Binary Division

Binary division, like the other three binary operations, addition, subtraction, and multiplication, involves division involving the 2 binary numbers 0 and 1. Its algorithm is similar to decimal division, except it has only 2 numbers instead of 10.

Using only two numbers makes binary division even simpler than decimal division. However, like other binary operations, binary division follows some rules:

Rules

The binary division follows the following 4 rules given in the table: 

Dividend	Divisor	Result
0	1	0
1	1	1
1	0	Meaningless
0	0	Meaningless

Thus, like the decimal system, division by 0 is meaningless in binary division.

Steps

The binary division is usually done using the long division method. There can be 2 possible situations involving the same number of digits in the dividend and the divisor.

Note: The number of digits in the dividend is counted from the right.

Case 1

Let us learn the steps by dividing the binary numbers 11111002 (the dividend) by 102 (the divisor.)

Step 1: Comparing the divisor with the dividend.

When the divisor is smaller than the dividend, we multiply the divisor with 1, and the result becomes the subtrahend. Finally, subtracting the subtrahend from the minuend gives us the remainder.

Here, on comparing the divisor with the dividend, we find the divisor 102 < 11001012, the dividend. Thus, the divisor will be multiplied by 1, resulting in the subtrahend.

Using the binary multiplication rule: 1 × 1 = 1, 1 × 0 = 0, 0 × 1 = 0, and 0 × 0 = 0, we get

Thus, 10 × 1 = 10 is the subtrahend.

Step 2: Subtracting the subtrahend 102 from the minuend 112 using the rules of binary subtraction: 0 – 1 = 1, 0 – 0 = 0, 1 – 1 = 0, and 1 – 0 = 0, we get (11 – 10) 1.

Step 3: Borrowing 1 from the next more significant bit and repeating step 1 and step 2 until the remainder becomes zero.

Thus, the binary division involving the divisor 10₂ and the dividend 1111100₂ is 111110₂

We can verify our answer by finding the decimal equivalent of 111110₂, which is 62₁₀. Learn binary to decimal conversion here.

Divisor: 10₂ → 2₁₀

Divisor: 1111100₂ → 124₁₀

Thus, the quotient is 124₁₀ ÷ 2₁₀ = 62₁₀

The decimal equivalent of 111110₂ is also 62₁₀. Thus, the answer is verified

Case 2

Let us divide 10010₂, the dividend, with the divisor 112

Here, the divisor (11) is larger than the dividend (10), and thus, we place 0 in the quotient and then consider the second bit of the dividend, which is 100. Now, the divisor is smaller than the dividend and thus performs division similar to the previous method until we get 0 in the remainder.

Thus, the binary division involving the divisor 11₂ and the dividend 10010₂ is 110₂

Division with Remainder Other Than 0

However, in some divisions, there can be remainders greater than zero, as in the example shown:

Solved Example