷ 2 0 1(LSB) Therefore, from the above table, we can write, (25)10 = (11001)2Decimal 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 128÷8 16 0(MSB) 16÷8 2 0 2÷8 0 2(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 128÷16 8 0(MSB) 8÷16 0 8(LSB) Therefore, the equivalent hexadecimal number is 8016 Here MSB stands for a Most significant bit and LSB stands for a least significant bit. (1011)2 = 138 1x20 = 1 1x21 = 2 0x22 = 0 1x23 = 8 Total = 1110 11/8 1 38 1/8 0