DATA REPRESENTATION IN COMPUTER
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Number System defines a set of values used to represent ‘quantity’.
Base or Radix: The total number of digits available in a number system .
Decimal Number System:
There are 10 digits i.e 1, 2, 3, 4, 5, 6, 7, 8, 9 available in decimal number system. It is known as Base 10 system. The value of a digit in a number depends upon its position in the number e.g. the number 546 in this system is represented as (546)_{10}
546 = (4*10^{2}) + (8*10^{1}) + (6*10^{0})
Binary Number System
The Binary Number System contains 2 unique digits 0 and 1. it is known as Base 2 system.
Octal Number System
There are 8 unique digits available in octal number system. These are 0, 1, 2, 3, 4, 5, 6, 7. thus, any number formed is the combination of these digits. It is known as Base 8 system.
Hexadecimal Number System
There are 16 unique digits available in Hexadecimal number system. These are 0, 1, 2, 3, 4, 5, 6, 7, 8, A, B, C, D, E, F where A denotes 10, B denotes 11………,F denotes 15. thus any number formed is a combination of these digits. It is known as Base 16 system.
Converting Between Number Bases
Converting Decimal to Binary, Octal and Hexadecimal
Integer Part
Remainder method
1. Divide the decimal number by the base of the target number system that is, to convert decimal to binary, divide the decimal number with 2 (the base of binary number system), 8 for octal and 16 for hexadecimal.
2. Note the remainder separately as the first digit from the right. In case of hexadecimal , if the remainder exceeds 9, convert the remainder into equivalent hexadecimal form. For e.g., if the remainder is 10 then note the remainder as A.
3. Continually repeat the process of dividing until the quotient is zero and keep writing the remainders after each step of division.
4. Finally, when no more division can occur, write down the remainders in reverse order.
Fractional Part
1. Multiply the fractional part by the value of the new base.
2. Record the integer part if it exits, else record 0.
3. Repeat step 1 with the result of the previous multiplication and then step 2, until the fractional part becomes 0. in case of infinite calculations, generally 6 digits are taken.
Converting Binary, Octal and Hexadecimal to Decimal
The method used for conversion of a binary, octal and hexadecimal number to decimal number involves each digit of the binary, octal or hexadecimal number to be multiplied by its weighted position, and then each of the weighted values are added to get the decimal number.
Example:
Binary Number | 1 | 1 | 0 | 1 | 0 |
Weight of each bit | 2^{4} | 2^{2} | 2^{2} | 2^{1} | 2^{0} |
Weghted Value | 2^{4} * 1 | 2^{3} * 1 | 2^{2} * 0 | 2^{1} * 1 | 2^{0} * 0 |
Solved Multiplication | 16 | 8 | 0 | 2 | 0 |
Sum of weight of all bits = 16 + 8 +0 + 2 + 0 = 26
Thus, the decimal equivalent of
(11010)_{2} is
(26)_{10}
Converting Between Octal and Hexadecimal
Steps of conversion
1. Convert each octal digit to 3-bit binary form
2. Combine all the 3-bit binary numbers.
3. Segregate the binary numbers into 4-bit binary form by starting the first number from the right bit (LSB) towards the number on the left bit (MSB).
4. Finally, convert these 4-bit blocks into their respective hexadecimal symbols.