Simple Methods to Add Binary Numbers with and without Regrouping
Binary numbers, also known as base-2 number systems, are represented using two digits namely 0 and 1. The numbers in a binary number system look like this - 1100011010. Each digit in the binary number system is known as ‘Bit’.
All digital devices use a binary number system in their electronic circuit. The input 0 indicates OFF state and whereas input 1 indicates the On state. Because of these implementations, binary number systems are most widely used in modern computer technology. Read the article below to know how to perform Binary addition with and without regrouping.
These may include addition, multiplication, division, and subtraction. Each binary operation is represented by a different symbol. Besides being used in Mathematics, these operations play an important role in computer technology also. They help us make operating systems and circuits for various electrical devices like computers, laptops, smartphones, etc.
Basic Binary Arithmetic Operations
In this article, we will discuss binary addition in detail along with binary addition examples so students can perform calculations faster.
What is Binary Addition?
Binary addition is the sum of two or more binary numbers. Binary addition is much similar to decimal addition, even a bit easier. In the decimal addition, if the sum of two numbers results in two digits, we carry the digit in the ten’s place to the next column to the left. Similarly in binary addition, if the sum of two numbers is greater than 1, we carry the 2’s digit over to the next column to the left For example, 1+ 1 = 10₂. In this case, we write 1’s digit (0) and carry the 2’s digit i.e. 1 of the result to the next column to the left. For this reason, the bit that is carried to the next column is known as the carry bit.
Binary Addition Rules
The addition of two binary numbers is as easy as the decimal number system. Just we have to take note of some rules while adding two binary numbers. There are four-five rules associated with binary addition. The binary addition rules are as follows.
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 =10 ( carry 1 to the next significant bit)
1 + 1 + 1 = 11( carry 1 to the next significant bit)
As binary numbers include only two digits i.e. 0 and 1, these four five rules are all the possible conditions for the addition of binary numbers.
Here is the stepwise procedure of how to add two binary numbers with regrouping and without regrouping.
Binary Addition without Regrouping
When the sum of two or more binary digits results in 0 or 1, then in such cases we don’t need any regrouping. Let’s add binary numbers \[101_{2}\] and \[10_{2}\] to understand it in a better way.
Step 1: Write all digits of both the binary numbers in a separate column according to their place values as shown below
1 0 1
+ 1 0
………..
Step 2: Starting from the rightmost column, add 1 and 0. Follow the binary addition rules which says 1 + 0 = 1.
1 0 1
+ 1 0
………..
1
Step 3: Moving to the next column to the left, add 0 and 1. Follow the binary addition rules which says 0 + 1 = 1.
1 0 1
+ 1 0
………..
1 1
………..
Step 4: Moving again to the next column to the left, we can see there is only one digit left i.e. 1. Hence, we can apply the rule 1 + 0 = 1.
1 0 1
+ 1 0
………..
1 1 1
………..
Therefore, \[101_{2} + 10_{2} = 111_{2}\].
Binary Addition with Regrouping
When the sum of two or more binary digits results in more than 0 or 1, then in such cases we need regrouping. Let’s add binary numbers 1001₂ and 111₂ to understand it in a better way.
Step 1: Write all digits of both the binary numbers in a separate column according to their place values as shown below
1 0 0 1
+ 1 1 1
………….
Step 2: Starting from the rightmost column, add 1 and 1. Follow the binary addition rules which says 1 + 1 = 10. This is equivalent to 2₁₀. Hence, we will write 0 at the bottom and two take 1 as a carryover to the next place value.
1
1 0 0 1
+ 1 1 1
………….
0
Step 3: Move to the next column to the left. Follow the binary addition rules which says 1 + 0 + 1 = 10. This is again equivalent to 2₁₀. Hence, we will write 0 at the bottom and two take 1 as a carryover to the next place value.
1 1
1 0 0 1
+ 1 1 1
………….
0 0
Step 4: Move again to the next column to the left. Follow the binary addition rules which says 1 + 1 + 0 = 10. This is again equivalent to 2₁₀.
1 1 1
1 0 0 1
+ 1 1 1
………….
0 0 0
Step 5: Move again to the next column to the left. Follow the binary addition rules which says 1 + 1 + 0 = 10. This is again equivalent to 2₁₀. As it is the last column left, we will not take 1 as carryover, instead, we will write 10 as the result at the bottom.
