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Binary Addition is similar to a normal addition in mathematics. The only difference is that it does the addition of using only two numbers that are ‘0’ and ‘1’ instead of other decimal numbers. Binary addition is the part of arithmetic operations performed in mathematics. Each number of binary addition is known as bit. If the system is ‘ON’ then it represents ‘1’. If the system is ‘OFF’ then it represents ’0’.
Key Terms: Binary addition, binary number system, Binary, binary bits, 1’s complements, 2’s complements, binary numbers, arithmetic operations, decimal number
Binary Number System
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The number system having a base or radix ’2’ is called a binary number system. Binary numbers are used to design digital electronic circuits. It is easy to construct digital circuits using binary numbers (using 0’s and 1’s). Also, computers understand only binary numbers or bits. All the operations performed on electronic devices such as computers, calculators, mobile phones etc. store the information in the form of binary numbers.
Binary Number System
Binary number system has only two digits such as ‘0’ and ‘1’. Each binary number is known as bit and each bit has its own weight or value. The extreme left bit of the above diagram is the Most Significant Bit (MSB) and it has the maximum value of 26. On the other hand, the extreme right bit of the above diagram shows the Least Significant Bit (LSB) and it has the minimum value or weight of 20.
Combination of Binary Bits
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Bit: Bit is the smallest unit of information and is the abbreviation of binary digit. It is either ‘0’ or ‘1’.
Nibble: Nibble is the combination of four binary bits. In the Hexadecimal number system nibble is used because both are in the form of a four digit number system.
For example, 1 nibble= 4 bits.
Combination of Binary Bits
Byte: Byte is the combination of eight binary bits. It is most commonly used in memory systems. Also, a byte contains two nibbles.
For example, 1 Byte= 2 Nibbles= 8 Bits.
Word: Word is the combination of sixteen binary bits. It contains 2 bytes or four nibbles. Also, it is used to represent memory.
For example; 1 Word = 2 Bytes = 4 Nibbles = 16 Bits.
Also Read: Permutations and Combinations
Basic Rules for Binary Addition
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There are four rules for binary addition. Below are the rules for binary addition:
Rule 1: If both the binary digits are zero then the result of addition is also zero with carry zero.
Mathematically,
0 + 0 = 0 ; Carry = 0
Rule 2: If the first binary number is 0 but the second binary number is 1 then the result of addition is 1 with carry 0.
Mathematically,
0 + 1 = 1 ; Carry = 0
Binary Addition Rules
Rule 3: If the first binary number is 1 but the second binary number is 0 then the result of addition is 1 with carry 0.
Mathematically,
1 + 0 = 1 ; Carry = 0
Rule 4: If both (first and second) binary numbers are 1 then the result of addition is 0 with carry 1. The carry is forwarded to the next bit.
Mathematically,
1 + 1 = 0 ; Carry = 1
Steps To Do Binary Addition
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All the steps to be followed for the addition of binary numbers follow the above rules. To understand the binary addition method clearly let us consider an example.
Here are two binary numbers: (1001)2 and (0101)2
Step 1: Binary addition starts from the one’s column. Add the first column of both the numbers. In both the numbers, the first right bit of both the columns is 1 then as per the binary fourth rule we consider 0 as a sum and 1 as a carry and forward it to the next column.
Mathematically,
Step 2: Now, add the bits of the 10’s column. Both the bits are 0 with carry 1. Then 0 + 1 = 1. Thus, sum = 1 and carry = 0.
Step 3: After that, add the bits of the 100’s column. The first bit is 0 and the second bit is 1 then 0 + 1= 1. Thus, the sum is 1 and carry is 0 because no bit is forwarded from the 10’s column.
Step 4: Now, add the 1000’s column. First binary bit = 1 and the second binary bit = 0. Also, no carry is forward from the previous column. So, 1 + 0 = 1.
Hence, (1001)2 + (0101)2 = (1110)2
Steps To Do Binary Addition (for example)
How To Cross Check The Result
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The result of the addition of any binary numbers is equal to the addition of any decimal numbers. To cross-check the result we need to convert the binary numbers into decimal numbers.
1001 = 1\(\times\)23 + 0\(\times\)22 + 0\(\times\)21 + 1\(\times\)20
1001 = 1\(\times\)8 + 0\(\times\)4 + 0\(\times\)2 + 1\(\times\)1
1001 = 8 + 0 + 0 + 1
(1001)2 = (9)10
And
0101 = 0\(\times\)23 + 1\(\times\)22 +0\(\times\)21 +1\(\times\)20
0101 = 0\(\times\)8 + 1\(\times\)4+0\(\times\)2+ 1\(\times\)1
0101 = 0+ 4+0+1
(0101)2 = (5)110
Also,
(1110 )2 = 1\(\times\)23 + 1\(\times\)22 +1\(\times\)21 +0\(\times\)20
(1110 )2 = 1\(\times\)8 + 1\(\times\)4+1\(\times\)2+ 0\(\times\)1
(1110 )2 = 8+ 4+2+0
(0101)2 = (14)10
Now, (9)10 + (5)10 = (14)10
Hence, in both the number system results remain the same.
