What is Binary Addition?

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What is Binary Addition? Check out the fundamentals of binary addition in this concise guide. Learn how ones and zeros combine to create a powerful mathematical tool for computers and digital systems.

What is Binary Addition?

Binary addition is a fundamental arithmetic operation in which two binary numbers (numbers expressed in the base-2 numeral system) are added together. In the binary numeral system, there are only two possible digits: 0 and 1. When performing binary addition, the same rules of carrying over that apply to decimal addition also apply, but with simpler possibilities due to the limited digit set.

Here’s how binary addition works:

• Start by adding the rightmost digits of the two binary numbers, along with any carry from the previous column (which starts as 0).
• If the sum of the digits is 0 or 1, write down that sum as the result for the current column and carry over 0 to the next column.
• If the sum of the digits is 2, write down 0 as the result for the current column and carry over 1 to the next column.
• If the sum of the digits is 3, write down 1 as the result for the current column and carry over 1 to the next column.
• Repeat the process for each column, moving from right to left.

What are the Rules of Binary Addition?

Binary addition follows similar principles to decimal addition, but it only involves two digits: 0 and 1. Here are the rules for binary addition:

Single-bit Addition:

0 + 0 = 0

0 + 1 = 1

1 + 0 = 1

1 + 1 = 0 (with a carry of 1)

Carry:

When adding two 1s, you get a sum of 0 and carry over 1 to the next higher position.

Column-wise Addition:

Start adding from the rightmost column (least significant bit) and move towards the left.

Here’s a step-by-step example of binary addition:

1101 (13 in decimal)

+ 1011 (11 in decimal)

_______

10000 (16 in decimal)

Explanation:

• In the rightmost column: 1 + 1 = 0 (carry 1)
• Second column: 1 + 0 (carry 1) = 0 (carry 1)
• Third column: 0 (carry 1) + 1 = 0 (carry 1)
• Fourth column: 1 + 0 (carry 1) = 0 (carry 1)
• Fifth column (leftmost): 0 + 1 = 1 (carry 0)

Note: The final carry after the leftmost column is typically ignored if there’s no additional column to add. Keep in mind that the concepts of binary addition are the same as decimal addition, but there are only two possible digits (0 and 1), and the concept of carrying over works a bit differently due to the limited digit range.

How to Solve Binary Addition?

Solving binary addition is similar to solving decimal addition, but it involves working with the binary numeral system, which uses only two digits: 0 and 1. Here’s a step-by-step guide on how to solve binary addition:

Let’s say you have two binary numbers to add: A = 1101 and B = 1010.

Step 1: Write down the two binary numbers, aligning them by their rightmost digits (least significant bits):

1101

+ 1010

Step 2: Start adding the corresponding digits from right to left, just like you would with decimal addition:

1101

+ 1010

——-

Step 3: Add the rightmost digits (LSBs).

1 + 0 = 1 (binary)

Write down the result under the line:

1101

+ 1010

——-

1

Step 4: Move one position to the left and add the next digits.

0 + 1 = 1 (binary)

Write down the result:

1101

+ 1010

——-

11

Step 5: Continue this process, moving one position to the left each time and adding the corresponding digits.

1 + 1 = 0 (carry 1) (binary)

1 + 0 + 1 = 0 (carry 1) (binary)

1 + 1 + 0 = 1 (carry 1) (binary)

0 + 0 + 1 = 1 (binary)

Write down the results and any carry digits:

1101

+ 1010

——-

10111

Step 6: Check if there’s any remaining carry from the last addition. In this case, there’s a carry of 1 from the leftmost addition.

1101

+ 1010

——-

10111

So, the sum of the binary numbers 1101 and 1010 is 10111.

Remember to carry over when the sum of the digits is 2 (10 in binary). Binary addition follows the same principles as decimal addition but with fewer possible digit values.

Binary Addition Examples with Solutions

Here are some binary addition examples along with their solutions. Binary addition works similarly to decimal addition, but it uses the base-2 numeral system (0 and 1) instead of the base-10 system (0 through 9). Let’s go through a few examples:

Example 1:

1010

+ 1101

——-

Step 1: Add the rightmost digits (1 + 0) which equals 1. Write down 1.

1010

+ 1101

——-

1

Step 2: Move to the next digits from the right (1 + 1), which equals 0 with a carry of 1. Write down 0.

1010

+ 1101

——-

01

Step 3: Add the next digits along with the carry (0 + 1 + 1), which equals 0 with a carry of 1. Write down 0.

1010

+ 1101

——-

001

Step 4: Add the leftmost digits along with the carry (1 + 1 + 1), which equals 1 with a carry of 1. Write down 1.

1010

+ 1101

——-

1001

The final result is 1001 in binary, which is equivalent to 9 in decimal.

Example 2:

11101

+ 10011

——–

Step 1: Add the rightmost digits (1 + 1) which equals 0 with a carry of 1. Write down 0.

11101

+ 10011

——–

0

Step 2: Move to the next digits (1 + 1), which equals 0 with a carry of 1. Write down 0.

11101

+ 10011

——–

00

Step 3: Add the next digits along with the carry (0 + 0 + 1), which equals 1. Write down 1.

11101

+ 10011

——–

100

Step 4: Continue with the next digits (1 + 0), which equals 1.

11101

+ 10011

——–

1100

Step 5: Finally, add the leftmost digits along with the carry (1 + 1), which equals 0 with a carry of 1. Write down 0.

11101

+ 10011

——–

111000

The final result is 111000 in binary, which is equivalent to 56 in decimal.

These examples illustrate how binary addition works. You add the digits from right to left, considering any carry from previous additions.

Binary Addition Table

Here’s a simple binary addition table that shows the results of adding various binary numbers:

In this table, Binary A and Binary B are the numbers being added in binary notation, and the Sum column shows the result of their addition in binary as well. The numbers in parentheses in the Sum column are the decimal equivalents of the binary sums.

Solved Examples on Binary Addition

Binary addition follows the same principles as decimal addition, but with only two possible digits: 0 and 1. Let’s go through a few examples step by step.

Problem 1: Add 1101 and 1011

1101

+ 1011

——-

Step 1: Start from the rightmost digits (least significant bits) and work your way to the left.

1 + 1 = 0 (carry 1)

0 + 1 + 1 = 0 (carry 1)

1 + 0 + 1 = 0 (carry 1)

1 + 1 + 0 = 0 (carry 1)

Now, we add the carry to the leftmost digits:

1 + 1 = 0 (carry 1)

So, the result is 10100 in binary, which is equivalent to 20 in decimal.

Problem 2: Add 10101 and 110

10101

+ 110

——-

Step 1: Start from the rightmost digits (least significant bits) and work your way to the left.

1 + 0 = 1

1 + 1 = 0 (carry 1)

0 + 1 = 1

1 + 1 = 0 (carry 1)

0 + 1 = 1

Now, we add the carry to the leftmost digits:

1 + 0 = 1

So, the result is 100011 in binary, which is equivalent to 35 in decimal.

Problem 3: Add 11111 and 10

11111

+ 10

——-

Step 1: Start from the rightmost digits (least significant bits) and work your way to the left.

1 + 0 = 1

1 + 1 = 0 (carry 1)

1 + 0 = 1

1 + 0 = 1

1 + 0 = 1

Now, we add the carry to the leftmost digit:

1 + 0 = 1

So, the result is 100001 in binary, which is equivalent to 33 in decimal.

Remember, binary addition is similar to decimal addition, but you only have two possible digits (0 and 1). If the sum of two digits is 2, you write down 0 and carry over 1 to the next column. This is similar to how carrying works in decimal addition when the sum is 10 or more.

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