1 1 1
1 0 0 1
+ 1 1 1
…………….
1 0 0 0 0
…………….
Therefore, \[1001_{2} + 111_{2} = 10000_{2}\]
Binary Addition Examples with Solutions
Example 1:
Add \[1010_{2} and 1111_{2}\]
Solution:
1 1
1 0 1 0
+ 1 1 1 1
---------------------------
1 1 0 0 1
---------------------------
Example 2:
Add: \[10011_{2} \, and \, 110001_{2}\]
Solution:
1 1 1
1 0 0 1 1
+ 1 1 0 0 0 1
------------------------------
1 0 0 0 1 0 0
------------------------------
Practice Problems
1. Add the binary numbers - 11001+10111
Ans: 0110000
2. What is the sum of 1111+0101?
Ans: 010100
Summary
Binary addition refers to adding more than one binary number. It is the same as the decimal system and covers binary numbers 0 and 1. For complex and fast calculations, we can use Binary addition converters. Binary numbers and their operations are used for various purposes, such as making electrical device circuits. Further, these operations are highly used in computer technology, where 0 indicates the OFF state of the circuit, and 1 indicates its ON state.
FAQs on Binary Addition Explained for Students
1. What is binary addition and why is it a fundamental concept in computing?
Binary addition is the process of adding numbers in the base-2 number system, which uses only two digits: 0 and 1. It is a fundamental concept because all modern digital computers and electronic devices perform calculations at their core level using binary arithmetic. Operations executed by a computer's Arithmetic Logic Unit (ALU), from simple calculations to complex data processing, are broken down into basic binary addition steps.
2. What are the four basic rules for performing binary addition?
The addition of binary numbers follows four simple rules, which are applied column by column, from right to left, just like in decimal addition. The rules are:
- 0 + 0 = 0
- 0 + 1 = 1
- 1 + 0 = 1
- 1 + 1 = 0 (with a carry-over of 1 to the next column)
3. Can you show an example of binary addition with a carry-over?
Certainly. Let's add the binary numbers 1011₂ and 0110₂.
Starting from the rightmost bit:
- Column 1 (Units place): 1 + 0 = 1.
- Column 2 (Twos place): 1 + 1 = 0, with a carry of 1.
- Column 3 (Fours place): 0 + 1 + (carry 1) = 1 + 1 = 0, with a carry of 1.
- Column 4 (Eights place): 1 + 0 + (carry 1) = 1 + 1 = 0, with a carry of 1.
- Final carry: The last carry of 1 is placed in the next column.
4. Why is 1 + 1 equal to 10 in binary (0 with a carry of 1) instead of 2?
This is a common point of confusion. The binary system only uses the digits 0 and 1; the digit '2' does not exist. The concept is similar to decimal addition. When we add 9 + 1 in decimal, the result is 10. We write '0' in the current place value and carry the '1' to the next. Similarly, in binary, 1 + 1 equals the decimal value of 2. To represent the number 2 in binary, we need two bits: 10₂. Therefore, the result in the current column is '0', and a '1' is carried over to the next higher place value.
5. How does binary addition differ from decimal addition?
The primary difference lies in the base of the number system. Decimal addition is performed in base-10 (using digits 0-9), while binary addition is performed in base-2 (using digits 0-1). While both systems use the principle of carrying over to the next place value, the threshold for this carry is different. In decimal, a carry is generated when a sum exceeds 9. In binary, a carry is generated when a sum exceeds 1.
6. What is the method for adding three or more binary numbers together?
To add multiple binary numbers, you can add them column by column. For each column, count the number of 1s. For example, if a column contains three 1s (1+1+1), the sum is 3 in decimal, which is 11₂ in binary. In this case, you would write '1' as the sum for that column and carry '1' over to the next column. Alternatively, you can add the first two numbers, take the result, and then add the third number to that result, continuing this process until all numbers are added.
7. Where is the principle of binary addition applied in everyday technology?
Binary addition is used constantly in almost all digital devices. For example, when you use a smartphone calculator, the processor performs binary addition. In a computer, tasks like running software, rendering graphics, or even processing sound involve countless binary calculations. It is also fundamental to network data transmission, where checksums are calculated using binary arithmetic to ensure data integrity during transfer.