1’s Complement Representation of Binary Numbers
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1’s complement of a binary number is written by simply replacing all the 0’s by 1’s and all the 1’s by 0’s. It is represented as A= -A or (A)’
Example: 1’s complement of (10010)2 = (01101)2
Also Read: Types of Events in Probability
2’s Complement Representation of Binary Numbers
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2’s complement of a binary number is obtained by simply adding 1 to the 1’s complement of the given number . 2’s complement is used during the addition of negative numbers.
Example: Consider 1010. Let us take the 1’s complement of this number.
1’s complement of (1010)2 = (0101)2
Now add (1111) to the result,
(0101)2 + (1111)2 = (0110)2
Hence, 2’s complement of (1010)2 = (0110)2
Special Cases In Binary Addition
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Before discussing special cases one thing that should be clear is the sign bits. In the binary system, MSB (Most significant bit) tells us whether the number is positive or negative.
- If the MSB is 0 then it represents the positive sign.
- If the MSB is 1 then it represents the negative sign.
Positive And Negative Binary Number Addition (Addition using 1’s complement)
Case 1: When the positive number has a greater magnitude.
In this case, if the positive number is greater than the negative number then first we need to take the 1’s complement of the negative number and add the carry to the resultant least significant bit of the 1’s complement. To make it clear. Let’s look at the below:
Example: +1110 and -1101
Let us consider the given number as a 5 bit signed number. Now, rewrite the given equations as follows:
+1110 = 01110
-1101 =11101
Take the 1’s complement of the negative number = 11101
Note that the signed bit remained the same and we cannot take its complement.
Now, add the positive number with the 1’s complement of the negative number.
The solution is 00001 = +0001
Therefore, the result of (+1110)2 and (-1101)2 = ( +0001)2
Case 2: When the negative number has the greater magnitude
In this case, if the negative number is greater than the positive number then first we need to take the 1’s complement of the negative number but in this case, non-end carry is generated. So, the addition is obtained by taking only 1’s complement of the result. The result of magnitude bits will always be negative.
Example: +1011 and -1111
Let us consider the given number as a 5 bit signed number. Now, rewrite the given equations as follows:
+1011 = 01011
-1111 = 11111
Take the 1’s complement of the negative number
Now, add the positive number with the 1’s complement of the negative number.
Now take the 1’s complement of the result.
Therefore, the result of (+1011)2 and (-1111)2 = ( -0001)2
Case 3: Addition Of Two Negative Binary Numbers
In this case, if both the numbers are negative then first we need to take the 1’s complement of both the negative numbers and always generate the carry. Add the carry which is generated from the MSB to the LSB of the resultant magnitude. So, the addition is obtained by taking only 1’s complement of the result. In this case, the result of magnitude bits will always be negative.
Example: -0110 and -0111
Let us consider the given number as a 5 bit signed number. Now, rewrite the given equations as follows:
-0110 = 10110
-0111 = 10111
Take the 1’s complement of the both the negative numbers
Now, add the 1’s complement of the negative numbers.
Now, take the 1’s complement of the sum to obtain the final result.
Therefore, the final result is(10010)2,=(-1101).
Things to Remember
- Binary number system has a base or radix of 2.
- Binary number systems perform all the arithmetic operations like addition, subtraction, multiplication and division.
- To add negative binary number 1’s complement is required.
- If the positive number has greater magnitude than negative then only 1’s complement is required and carry added to the result to get the required sum.
- If the negative number has greater magnitude than positive then 2’s complement is required and it generates non-end carry.
- If MSB(most significant bit) is 0 then the number is positive.
- When MSB(most significant bit) is 1 then the number is negative.
Also Read:
Sample Questions
Ques. What is the base or radix in the number system? [2 marks]
Ans. Base or Radix is defined as the number of different symbols used in the number system. In simpler words, the number of values that a character or digit can assume is known as the bae or radix. It is represented by the ‘r’ or ‘b’. Mathematically,
Base or Radix, r = r-1
For example, the highest number system in binary is (2-1)=1. So, the radix or base of the binary number system is 2. Also, it holds either ‘0’ or’1’.
Ques. Define bit, byte, nibble and word. [3 marks]
Ans. All these terms are used to define the combination in the binary number system. It is
Bit: Bit is the smallest unit of information and abbreviation of binary digit. It is either ‘0’ or ‘1’.
Nibble: Nibble is the combination of four binary bits. In BCD and Hexadecimal number system nibble is used because both are four digit number systems.
For example; 1 nibble= 4 bits
Byte: Byte is the combination of eight binary bits. It is most commonly used in memory. Also, a byte contains two nibbles.
