Binary Addition

Binary addition involves the addition of two binary numbers, 0 and 1. Each digit is called a bit.

• 0 denotes LOW or OFF
• 1 denotes HIGH or ON

Rules

The addition in binary is similar to decimal addition involving 2 numbers. It follows 4 rules:

1. 0 + 0 = 0
2. 0 + 1 = 1
3. 1 + 0 = 1
4. 1 + 1 = 10 (0 carry 1)

We can summarize it as:

A + B	Sum	Carry
0 + 0	0	0
0 + 1	1	0
1 + 0	1	0
1 + 1	0	1

The result of binary addition is the same as we obtain from decimal addition, with the difference in the place values of the digits.

Addition with Regrouping

Regrouping is done when the sum of 2 or more binary digits is greater than 1. 

Let us add the binary numbers 1101₂ and 101₂. Here, we show the overflow of the process.

Step 1: Aligning the numbers based on their place values

	1	1	0	1
+		1	0	1

Step 2: Adding the digits in ones place using the binary addition rules.

1 + 1 = 10

Thus,

			1
	1	1	0	1
+		1	0	1
				0

Step 3: Moving on to the next column.

1 + 0 + 0 = 1

			1
	1	1	0	1
+		1	0	1
			1	0

Step 4: In the 3^rd column

1 + 1 + 10, so carry 1

	1		1
	1	1	0	1
+		1	0	1
		0	1	0

Step 5: Finally,

1 + 1 = 10

			1
	1	1	0	1
+		1	0	1
1	0	0	1	0

Thus, 1101₂ + 101₂ = 010010₂

Addition without Regrouping

When the 2 binary numbers being added are 0 and 1, regrouping is unnecessary.

Let us add the binary numbers 101₂ and 10₂

Step 1: Aligning the numbers based on their place values

	1	0	1
+		1	0

Step 2: Adding the digits in ones place using the binary addition rules.

1 + 0 = 1

		1	0	1
+			1	0
				1

Step 3: Moving on to the second column.

Similarly, 0 + 1 = 1

Here, also there is no carry

		1	0	1
+			1	0
			1	1

Step 4: Finally, in the last column

1 + 0 = 1

		1	0	1
+			1	0
	0	1	1	1

Thus, 101₂ + 10₂ = 0111₂

Binary Addition Using 1’s Complement

There are 2 types of complement of a binary number: 1’s complement and 2’ complement. In 1’s complement, we add two negative binary numbers, while 2’s complement of a binary number is always 1, added to the 1’s complement of the binary number.

Rule

1. 0 represents a positive sign
2. represents a negative sign

Addition of Positive and Negative Numbers

There are 2 possible situations:

Case 1: When the Positive Number is Greater in Magnitude

Here, we take the 1’s complement of the negative number, and the carried number is added to the sum of the numbers at the 1’s place. To find the 1’s complement of a number, we alter every 0 by 1 and every 1 by 0.

For example, 1’s complement of the binary number 100 is 011.

Let us add the numbers: 1010₂ + (-1101₂)

Rewriting, we get:

1110₂ – 1101₂

+ 1110₂ → 01110₂

-1101₂ → 10010₂ (Taking 1’ complement)

	0	1	1	1	0
+	1	0	0	1	0
	0	0	0	0	0
					1
	0	0	0	0	1

Thus, the required sum is + 0001.

Case 2: When the Negative Number is Greater in Magnitude

In this case, we take the 1’s complement of the negative number, and there is no end-around carrying. Finally, the result is obtained by taking the 1’s complement of the sum.

The result will always be a negative number.

Let us add the numbers: 0111₂ + (-1000)₂

1’s complement of -1000 → 0111₂

Now, adding the 1’s complement to 0111₂

	0	1	1	1
+	0	1	1	1
	1	1	1	0

Finally, by taking the 1’s complement of the sum, we get

 1 1 1 0 → 0 0 0 1

Here, we will add a negative sign before the number (as the negative number was greater) to get the final result. 

0 0 0 1 → – 0 0 0 1

Addition of Two Negative Numbers

Here, we find the 1’s complement of both the negative numbers and then add both the complement numbers. Here, we will get the end-around carry to get the final result.

Let us add the numbers: -1010₂ +(- 0011₂)

Taking 1’s complement of both the numbers

1010₂ → 10101₂

0011₂ → 11100₂

	1	0	1	0	1
+	1	1	1	0	0
	1	0	0	0	1
					1
	1	0	0	1	0

Taking the 1’s complement of 10010₂ to get the result10010₂ → 11101₂