apply the non-restoring division algorithm for unsignednumbers on the following problem and show all steps with commentsand identify answer and remainder .
The Correct Answer and Explanation is :
To apply the non-restoring division algorithm to the problem, let’s first explain the process and then demonstrate it with an example.
Non-Restoring Division Algorithm Overview:
The non-restoring division algorithm is a method of dividing two unsigned binary numbers. This algorithm divides a dividend by a divisor and provides the quotient and remainder. It works by iterating through a series of steps using subtraction and addition without restoring the partial remainder during intermediate steps.
Steps Involved:
- Initial Setup:
- The dividend ( D ) (numerator) and divisor ( Q ) (denominator) are both represented in binary.
- The quotient ( Q ) is initialized to 0, and the remainder ( R ) is initially set to 0.
- Shift and Subtract:
- The dividend ( D ) is shifted left by 1 bit, and a 0 is added to the least significant bit (LSB) of the remainder ( R ).
- Subtract the divisor ( Q ) from ( R ). If the result is negative, we “restore” by adding ( Q ) back to ( R ) and set the bit of the quotient to 0.
- If the result is non-negative, the bit of the quotient is set to 1.
- Repeat:
- Repeat the shifting and subtraction process until the required number of steps have been completed (usually the number of bits in the dividend).
- Final Adjustment:
- After all steps, the quotient is the binary value obtained, and the remainder is the final value in ( R ).
Example:
Let’s divide ( D = 13 ) (dividend) by ( Q = 3 ) (divisor), both in binary.
- Convert to Binary:
- Dividend: ( 13_{10} = 1101_2 )
- Divisor: ( 3_{10} = 0011_2 )
- Initialize quotient ( Q = 0000 ) and remainder ( R = 0000 ).
- Step 1: Shift ( D ) left by 1 position, remainder ( R ) becomes ( 0000 ).
- Subtract divisor from remainder: ( R – Q = 0000 – 0011 = -0011 ) (negative).
- Restore by adding ( Q ) back: ( R = 0000 + 0011 = 0011 ).
- Set quotient bit to 0.
- Step 2: Shift ( D ) left by 1 position again, remainder ( R ) becomes ( 0110 ).
- Subtract divisor from remainder: ( R – Q = 0110 – 0011 = 0011 ) (positive).
- Set quotient bit to 1.
- Step 3: Shift ( D ) left again, remainder ( R ) becomes ( 0111 ).
- Subtract divisor from remainder: ( R – Q = 0111 – 0011 = 0100 ) (positive).
- Set quotient bit to 1.
- Step 4: Shift ( D ) left again, remainder ( R ) becomes ( 1000 ).
- Subtract divisor from remainder: ( R – Q = 1000 – 0011 = 0101 ) (positive).
- Set quotient bit to 1.
- Final Result: The quotient is ( 1111_2 = 15_{10} ), and the remainder is ( 0101_2 = 5_{10} ).
Thus, ( 13 \div 3 = 5 ) with a remainder of 5.
Explanation:
The algorithm proceeds by iterating through bit shifts and performing either subtraction or addition based on the sign of the result. The quotient bits are updated with 0 or 1 depending on whether the division step results in a negative or non-negative remainder. This algorithm avoids the need to “restore” the remainder during intermediate steps, making it more efficient than the restoring division method. The final quotient and remainder provide the answer to the division.
In summary:
- Quotient: 5 (binary: 1111)
- Remainder: 5 (binary: 0101)
The non-restoring division algorithm is a robust and efficient method for binary division, commonly used in hardware implementations where speed is critical.