For example, 1 Byte= 2 Nibbles= 8 Bits
Word: Word is the combination of sixteen binary bits. It contains 2 bytes or four nibbles. Also, it is used to represent memory.
For example; 1 Word=2 Bytes= 4 Nibbles= 16 Bits
Ques.Add the following binary numbers: (11010)2 and (01010)2 . [2 marks]
Ans. The addition of (11010)2 and (01010)2 is 100100
Ques. Add the following binary numbers: [3 marks]
(101.01)2 and (11.10)2
(110101.101)2 and (1101.11)2
Ans.
(a) The addition of (101.01)2 and (11.10)2 is
(b) The addition of (110101.101)2 and (1101.11)2 is
Ques. Add the following number and also give its equivalent decimal addition. [5 marks]
(a) (01101)2 and (01110)2
(b) (11100)2 and (10011)2
Ans. (a) Given,
Binary Addition:
Decimal Addition:
(01101)2 ⇒ 0*24+1*23+ 1*22 +0*21+1*20
(01101)2 ⇒ 0+1*8+ 1*4 +0*2+1*1
(01101)2 ⇒ 0+8+ 4 +0+1
(01101)2 ⇒ 13
(01101)2 ⇒ (13)10
And
(01110)2 ⇒ 0*24+1*23+ 1*22 +1*21+0*20
(01110)2 ⇒ 0+1*8+ 1*4 +1*2+0
(01110)2 ⇒ 0+8+ 4 +2+0
(01110)2 ⇒ 14
(01110)2 ⇒ (14)10
Now, (13)10 + (14)10 = (27)10
(11011)2 ⇒ 1*24+1*23+ 0*22 +1*21+1*20
(11011)2 ⇒ 1*16+1*8+ 0*4 +1*2+1*1
(11011)2⇒ 16+8+ 0 +2+1
(11011)2 ⇒ 27
(11011)2 ⇒ (27)10
Hence, (01101)2 + (01110)2 = (11011)2
(b) Binary Addition:
Decimal Addition:
(11100)2 ⇒ 1*24+1*23+ 1*22 +0*21+0*20
(11100)2 ⇒ 1*16+1*8+ 1*4 +0*2+0*1
(11100)2 ⇒ 16+8+ 4 +0+0
(11100)2 ⇒ 28
(11100)2 ⇒ (28)10
And
(10011)2 ⇒ 1*24+0*23+ 0*22 +1*21+1*20
(10011)2 ⇒ 1*16+0*8+ 0*4 +1*2+1*1
(10011)2 ⇒ 16+0+ 0 +2+1
(10011)2 ⇒ 19
(10011)2 ⇒ (19)10
Now, (28)10 + (19)10 = (47)10
(101111)2 ⇒ 1*25+1*24+0*23+ 1*22 +1*21+1*20
(101111)2 ⇒ 1*32+0*16+1*8+ 1*4 +1*2+1*1
(101111)2 ⇒ 32+0+8+ 4+2+1
(101111)2 ⇒ 47
(101111)2 ⇒ (47)10
Hence, (11100)2 + (10011)2 = (101111)2
Ques. What are the fundamental rules to do binary addition in mathematics? [2 marks]
Ans. There are four rules for binary addition. The below are the rules for binary addition:
Rule 1: if both the binary digits are zero then the result of addition is also zero with carry zero.
Mathematically, 0+0 = 0 ; Carry=0
Rule 2: If the first binary number is 0 but the second binary number is 1 then the result of addition is 1 with carry 0.
Mathematically; 0+1= 1 ; Carry=0
Rule 3: If the first binary number is 1 but the second binary number is 0 then the result of addition is 1 with carry 0.
Mathematically; 1+0= 1 ; Carry=0
Rule 4: If the both (first and second) binary number is 1 then the result of addition is 0 with carry 1.
Mathematically; 1+1= 0 ; Carry=1 ( since carry is forwarded to the next bit)
Ques. Add the +1101 and - 1011. [2 marks]
Ans. We all know that when the positive number has a greater magnitude than negative. Then, the 1’s complement of the negative number is used to do binary addition.
Now, -1011 1’s complement is ⇒ 10100 ( sign of sign bit remain same)
+1101 ⇒01101
Addition of these two number is
Therefore, the required sum is (00010)2
Where, 0010 is the magnitude and 0 indicates the sign bit.
Ques. Add the (-1010)2 and (-0101)2 . [2 marks]
Ans. First we take 1’s complement then add them.
1’s complement of (-1010)2 and (-0101)2 are as follows:
-1010 ⇒ 10101
-0101⇒ 11010
Now, to get the sum of two binary numbers is obtained after taking the 1’s complement of the above result. We get,
10000 1’s complement is ⇒ 11111
Therefore, the sum of two negative binary numbers is -1111 and 1 is the MSB(most significant bit) which indicates sign bit.
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