Subtraction of Binary Numbers

Subtracting binary numbers involves using the concept of borrowing, similar to decimal subtraction. In binary, subtraction follows specific rules due to the presence of only two digits, 0 and 1.

The process involves comparing bits and borrowing from higher-order bits when necessary. The main components of binary subtraction include:

Borrowing: A process similar to borrowing in decimal subtraction.

Binary digits: Only 0 and 1 are used in binary numbers.

Operators: The subtraction operation primarily uses the minus (-) symbol.

When subtracting binary numbers, follow these steps:

Borrow when necessary: If the current bit of the minuend is smaller than the bit of the subtrahend, borrow from the next higher bit.

Subtract bits: Subtract each bit of the subtrahend from the corresponding bit of the minuend.

Simplify result: After performing the subtraction, simplify the result by removing any unnecessary leading zeros.

The following are the methods for binary subtraction:

Method 1: Bit-by-Bit Subtraction

Step 1: Align the numbers at the right.

Step 2: Subtract each bit, borrowing from the next bit if needed.

Step 3: Write the result for each bit from right to left.

Example:

Question: Subtract 1101 from 10100 - Align the numbers:  10100 - 1101

Step 1: Borrow from higher bits if needed.

Step 2: Perform subtraction. Answer: 01111

Method 2: Complement Method

In this method, we use the 2’s complement to perform subtraction:

Step 1: Find the 2’s complement of the subtrahend.

Step 2: Add it to the minuend.

Step 3: If there’s an overflow, discard it. Example: Subtract 1101 from 10100 using 2’s complement.

Solution:

Step 1: 2’s complement of 1101: 0011

Step 2: Add to 10100:  10100 + 0011

Result: 01111

In binary arithmetic, subtraction has the following properties:

• Subtraction is not commutative In subtraction, changing the order of the numbers changes the result, i.e., A - B ≠ B - A

• Subtraction is not associative Subtraction cannot be regrouped like addition. Changing the grouping alters the result. (A − B) − C ≠ A − (B − C)

• Subtraction as addition of complements Subtracting a number is equivalent to adding its complement, allowing subtraction to be performed as an addition operation. A − B = A + (−B)

• Subtracting zero from a number leaves the number unchanged Subtracting zero from any binary number results in the same binary number: A - 0 = A

These tips can help make binary subtraction simpler and more efficient:

Tip 1: Pay attention to borrowing during subtraction, as it can affect the entire process.

Tip 2: For large numbers, use the complement method to simplify calculations, as it minimizes manual borrowing.

Tip 3: Visual aids, such as arranging numbers in columns, can help track borrowing and ensure accuracy.

Binary subtraction can be tricky, leading to common mistakes. Awareness of these can help prevent errors.