- Binary Division Definition: Binary division is defined as an arithmetic operation involving the division of binary numbers.
- Initial Step: The first step in binary division is to consider the left-most digits of the dividend and multiply the divisor by 1.
- Binary Subtraction: Subtract the divisor from the selected digits of the dividend to proceed to the next step.
- Updating Quotient: Insert 0 into the quotient if the multiplication result exceeds the minuend and bring down the next bit.
- Final Result: Complete the division process to find the quotient and remainder.
How to Divide Binary Numbers
Binary division finds how many times a nonzero binary divisor fits into a binary dividend and reports any remainder. It is one operation in binary arithmetic.
Binary addition, binary subtraction, binary multiplication and binary division are the four basic arithmetic operations.
Unsigned binary long division follows the same compare, multiply, subtract and bring-down cycle as decimal long division.
Set each quotient digit to 1 when the current partial dividend is at least as large as the divisor, and set it to 0 when the partial dividend is smaller.
After choosing a quotient bit, multiply the divisor by that bit, subtract the product from the partial dividend and bring down the next dividend bit.
For binary division, repeat this specific cycle until every dividend bit has been used. The final partial dividend is the remainder, which must be smaller than the divisor.
The following binary division example problem shows each quotient decision.
Let A = 11010 and B = 101, with A as the dividend and B as the nonzero divisor. In decimal, this is 26 divided by 5, so the expected quotient is 5 with remainder 1.
Start with the shortest left-hand part of A that is at least as large as B. The first two bits, 11, are smaller than 101, so include the next bit and use 110.
Because 110 is at least 101, write 1 as the first quotient bit. Multiply the divisor by 1 and write the product 101 below the selected dividend bits.
The one-bit products are 1 × 1 = 1, 1 × 0 = 0 and 1 × 1 = 1, so the product remains 101:
Subtract 101 from 110 to obtain 1. The binary subtraction guide explains the borrow method used here.
Bring down the next dividend bit, 1, to make the partial dividend 11.
Because 11 is smaller than 101, write 0 as the next quotient bit. Do not subtract the divisor in this cycle.
Bring down the final dividend bit, 0, so the new partial dividend is 110. Because 110 is at least 101, write 1 as the final quotient bit.
Multiply the divisor by 1 and write 101 below 110, ready for the last subtraction.
No dividend bits remain after this subtraction, so the value left below the line will be the remainder.
The final binary subtraction is 110 – 101 = 1. This completes the binary division.
Quotient = 101 and remainder = 1. The result satisfies the division identity: 11010 = (101 × 101) + 1, and the remainder 1 is smaller than the divisor 101.
Division by zero is undefined, so the divisor must be checked before starting the procedure.
This example covers nonnegative integers. Signed numbers, fixed-width overflow and fractional binary division require additional representation rules.
