Binary numbers are special numbers that use only two digits: 0 and 1. You won’t find digits like 2, 3, 4, or higher in binary. These numbers are used in computers and digital systems. Just like regular numbers, binary numbers can be added, subtracted, multiplied, and divided. Binary subtraction means taking one binary number away from another. The method is similar to normal subtraction, but since only 0 and 1 are used, there are special rules to follow, especially when borrowing. Binary subtraction is an important concept in computer and digital electronics.

In this mathematics article, we will learn what binary subtraction is, what the rules of binary subtraction are, steps to do binary subtraction, binary subtraction using 2’s and 1’s complement, and solve problems based on binary subtraction.

Binary Subtraction

Binary subtraction is one of the basic arithmetic operations in the binary number system, which uses only the digits 0 and 1. It works much like decimal (base-10) subtraction but with simpler rules because there are fewer digits. For example, in base 10 we write 1 + 1 + 1 = 3, but in binary we write 1 + 1 + 1 = 11. When subtracting binary numbers, you must handle borrowing carefully, as it occurs more often than in decimal subtraction. If you subtract 1 from 0 in binary, you borrow 1 from the next higher bit and the result becomes 1. Borrowing always comes from a higher-order digit. When subtracting long binary numbers, work column by column and track each borrow to get the correct answer. Binary subtraction is widely used in digital electronics and computing.

Binary subtraction becomes much easier than decimal subtraction when the following rules are understood:

• \(0 – 0 = 0\)
• \(0 – 1 = 1\), ( with a borrow of 1)
• \(1 – 0 = 1\)
• \(1 – 1 = 0\)

Binary Subtraction Rules Table

In binary subtraction, we only deal with two digits: 0 and 1. Here are the basic rules you need to follow:

Binary Subtraction Examples

Example 1: Perform binary subtraction: \(101111 – 010101 = ?\)

Solution: Using the rules of binary subtraction we have

Therefore, The subtraction of \(101111 – 010101 = 011010\).

Example 2: Perform binary subtraction: \(100101 – 011110 = ?\)

Solution: Using the rules of binary subtraction we have

Therefore, the subtraction of \(100101 – 011110 = 000111\).

Steps To Do Binary Subtraction

Let us learn stepwise procedures that how to do binary subtraction with borrowing and without borrowing using the solved examples.

To learn about binary to decimal conversion, please click here.

Binary Subtraction With Borrowing:

Example: Subtract \(101\) from \(1001\).

Here the decimal equivalent of \(101\) is \(5\) and \(1001\) is \(9\).

Step 1: Arrange the numbers as shown below.

Step 2: Use the rules of binary subtraction to subtract \(101\) from \(1001\).

Here let us subtract the numbers starting from the right and move to the next higher order digit. The first step is to subtract \((1-1)\), which is equal to \(0\). Similarly, we move on to the next higher order digit and subtract \((0 – 0)\), which is equal to \(0\). In the next step, we have to subtract \((0 – 1)\), so we borrow a \(1\) from the next higher order digit. Therefore, the result of subtracting \((0 – 1)\) is \(1\).

Step 3: Thus the difference of \(101\) from \(1001\) is \(100\).

To verify this, check the decimal equivalent of \(100\), which is \(4\) and \(9 – 5 = 4\). Hence the answer is correct.

To learn about addition and subtraction of decimals, please click here.

Binary Subtraction Without Borrowing:

Example: Subtract \(100\) from \(1111\).

Here the decimal equivalent of \(100\) is \(4\) and \(1111\) is \(15\).

Step 1: Arrange the numbers as shown below.

Step 2: Use the rules of binary subtraction to subtract \(100\) from \(1111\). In this subtraction, we do not encounter the subtraction of \(1\) from \(0\). Hence, the difference is \(1011\).

Step 3: Thus the difference of \(100\) from \(1111\) is \(1011\).The decimal equivalent of \(1011\) is \(11\). Hence the answer is correct.

To learn about the multiplication of binary numbers, please click here.

Binary Subtraction Using 1’s Complement

Binary subtraction using 1’s complement is a method to subtract two binary numbers. This method allows subtraction of two binary numbers by addition. The 1’s complement of a binary number can be obtained by replacing all \(0\) to \(1\) and all \(1\) to \(0\). For example, the 1’s complement of the binary number \(1101\) is \(0010\).

To perform binary subtraction using 1’s complement, please follow the steps given below:

Step 1: Determine the 1’s complement of the subtrahend (which means the second number of subtraction).

Step 2: Add it with the minuend or the first number.

Step 3: If there is a carryover left, then add it with the result obtained from step 2.

Step 4: If there are no carryovers, then the result obtained in step 2 is the difference of the two numbers using 1’s complement binary subtraction.

For example, Subtract \(110010 – 100101\) using 1’s complement.

