What is Binary Subtraction?

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What is Binary subtraction? Learn the basics of the fundamental operation in digital computing. Discover how 1s and 0s are manipulated to perform arithmetic in the binary number system.

What is Binary Subtraction?

Binary subtraction is a mathematical operation performed on binary numbers, which are numbers expressed in the base-2 numeral system. In the binary numeral system, there are only two possible digits: 0 and 1. Binary subtraction involves subtracting one binary number from another, similar to how subtraction is performed in the decimal (base-10) numeral system.

The process of binary subtraction follows similar principles to decimal subtraction, but with a focus on the binary digits. Here’s a basic overview of the process:

Start from the rightmost digit of the numbers being subtracted.

If the digit being subtracted (the subtrahend) is smaller than the corresponding digit in the number being subtracted from (the minuend), a “borrow” is needed from the next higher place value.

If a borrow is performed, the borrowed amount is subtracted from the digit in the next higher place value of the minuend.

Subtract the digits and any borrow, and write the result in the corresponding place value of the difference.

Repeat the process for all digits from right to left.

It’s important to note that binary subtraction can result in borrowing through multiple place values, just like in decimal subtraction.

Here’s a simple example of binary subtraction:

1011 (11 in decimal)

– 110 (6 in decimal)

——-

101 (5 in decimal)

In this example, borrowing occurred in the second and third place values, resulting in the correct difference of 101 (which is 5 in decimal).

Binary subtraction is a fundamental operation in digital circuits and computer arithmetic, as binary numbers are commonly used in computer systems to represent and manipulate data.

What are the Rules of Binary Subtraction?

Binary subtraction follows similar rules to decimal subtraction, but with the added complexity of dealing with binary digits (0 and 1). Here are the rules for binary subtraction:

Single Digit Subtraction:

Begin by subtracting the rightmost (least significant) digits, just like in decimal subtraction. If the minuend (the number you’re subtracting from) is greater than or equal to the subtrahend (the number you’re subtracting), subtract the subtrahend from the minuend and write down the result. If the minuend is smaller than the subtrahend, you’ll need to borrow a 1 from a higher significant digit in the minuend.

Borrowing (Regrouping):

When borrowing is needed, look at the next higher significant digit in the minuend. If it’s a 1, borrow it and add 2 to the current digit. This is because you’re essentially borrowing 2 in binary (10) rather than just 1 in decimal. After borrowing, subtract the subtrahend from the adjusted minuend digit.

Repeat:

Continue this process, moving from right to left through both numbers, performing the necessary borrowing and subtraction as needed.

Zero Borrowing:

If borrowing would involve subtracting from a zero, you’ll need to look for the nearest nonzero digit to borrow from. This can lead to multiple borrow operations in a row.

Here’s an example of binary subtraction:

1101 (13 in decimal)

– 101 (5 in decimal)

——

100 (4 in decimal)

Let’s go through it step by step:

Subtract the rightmost digits: 1 – 1 = 0.

Move to the next digit: 0 (borrow 1 from the left).

The leftmost digit is 1, so subtract 1 (borrowed) from 1, which becomes 0. Then subtract 0 – 1 = 1.

The result is 100, which is 4 in decimal.

Remember that practicing binary subtraction will help you become more comfortable with the process. It’s also essential to be careful when borrowing and performing subtraction to avoid mistakes in the calculation.

How to Subtract Binary Numbers?

Subtracting binary numbers is similar to subtracting decimal numbers, but it’s done using the binary base (base 2) instead of the decimal base (base 10). Here’s a step-by-step guide on how to subtract binary numbers:

Let’s say you’re subtracting binary number B from binary number A.

Set Up:

Make sure both binary numbers have the same number of digits by adding leading zeros if necessary.

Perform Borrowing (if needed):

If any digit in the lower number (B) is larger than the corresponding digit in the upper number (A), you need to borrow 1 from the higher digit in A. Change the higher digit in A to one less and then perform the subtraction.

Subtract Digits:

Start from the rightmost digit (the least significant digit) and subtract each corresponding pair of digits from B and A. If the digit in B is larger than the digit in A, and you didn’t already perform borrowing, perform borrowing now.

Write Down the Result:

Write down the result of each subtraction below the line, just like you would in decimal subtraction.

Carry Over Borrow (if performed):

If you had to perform borrowing in step 2, add the borrowed value to the corresponding digit in B, then continue subtracting.

Continue Subtraction:

Continue subtracting digits from right to left, carrying over any borrows as needed.

Ignore Extra Borrow:

If you have any digits in A that haven’t been subtracted due to borrows, ignore those digits in the final result.

Here’s an example to illustrate the process:

Let’s subtract binary number 1101 from binary number 10110.

10110 (A)

– 1101 (B)

———

1001 (Result)

In this example:

The rightmost digits are 1 and 1, so 1 – 1 equals 0.

The next digit is 1 in A and 0 in B. Since 0 is smaller than 1, you need to borrow 1 from the leftmost digit in A. So A becomes 10001 and the subtraction continues.

Now, you subtract 1 (borrowed) from 1, which gives you 0.

Finally, subtract 0 from 1, giving you 1.

The result is 1001, which is the binary representation of the decimal number 9.

Remember that practice makes perfect, so try a few more examples to get comfortable with binary subtraction!

Binary Subtraction Table

Binary Subtraction Using 1’s Complement

Binary subtraction using the 1’s complement involves a two-step process:

Find the 1’s complement of the subtrahend (the number you’re subtracting): To get the 1’s complement of a binary number, you need to flip all the bits (0s become 1s and 1s become 0s).

Add the 1’s complement to the minuend (the number you’re subtracting from): This addition is performed using binary addition rules. If there’s a carry-out after adding, you need to add it back to the result.

Let’s go through an example to illustrate the process. We’ll subtract binary number 10101 from binary number 11011.

Step 1: Find the 1’s complement of the subtrahend (10101):

Flipping the bits gives you: 01010

Step 2: Add the 1’s complement to the minuend (11011 + 01010):

11011 (minuend)

+ 01010 (1’s complement of subtrahend)

——–

100001 (result)

Here’s the binary subtraction using 1’s complement:

11011 – 10101 = 100001 in binary.

Keep in mind that in binary subtraction using 1’s complement, you might need to adjust for any overflow or carry-out bits in the final result. If there’s a carry-out, you need to add it back to the result to get the correct answer.

Remember to double-check your work and practise with more examples to become comfortable with the process!

Solved Examples on Binary Subtraction

Here are a few examples of solved binary subtraction problems:

Problem 1:

Perform binary subtraction: 11010 – 1011

Solution:

Start from the rightmost bit and move towards the left, performing subtraction at each bit position.

11010

– 1011

——–

10011

Problem 2:

Perform binary subtraction: 101001 – 11011

Solution:

Start from the rightmost bit and move towards the left, performing subtraction at each bit position.

101001

– 11011

——–

100110

Problem 3:

Perform binary subtraction: 1001011 – 110010

Solution:

Start from the rightmost bit and move towards the left, performing subtraction at each bit position.

1001011

– 110010

———-

1011001

Remember that when subtracting binary numbers, if the number being subtracted is larger than the number you’re subtracting from, you might need to perform a “borrow” from the higher bit positions. This is similar to how borrowing works in decimal subtraction.

Also, keep in mind that in the case of binary subtraction, borrowing is done by borrowing a 1 from the next higher bit rather than a 10 in decimal subtraction.

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