Binary Long Division Method Explained

Binary long division is the standard mathematical procedure used to divide numbers in base-2, operating on the same fundamental principles as decimal long division but with much simpler arithmetic. Because the binary system contains only the digits 0 and 1, determining the quotient at each step requires only comparison and subtraction rather than trial multiplication. This article outlines the step-by-step pencil-and-paper method used for binary division, followed by how this logic translates to computer hardware algorithms.

The Standard Paper-and-Pencil Method

The standard method for binary long division involves four repeating operations: compare, place quotient bit, subtract, and bring down.

1. Setup: Write the divisor to the left and the dividend under the division bracket.
2. Compare: Look at the leftmost bits of the dividend. Determine if this segment is greater than or equal to the divisor.
3. Determine the Quotient Bit:
   • If the dividend segment is greater than or equal to the divisor, write a 1 in the quotient above the current column.
   • If the dividend segment is smaller than the divisor, write a 0 in the quotient.
4. Subtract: If the quotient bit is 1, subtract the divisor from the current dividend segment using binary subtraction rules (\(1 - 0 = 1\), \(1 - 1 = 0\), and \(0 - 1 = 1\) with a borrow). If the quotient bit is 0, skip subtraction.
5. Bring Down: Bring down the next bit of the dividend to the right of the current remainder.
6. Repeat: Repeat the process until all bits of the dividend have been processed. The final value at the bottom is the remainder.

Step-by-Step Example

Problem: Divide \(1101_2\) (13 in decimal) by \(11_2\) (3 in decimal).

1. Compare First Two Bits: The divisor is two bits long (11). Compare it with the first two bits of the dividend (11).
   • Since \(11_2 \ge 11_2\), the first quotient bit is 1.
   • Subtract: \(11_2 - 11_2 = 0\).
2. Bring Down Next Bit: Bring down the next digit (0), making the current segment 00.
   • Compare: \(00_2 < 11_2\), so the next quotient bit is 0.
3. Bring Down Next Bit: Bring down the final digit (1), making the current segment 01.
   • Compare: \(01_2 < 11_2\), so the final quotient bit is 0.

Result: * Quotient: \(100_2\) (4 in decimal) * Remainder: \(1_2\) (1 in decimal)

Checking with decimal values: \(13 \div 3 = 4\) with a remainder of \(1\).

Hardware Implementation Methods

In computer processors, binary long division is automated using digital circuits through specialized division algorithms:

• Restoring Division: The circuit subtracts the divisor from the dividend register at each cycle. If the result is negative, it sets the quotient bit to 0 and adds the divisor back (restores the original value) before shifting.
• Non-Restoring Division: Avoids the restoration step by leaving the negative result in place and adding the divisor in the subsequent cycle instead of subtracting, significantly increasing execution speed.
• SRT Division: A fast hardware method utilizing redundant representations and lookup tables to guess multiple quotient bits per cycle without performing full subtractions.