Binary Addition: Conversion, Definition, Examples

What Is Binary Addition?

The number system that has the base “2” is called a binary number system. It uses only two numbers, 0 and 1. Each digit in the binary number system is known as a “bit.” Binary bits “0” and “1” stand for “LOW” and “HIGH,” or “OFF” and “ON” respectively. Let’s take a look at some equivalent values of decimal and binary addition.

Binary addition is done in the same way as we add two or more numbers in a decimal number system. We start adding digits from the right. We can use an online Binary Calculator for instant binary addition.

Let’s take a look at binary equivalents of $1 –10$ numbers, which will be useful as we start learning binary addition.

Definition of Binary Addition

Binary addition can be defined as the method of adding two binary numbers. It follows a set of rules. The rules are framed; for example, the answer to the binary addition and decimal addition is the same. Consider an example.

$2 + 1 = 3$ in decimal addition.

In binary addition, we get

$10_{2} + 1_{2} = 11_{2}$, which represents 3 in the decimal number system.

Binary addition is pretty similar to decimal addition. However, since the base is 2, we have to understand how to use a carryover. In decimal addition, when we get $7 + 3 = 10$, we write 0 and take 1 as a carryover. Similarly, in binary addition, when you get $1 + 1 = 2$, which is 10 in the decimal system, we write 0 and take a carry of 1. Let’s understand this in detail!

Rules of Binary Addition 

Adding in binary is significantly simpler than adding in decimal when you keep in mind the following tips or guidelines. Let’s take a look at the rules of binary addition.

Rule 1: $0 + 0 = 0$; Carry $= 0$

Rule 2:  $0 + 1 = 1$; Carry $= 0$

Rule 3: $1 + 0 = 1$; Carry $= 0$

Rule 4: $1 + 1 = 0$; Carry $= 1$

We can summarize this as:

Binary Addition Table

The table of adding two binary numbers 0 and 1 is given below:

In this table, x and y are the two binary numbers. So when we give the input for $x = 0$ and $y = 0$, 

then the output is equal to 0. When $x = 0$ or 1 and $y = 1$ or 0, then $x + y = 1$. But when both x and y are equal to 1, then their addition equals 0, but the carryover number will equal to 1, which means basically $1 + 1 = 10$ in binary addition, where 1 is carried forward to the next digit.

How to Do Binary Addition

Addition of binary numbers includes two cases:

• Binary addition with regrouping
• Binary addition without regrouping

Let us understand binary addition step by step.

Binary Addition with Regrouping

Regrouping is necessary when the sum of two or more binary digits is greater than 1. Let’s add binary numbers $1011_{2}$ and $111_{2}$ to understand it better.

Step 1:  Write the numbers such that their respective place values are aligned. 

Step 2: Add the digits at ones place using the binary addition rules.

Now, according to binary addition rules, $1 + 1 = 10$.

Thus, we will write 0 at the bottom and take 1 as a carryover to the next place value. 

Step 3: Move to the next column on the left, following the rules, $1 + 1 + 1 = 10 + 1$. We will again write 1 at the bottom and take 1 as a carryover to the next place value. 

Step 4: Completing the next column in the similar way, we get

Step 5: Now coming to the final column, 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.

Hence, $1011_{2} +  111_{2} = 10010_{2}$

Binary Addition without Regrouping

When the addition of two or more binary digits is 0 or 1, regrouping is not necessary. Let’s add binary numbers $100_{2}$ and $11_{2}$ to understand it better. 

Step 1: Arrange the numbers such that the place values are aligned.

Step 2: Start adding from the rightmost column. Here, we first add 0 and 1. According to binary addition rules, $0 + 1 = 1$. There is no carryover here.

Step 3: Similarly, there is no carryover when we add the digits in the next columns. We get:

 Hence, $100_{2} +  11_{2} = 111_{2}$

Binary Addition Using 1’s Complement

There are two types of complements of any binary number—1’s complement and 2’s complement. 1’s complement is used in signed binary representation. By using 1’s complement, we can add two negative binary numbers; we can also add one negative number and a positive number.

Two important things to remember in binary addition are:

• The number 0 represents the positive sign
• The number 1 represents the negative sign

To find the 1’s complement of a number we replace every 0 by 1 and every 1 by 0 in a binary number. 

For example, the 1’s complement of the binary number 101 is 010. 

Let us learn how to add a positive number to a negative number.

When the positive number is greater, we find the 1’s complement of the negative number.

For example, let us add $1011$ to $(-1010)$. We will calculate $1011\;-\;1010$.

Here, $+ 1011 \rightarrow 0 1011$

          $-1010 \rightarrow 1 0101$   (taking 1s complement)

Here, the carryover 1 that we got in the end (end-around carry) is again added to the result 0000. We get $00000 + 1 = 00001$

$1011_{2} + 0101_{2} = 0001_{2}$

When the negative number has greater magnitude, we take the 1’s complement of the negative number. There will be no end-around carry (the carryover we get in the end) in this case. Finally, the sum is obtained by taking the 1’s complement of the result.

Let’s add $1101$ and $-1110$.

Here,

 $+ 1101 \rightarrow 0 1101$

 $-1110\rightarrow 1 0001$   (taking 1’s complement)

Now, adding both numbers, we get

Final answer $=  -0001_{2}$ … taking 1’s complement and using the sign bit

Fun Facts!

• Binary as a system can be traced all the way back to the 1560s. It was Thomas Harriot, an English mathematician, who discovered the binary system.

• A binary code is adopted into various languages used today, including braille. 

• Binary numbers and their operations are used for various purposes, such as making electrical device circuits. These operations are highly used in computer technology, where 0 indicates the OFF state of the circuit, and 1 indicates its ON state.

• Binary number systems perform all the arithmetic operations like addition, subtraction, multiplication, and division.

• If the last digit of a binary number is 1, the number is odd; if it’s 0, the number is even. Ex: 1101 represents an odd number (13); 10010 represents an even number (18).

Conclusion

In this article, we learned in detail about the binary addition definition, rules, and facts. Let’s solve some examples and practice problems to understand the concept better.

Solved Examples

1. Solve: $1 1 1 0 0_{2} +  1 0 1 0 1_{2}$

Solution: 

Therefore, $1 1 1 0 0_{2} +  1 0 1 0 1_{2} = 1 1 0 0 0 1_{2}$

2. Find $1 1 0 1 1 0_{2} + 1 1 1 1_{2}$.

Solution: 

Therefore, $1 1 0 1 1 0_{2} + 1 1 1 1_{2} = 1 0 0 0 1 0 1_{2}$

3. Add  $110_{2} + 111_{2} + 101_{2}$

Solution: 

To perform the addition of three binary numbers, add two binary numbers together and then add the resultant value with the third one.

Let’s add $110 + 111$ first.

Now, let’s add the third binary number to the resultant value of addition of the first two.

Therefore, $110_{2} + 111_{2} + 101_{2} = 1 0 0 1 0_{2}$

Practice Problems

Frequently Asked Questions