Binary Subtraction: Rules, 1's Complement & Solved Examples

Binary Subtraction is one of the four binary operations in Mathematics. Binary subtraction includes subtracting two binary numbers (0 and 1). It is identical to the fundamental arithmetic subtraction of decimal numbers usually done in basic Maths. Only the integers 0 and 1 are used in binary numbers. Because just 0 and 1 are involved, we may need to remove 0 from 1 on some occasions. The concept of borrowing is used in such situations from the next higher-order digit. Borrowing in binary subtraction is the same as in arithmetic subtraction. The binary subtraction rules are:

• 0 – 0 = 0
• 1 – 0 = 1
• 1 – 1 = 0
• 0 – 1 = 1 (Borrow 1)

Read More: Binary Multiplication

Key Terms: Binary Subtraction, Number System, Binary Operations, Decimal, Integers, Subtraction, Arithmetic Operation, Borrowing 

---

What is Binary Subtraction?

[Click Here for Sample Questions]

Binary Subtraction is an arithmetic operation that is similar to the subtraction of base 10 numbers or decimal numbers. 

• In the base 10 number system, 1 + 1 + 1 equals 3; in the binary number system, 1 + 1 + 1 equals 11. 
• When adding and subtracting binary numbers, one must be cautious when borrowing because it will happen more frequently.
• One must account for borrowing when subtracting numerous columns of binary digits.
• When one is subtracted from zero, the outcome is one, with one borrowed from the highest-order bit or digit.

Binary Subtraction

Read More:

---

Binary Subtraction Table

[Click Here for Previous Year Questions]

The subtraction of binary numbers is done with the help of the given rules:

• The sum of two binary numbers 1 and 1 equals 10, where 0 is considered and 1 is carried forward to the next high order.
• However, when 1 and 1 are subtracted, the result is 0, and nothing is carried forward.
• When 1 is subtracted from 0 in decimal subtraction, we borrow 1 from the next previous number to make it 10.
• After subtraction, we get 9, i.e. 10 – 1 = 9.
• Binary subtraction, on the other hand, yields only one result.

Read More: NCERT Solutions for Class 12 Mathematics Relations and Functions

---

Rules for Binary Subtraction

[Click Here for Sample Questions]

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

• 0 – 0 = 0
• 0 – 1 equals 1 along with 1 as a borrow
• 1 – 0 = 1
• 1 – 1 = 0

---

Binary Subtraction Steps

[Click Here for Previous Year Questions]

In order to learn Binary Subtraction, we will use the example: 1010 – 101.

\(\begin{equation} \frac{ \begin{array}[b]{r} 1010\\ - 101 \end{array} }{ } \end{equation}\)

Step 1: Take the 1's column and subtract it, (0 – 1), yielding 1 as per the binary subtraction condition with a borrow of 1 from the 10's position.

Step 2: The value 1 in the 10's column is changed to the value 0 after borrowing 1 from the 10's column.

1 Borrow

\(\begin{equation} \frac{ \begin{array}[b]{r} 1010\\ - 101 \end{array} }{\:\:1 } \end{equation}\)

Step 3: Subtract the value in the tenth place, resulting in (0 – 0) = 0.

1 Borrow

\(\begin{equation} \frac{ \begin{array}[b]{r} 1010\\ - 101 \end{array} }{\:\:\:\:01 } \end{equation}\)

Step 4: Subtract the values in the hundredth place. Take 1 from the 1000th position (0 – 1) = 1.

1 1 Borrow

\(\begin{equation} \frac{ \begin{array}[b]{r} 1010\\ - 101 \end{array} }{\:0101 } \end{equation}\)

When we compare the binary subtraction resulting value to the decimal value, we should get the same result. The decimal number 10 is equal to the binary value 1010, while the binary value 101 is equal to 5.

So, 10 – 5 = 5

As a result, the binary number 0101 equals the decimal number 5.

Binary Subtraction Rules

---

Binary Subtraction Using 1's Complement

[Click Here for Sample Questions]

1's complement of a number is obtained by interchanging every 0 to 1 and every 1 to 0 in a binary number. For instance, the 1's complement of the binary number 11021102 is 00120012.

• The positive sign is represented by the number 0.
• The negative sign is represented by number one.

Procedure for Binary Subtraction by 1's Complement 

The steps to perform binary subtraction using 1's complement are as follows:

• Write the subtrahend's 1's complement.
• Then, using the minuend, subtract the 1's complement subtrahend.
• If there is a carryover in the result, add it in the least significant bit.
• If there is no carryover, take the resultant's 1's complement, which is negative.

Read More: Addition of Two Real Functions

---

Binary Subtraction Examples

[Click Here for Previous Year Questions]

Here are a few examples to understand the concept of Binary Subtraction better:

Example 1: Subtract 0011010 – 001100. 

Solution: The difference will be calculated as:

1 1 Borrow

0 0 1 1 0 1 0

- 0 0 1 1 0 0

——————

0 0 0 1 1 1 0

Decimal Equivalent:

0 0 1 1 0 1 0 = 26

0 0 1 1 0 0 = 12

Therefore, 26 – 12 = 14

The binary resultant 0011010 is equivalent to the 14.

Example 2: Subtract 11010.101 from 101100.011.  

Ans. To subtract 11010.101 from 101100.011,

1   1     1 Borrow

1 0 1 1 0 0 . 0 1 1
 - 1 1 0 1 0 . 1 0 1    
   1 0 0 0 1 . 1 1 0

Read More: 

---

Things to Remember

• Binary subtraction is a fundamental part of binary arithmetic calculations.
• The four types of binary operations are binary addition, binary subtraction, binary multiplication, and binary division.
• The sum of two binary numbers 1 and 1 equals 10, where 0 is ignored and 1 is carried forward to the next high order.
• When 1 and 1 are subtracted, the result is 0, and nothing is carried forward.
• When 1 is subtracted from 0 in decimal subtraction, we borrow 1 to make it 10, and after subtraction, we get 9, i.e. 10 – 1 = 9.
• Binary subtraction yields only one result.

---

Previous Year Questions

1. If R is a relation on a set R of all real numbers defined by aRb, if |a-b| ≤ 1. Then R is? [BITSAT 2013]
2. Prove the associative and commutative properties for the binary operations. [KCET 2006]
3. In the group G={1,3,7,9} under multiplication, the inverse of 3 is? [KCET 2005]
4. If A={a,b,c} then a number of binary operations on A is? [KCET 2020]
5. A set A has 5 elements. Then the maximum number of relations on… [COMEDK UGET 2007]
6. In Z, the set of all integers, the inverse of -7 with respect… [COMEDK UGET 2011]
7. If a set has n elements, then the number of relations… [COMEDK UGET 2008]
8. If N is a set of natural numbers, then under binary operation…
9. If F is a function such that F (0) = 2, F (1) = 3, F (x+2) = 2F (x) - F (x+1)…
10. The number of bijective functions from the set A to itself… [COMEDK UGET 2015]
11. The number of functions that can be defined from the set A… [KEAM]
12. The domain of the function\(f \left(x\right)=\frac{\log_{2}\left(x+3\right)}{x^{2}+3x+2} \)… [KEAM]
13. On the set of positive rationals, a binary operation… [KCET 2019]
14. The set of zeros of f(x) = 0 is a non-empty set… [AMUEEE 2013]