Binary Division

Binary division works just like normal decimal division, but it uses only the digits 0 and 1. The steps are very similar: you divide, multiply, and subtract, but you can follow the rules of binary arithmetic. In each step, you use binary subtraction and binary multiplication to find the quotient. Because binary has only two digits, 0 and 1, division is often simpler than with decimal numbers.

Binary division follows the same method as decimal division, but only uses 0s and 1s. This form of division follows certain rules that must be understood. It focuses on just two symbols, 0 and 1, and 2 is the base value of this technique. The four basic rules of binary division are:

Using the long division method, we can easily divide binary numbers and find the result. We can perform binary division by following these steps:

Step 1: Before performing calculations, identify the dividend and the divisor. If the divisor is larger than the dividend, put 0 as the quotient and bring down the second bit of the dividend.

If the divisor is smaller than the dividend, multiply the divisor by 1, and the product becomes the subtrahend. After that, to get the remainder, subtract the subtrahend (the number we subtract) from the minuend (current part of the dividend we are working with). 

Step 2: After bringing down the next bit from the dividend, repeat step 1. 

Step 3: Continue the same steps until the whole dividend has been processed, or the remainder becomes zero.  
 

Let us take an example to understand the binary division in detail. For example, \(11010_2 \div 101_2 \)

Here, the given binary numbers are \(011010_2 \) and \(0101_2\). The leading zeros of the given numbers do not change the value, so we can simplify the numbers to \(11010_2 \) and \(101_2\). 

Dividend = \(11010_2 \)

Divisor = \(101_2\)

Step 1: Since the divisor is smaller than the dividend, we must multiply the divisor by 1. Hence, the product is
\(101_2 \; (\,101_2 \times 1 = 101_2\,) \)).
 

The product (1012) becomes the subtrahend. Subtract the subtrahend from the current part of the dividend

\((110_2): \; 110_2 - 101_2 = 001_2 \).

Here, the quotient starts with 1. 

Step 2: Bring down the next bit (1) from the dividend, which makes the current portion of the dividend \(0011_2 \). \(101_2 \) is greater than \(0011_2\), we can put 0 in the quotient and bring down the next bit (0), making it \(0110_2\).

Step 3: Multiply the divisor by 1, and \(101_2 \times 1 = 101_2 \).

Step 4: Subtract \(101_2\) from \(110_2\). 

\(110_2 - 101_2 = 001_2 \)
 

The quotient is, \(101_2\) and the remainder is \(1_2\) in binary (1 in decimal).

Binary and decimal division give the same result. Converting between both systems confirms that binary calculations match their decimal equivalents accurately.
 

Let’s see an example:
First, convert the binary numbers to decimal:
 

(101000)₂ = 40₁₀

(010)₂ = 2₁₀
 

Now divide the decimal values:
 

40 ÷ 2 = 20
 

Next, convert 20 back to binary:
 

20₁₀ = (10100)₂
 

This means:
 

(10100)₂ = 20₁₀
 

Final Conclusion
 

Both binary and decimal division give the same result. So, in this new example, both binary and decimal division yield 20.
 

Here are some tips to keep in mind when performing binary division:
 

• Understand the four basic rules of binary division. The four rules of binary division are: 
  
  \(0 ÷ 0 =\) Undefined 
  
  \(0 ÷ 1 = 0 \)
  
  \(1 ÷ 1 = 1 \)
  
  \(1 ÷ 0 =\) Undefined 
  
   
• If the divisor is greater than the dividend, write 0 in the quotient. If the divisor is smaller than the dividend, multiply the divisor by 1.
  
   
• Bring down the next bit from the dividend after each subtraction. 
  
   
• Remove the leading zeros from the dividend and divisor, as they will not change the value of the binary numbers.
  
   
• Use the long division method for binary division, as it is a simple and effective way to find the answer. 
   

Binary division has many real life applications and some of them are mentioned below. 

• Computer arithmetic: Processors carry out division at the binary level when executing instructions like 'divide'. In CPUs and Arithmetic Logic Units(ALU), binary division circuits implement quotient and remainder operations.
  
   
• Cryptography and encryption: Many cryptographic algorithms rely on arithmetic binary. For example, when implementing RSA or elliptic curve algorithms, binary division and modular inverses in binary form are essential for key generation, encryption and decryption.
  
   
• Data storage and error-correcting codes: In RAID systems, ECC (Error Correcting Codes), and Storage Controllers, division in binary is used to compute parity bits, checksums and to correct data. For instance, dividing binary polynomials helps generate error-checking codes that detect or fix bit errors in hard discs, memory and network transmissions.
  
   
•    IP address calculation: IP addresses and subnet masks are managed in binary formats. When dividing networks into subnets, routers compute divisions of binary address blocks to allocate addresses and define ranges. Binary division helps in dividing address space evenly and efficiently. 
  
    
• Graphics, scaling and texture mapping: When computers scale images or textures, they often perform divisions in binary to adjust resolution or brightness. For instance, dividing pixel values by scaling factors. In rendering, shaders may divide binary color values so that things like blending or interpolation fast and precise.