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BINARY SUBTRACTION

Binary subtraction is the process of subtracting binary numbers. Binary numbers
include only 0 and 1. The process of binary subtraction is the same as the
arithmetic operation of subtraction that we do with numbers. Since only 0 and 1
are involved here, we may sometimes need to subtract 0 from 1. In such cases, we
use the concept of borrowing as we do in an arithmetic subtraction. A binary
number is expressed with a base-2. For example, a binary number is written as
10121012

1. Rules of Binary Subtraction 2. How To Do Binary Subtraction? 3. Binary
Subtraction Using 1's Complement 4. Binary Subtraction Examples 5. Practice
Questions on Binary Subtraction 6. FAQs on Binary Subtraction


RULES OF BINARY SUBTRACTION

There are some rules in which binary numbers are subtracted. They are,




HOW TO DO BINARY SUBTRACTION?

Decimal or base-10 numbers can be expressed as binary numbers. Binary numbers
are used in computers to represent data since they understand only binary
digits, 0 and 1. Let us understand how to subtract binary numbers with the
example shown below.

Case i) - Binary subtraction without borrowing

Subtract 10021002 from 1111211112 .Here number 4 is represented in binary as
10021002 and number 15 is represented as 1111211112.

Step 1: Arrange the numbers as shown in the figure below.



Step 2: Follow the binary subtraction rules to subtract the numbers. In this
subtraction, we do not encounter the subtraction of 1 from 0. Hence, the
difference is 1011210112.



Step 3: The decimal equivalent of 1011210112 is 11. Hence the difference is
correct.

Case ii) Binary subtraction with borrowing

Subtract 10121012 from 1001210012. Here number 5 is represented in binary as
10121012 and number 9 is represented as 1001210012.

Step 1: Arrange the numbers as shown below.


Step 2: Follow the binary subtraction rules to subtract the numbers. In this
subtraction, first, 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). This is
equal to 0. Similarly, we move on to the next higher order digit and subtract (0
- 0), which is 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: Therefore the dfference of 1001210012 and 10121012 is 10021002. To
verify this, let us find the decimal equivalent of 10021002, which is 4,
Therefore, 9 - 5 = 4.


BINARY SUBTRACTION USING 1'S COMPLEMENT

The 1's complement of a number is obtained by interchanging every 0 to 1 and
every 1 to 0 in a binary number. For example, the 1's complement of the binary
number 11021102 is 00120012. To perform binary subtraction using 1's complement,
please follow the steps mentioned below.

 * Step 1: Find 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.

Let us understand this with an example.Subtract 11001021100102 - 10010121001012
using 1's complement. Here the binary equivalent of 50 is 11010121101012 and the
binary equivalent of 37 is 10010121001012.

Step 1: Find out the 1's complement of the subtrahend (37), which is
01101020110102.

Step 2: Add it with the minuend(50), which is 11001021100102.

Step 3: Arrange the numbers as follows and add them.



Step 4: The left-most digit 1 is a carryover of this addition. Since there is a
carryover we add it with the result, which is 00110020011002.



Therefore, the result is 1101211012. Also, the difference of 50 - 37 is 13. The
binary equivalent of 13 is 1101211012.

Related Topics

Check out some interesting topics related to binary subtraction.

 * 22 in binary
 * 12 in binary
 * Numbers
 * Number Systems
 * Subtraction

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BINARY SUBTRACTION EXAMPLES

 1. Example 1: Subtract 1101211012 from 101102101102.
    
    Solution:
    
    Step 1: Arrange the numbers as shown below.
                  1 0 1 1 0 
                   - 1 1 0 1 
                ---------------
                _________
    
    Step 2: Start subtracting from the right. Starting to subtract from the
    rightmost position, we begin with (0 - 1). Since 0 - 1 can not be done, we
    borrow a 1 from the next higher order digit. So the result of (0 - 1) is now
    1. Since we have borrowed the 1 from the next higher order digit, the 1 has
    become 0. We know that (0 - 0) is 0. Now, move on to the next higher order
    digit, wherein we have to subtract (1 - 1), which is 0. The next higher
    order digit has (0 - 1), for which we need to borrow a 1. On borrowing, we
    obtain the result as 1. Therefore, the difference is equal to 1001210012.
    
                  1 0 1 1 0 
                   - 1 1 0 1 
                -----------------
                    1 0 0 1
                __________
                 
    
    Step 3: The decimal equivalent of 101102101102 is 22 and the decimal
    equivalent of 1101211012 is 13. Therefore the result is 9. The binary
    equivalent of 9 is 1001210012.

 2. Example 2: Subtract 10001021000102 from 11010121101012 using 1's complement
    method of binary subtraction.
    
    Solution:
    
    Step 1: Find the 1's complement of the subtrahend, which is 10001021000102.
    The 1's complement is 01110120111012 .
    
    Step 2: Add it with the minuend.
    
    Step 3:
    
              0 1 1 1 0 1 
           + 1 1 0 1 0 1 
         -------------------
           1 0 1 0 0 1 0
        --------------------
    
    Step 4: Here we get the last left-most digit 1 as a carryover. Now, we add
    it with the result obtained in step 3.
    
    Step 5: Therefore, we get,
    
                 0 1 0 0 1 0 
                             + 1
                ----------------
                 0 1 0 0 1 1
                ----------------
    
    The decimal equivalent of 10001021000102 is 53 and the decimal equivalent of
    11010121101012 is 34. Therefore the result is 19. The binary equivalent of
    19 is 100112100112.

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PRACTICE QUESTIONS ON BINARY SUBTRACTION

 1.  
 2.  
 3.  

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FAQS ON BINARY SUBTRACTION


HOW TO SUBTRACT BINARY NUMBERS?

Binary subtraction can be performed by the normal borrow method of arithmetic
subtraction or by finding the 1's complement of the subtrahend and adding it
with the minuend and add carryovers if any with the sum.


WHAT ARE THE RULES FOR BINARY SUBTRACTION?

The following rules are to be followed before we subtract two binary numbers.
They are,

 * 1 - 0 = 1
 * 1 - 1 = 0
 * 0 - 0 = 0
 * 0 - 1 = 1 (This can not be done directly, hence we borrow a 1 from the next
   higher order digit.)


WHAT DO YOU MEAN BY 1'S COMPLEMENT?

1's complement of a number is replacing every 0 with a 1 and every 1 with a 0 in
a binary number. For example, the 1's complement of 10121012 is 01020102.


HOW DO YOU BORROW IN BINARY SUBTRACTION?

Borrowing is done in binary subtraction when we encounter a situation of
subtracting 1 from 0. In that case, we borrow a 1 from the next higher digit and
perform the subtraction. Therefore, 0 - 1 gives us 1.


WHAT IS BINARY NUMBER SYSTEM?

The Binary number system is a type of number system which uses only two digits,
0 and 1. Computers use binary digits to store all types of information. We can
perform arithmetic operations like addition and subtraction in the binary number
system.


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