Number Conversions : Important for ISC Class XI

Introduction Decimal Number System Binary Number System Why Binary? Octal Number System Hexadecimal Number System Relationship between Hexadecimal, Octal, Decimal, and Binary Number Conversions In early days when there were no means of counting, people use to count with the help of fingers, stones, sticks, etc. These methods were not adequate and had many limitations. Many number system were introduced with the passage of time like: Decimal Number System Binary Number System Octal Number System Hexadecimal Number System • It consist of ten digit i.e. 0, 1, 2, 3, 4, 5, 6, 7, 8, 9 with the base 10. • Each number can be used individually or they can be grouped to form a numeric value as 85,48,35,456 etc. • The Binary Number System consist of only two digits– 0 and 1. • Since this system use two digits, it has the base 2. • All digital computer use this number system and convert the data input from the decimal format into its binary equivalent. Since the computer is made up of electronic components; it can have only two states, either • On(1) • Off(0) The data which is given to the computer is converted into binary form because a computer understand only binary language. It further converts the binary results into their decimal equivalents for output. In the Octal Number System it consist of 8 digits i.e. 0, 1, 2, 3, 4, 5, 6, 7 with a base 8. The sequence of octal number goes as 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 16, 17, 20, 21, 22, …..as go on. See each successive number after 7 is a combination of two or more unique symbols of octal system. The Hexadecimal system use base 16. It has 16 possible digit symbol. It use the digit 0 through 9 plus the letters A, B, C, D, E, and F as the 16 digit symbols. Hexad eci mal 0 1 2 3 4 5 6 7 8 9 A B C D E F Octal 0 1 2 3 4 5 6 7 10 11 12 13 14 15 16 17 Decimal Binary 0 0000 1 0001 2 0010 3 0011 4 0100 5 0101 6 0110 7 0111 8 1000 9 1001 10 1010 11 1011 12 1100 13 1101 14 1110 15 1111 Notice that each hexadecimal digit represent a group of four binary digit. It Is important to remember that Hex(Abbreviation for Hexadecimal) digit A through F are equivalent to the decimal value 10 through 15. The method of converting Decimal to binary is repeated-division method. For conversion follow the rules: 1. Divide the given decimal number with the base 2. 2. Write down the remainder and divide the quotient by 2. 3. Repeat step 2 till the quotient is zero. (25)10=(11001)2 To convert a binary number follow the steps: 1. Multiply each binary number with 2 having the power 0 for last position, starting from the right digit. 2. Increase the power one by one, with base as 2. 3. Sum up all the products to get decimal number. The method of converting Decimal to Octal is repeated-division method. For conversion follow the rules: 1. Divide the given decimal number with the base 8, 2. Write down the remainder and divide the quotient by 8, 3. Repeat step 2 till the quotient is zero. (33)10=(41)8 To convert a octal number follow the steps: 1. Multiply each Octal number with 8 having the power 0 for last position, starting from the right digit. 2. Increase the power one by one, with base as 8. 3. Sum up all the products to get decimal number. The conversion from octal to binary is performed by converting each octal digit to its 3-bit binary equivalent. The eight possible digits are converted as indicated below: Octal Digit Binary Equivalent 01234567 000 001 010 011 100 101 110 111 Using these conversions, any octal number is converted to binary by individually converting each digit. 5 4 3 1 101 100 011 001 Thus (5431)8= (101 100 011 001) 2 1. Group the binary integer into a3 bits starting at the least significant bit(LSB) 2. If unable to form group then, add one or two 0's 3. Each group is converted to its octal equivalent. For example Binary number 11010110 into octal start making a group of three digits 011 010 110 326 thus (11010110)2=(326)10 The method of converting Decimal to Hexadecimal is repeated-division method. For conversion follow the rules: 1. Divide the given decimal number with the base 16. 2. Write down the remainder and divide the quotient by 16. 3. Repeat step 2 till the quotient is zero. 16 423 Remainders 16 26 7 LSB 16 1 A 0 1 MSB To convert a Hexadecimal number follow the steps: 1. Multiply each hexadecimal number with 16 having the power 0 for last position, starting from the right digit. 2. Increase the power one by one, with base as 16. 3. Sum up all the products to get decimal number. Make a group of four digits starting from Least Significant Bit(LSB) If unable to form group then, add one or two 0's Each group is converted to its Hexadecimal equivalent. For example 1010111010 Divide the number in group of four digits starting from the last digit 0010 1011 1010 2 BA thus (1010111010)2=(2BA)16 Lets convert (9F2)16= 9 F 2 1001 1111 0010 Thus (9F2) 16=(1001 1111 0010)2 Multiply the fractional decimal number by 2. Integral part of resultant decimal number will be first digit of fraction binary number. Repeat step 1 using only fractional part of decimal number and then step 2. 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 20 = 0.0 1 x 21 = 2.0 2.6875 Decimal to binary 36.22 binary to decimal 11001.1001 decimal to octal 76.29 Octal to binary Binary to hexadecimal (721.262)8=()16 7 2 1. 111 010 001 00011101 0001 1 13 1 26 2 010 110 010 0101 1001 0 make a group of four 590 (1D1 .590)16 Hexadecimal to Binary + Binary to Octal Convert Hexadecimal number to binary (48)16=()8 48 0100 1000 (48)16 = (01001000)2 Convert Binary Number into Octal Number Split the binary number from left to right each group 3 bits 001 001 000 110 (01001000)2 = (110)8 A B A + B =C 0 0 0 + 0 =0 0 1 0 + 1 =1 1 0 1 + 0 =1 1 1 1 + 1 = 10 Example – Compute (11111)₂ + (1011)₂ 1 1 11 Carry Over 1 1 1 11 0 1 0 11 1 0 1 0 10 Therefore, (11111)₂ + (1011)₂ = (101010)₂ A B 0 0 1 0 1 1 0 1 Example :- Compute (1100)₂ - (1011)₂ 10 -1 = 1 1 1Borrowed 1 0 1 -1 0 1 1 0 0 1 0 A -B =C 0–0=0 1–0=1 1–1=0 0–1=1 (With a borrow taken from next place i.e., 10-1=1) The answer is = (0010)₂ A B A * B =C 0 0 0*0=0 0 1 0*1=0 1 0 1*0=0 1 1 1*1=1 Example- Compute (101)₂ × (11)₂ 101 × 11 101 + 101× 1111 Therefore, the answer is (1111)₂ Example- Compute (110)₂ ÷ (10)₂ 11 10110 10 0 10 10 00 Quotient Dividend Remainder