Here the decimal equivalent of \(110010\) is \(50\) and \(100101\) is \(37\).

Step 1: 1’s complement of \(100101\) is \(011010\).

Step 2: Add this with \(110010\).

Step 3: Arrange the numbers as given below and add them.

Step 4: So, the leftmost digit \(1\) is a carryover of this addition. Since there is a carryover we add it with the result, which is \(001100\).

Therefore, the answer is \(1101\). And the difference of \(50 – 37 = 13\). The decimal equivalent of \(1101\) is \(13\). So, the answer is verified.

To learn about binary addition using 1’s complement and 2’s complement, please click here.

Procedure for Binary Subtraction using 1’s Complement

1. Identify the minuend and the subtrahend
   • Minuend is the number from which you subtract.
   • Subtrahend is the number you are subtracting.
2. Find the 1’s complement of the subtrahend
   • Flip all the bits: change all 0s to 1s and all 1s to 0s.
3. Add the 1’s complement of the subtrahend to the minuend
4. Check for a carry bit:
   • If there is a carry, add that carry (1) to the least significant bit (rightmost digit) of the result.
   • If there is no carry, take the 1’s complement of the result and add a negative sign to the answer.

Binary subtraction Using 2’s Complement

Binary subtraction using 2’s complement is a method to subtract two binary numbers. For finding 2’s complement of the binary number, we will first find the 1’s complement of the binary number and then add \(1\) to the least significant bit of it. For example, the 2’s complement of the binary number \(110100\). The 1’s complement of the number \(110100\) is \(001011\). Now add \(1\) to the LSB of this number, i.e., \((001011)+1=001100\).

To perform binary subtraction using 2’s complement, please follow the steps given below:

Step 1: In the first step, find the 2’s complement of the subtrahend.

Step 2: Add the complement number with the minuend.

Step 3: If we get the carry by adding both the numbers, then we discard this carry and the result is positive, else take 2’s complement of the result which will be negative.

For example, Subtract \(10101 – 00111\) using 2’s complement.

Here the decimal equivalent of \(10101\) is \(21\) and \(00111\) is \(7\).

Step 1: 2’s complement of \(00111\) is \(11001\).

Step 2: Add this with \(10101\).

Step 3: We get the carry bit 1. So we discard this carry bit and remaining is the final result and a positive number. Therefore the answer is \(1110\).

Since the difference of \(21 – 7 = 14\). The decimal equivalent of \(1110\) is \(14\). So, the answer is verified.

To learn about different types of numbers with definitions and examples, please click here.

Procedure for Binary Subtraction using 2’s Complement

1. Identify the minuend and subtrahend
2. Find the 2’s complement of the subtrahend:
   • First, take the 1’s complement (flip all bits)
   • Then add 1 to the result
3. Add this 2’s complement to the minuend
4. Check for a carry bit:
   • If there is a carry, ignore it. The result is positive and correct.
   • If there is no carry, the result is in 2’s complement form, so take the 2’s complement of the result and add a negative sign.

Binary Subtraction Solved Example

1. Subtract the following:

(i). \(101\) from \(1001\)

(ii). \(111\) from \(1000\)

Solution: Using the rules for subtraction of binary numbers, solve the following.

(i). Subtract \(101\) from \(1001\):

Thus the answer is \(0100\).

(ii). Subtract \(111\) from \(1000\):

Thus the answer is \(0001\).

2. Subtract \(110101 – 100010\) using 1’s complement.

Solution: Here the decimal equivalent of \(100010\) is \(34\) and \(110101\) is \(53\).

Now 1’s complement of subtrahend, \(100010\) is \(011101\).

Add this with minuend, \(110101\), then arrange the numbers as given below and add them.

So, the leftmost digit \(1\) is a carryover of this addition. Since there is a carryover we add it with the result, which is \(010010\).

Therefore, the answer is \(10011\). And the difference of \(53 – 34 = 19\). The decimal equivalent of \(10011\) is \(19\). So, the answer is verified.

3. Subtract 1001₂ – 0110₂ using 2’s Complement Method

Step 1: Identify the Minuend and Subtrahend

• Minuend = 1001₂ (this is the number we subtract from)
• Subtrahend = 0110₂ (this is the number we subtract)

Step 2: Find the 2’s Complement of the Subtrahend

1. Write the subtrahend: 0110₂
2. Find the 1’s complement (flip the bits): 1001₂
3. Add 1 to the 1’s complement:
   1001

• 1
  ———
  1010 → This is the 2’s complement of 0110₂

Step 3: Add the Minuend and the 2’s Complement

 1001 (Minuend)

+1010 (2’s complement of Subtrahend)

------

 10011

Step 4: Ignore the carry (the leftmost 1)

The result is: 0011₂

1001₂ – 0110₂ = 0011₂

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