Understanding Binary Addition: Sum of 1101111 and 1100101

This problem asks us to find the sum of two binary numbers: 1101111 and 1100101. Binary addition is similar to decimal addition, but instead of carrying over when the sum reaches 10, you carry over when the sum reaches 2 (which is 10 in binary).

Rules of Binary Addition

Here are the basic rules for adding two binary digits:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 0 = 1
• 1 + 1 = 10 (write down 0, carry over 1)

When adding three binary digits (like two bits plus a carry bit):

• 0 + 0 + 0 = 0
• 0 + 0 + 1 = 1
• 0 + 1 + 1 = 10 (write down 0, carry over 1)
• 1 + 1 + 1 = 11 (write down 1, carry over 1)

Step-by-Step Binary Addition

Let's add the two given binary numbers, 1101111 and 1100101, column by column from right to left, just like in decimal addition.

Reading the result bits from bottom to top (left to right in the sum) and including the final carry, we get:

Final Carry: 1

Column 7: 1

Column 6: 0

Column 5: 1

Column 4: 0

Column 3: 1

Column 2: 0

Column 1: 0

Combining these digits gives the sum: 11010100.

Let's visualize the addition:

  1 1 0 1 1 1 1
+ 1 1 0 0 1 0 1
-----------------
1 1 0 1 0 1 0 0
  \(\text{carry:}\) 1 1   1 1   1 0

Starting from the right:

• \(1 + 1 = 10_2\). Write 0, carry 1.
• \(1 + 0 + \text{carry } 1 = 10_2\). Write 0, carry 1.
• \(1 + 1 + \text{carry } 1 = 11_2\). Write 1, carry 1.
• \(1 + 0 + \text{carry } 1 = 10_2\). Write 0, carry 1.
• \(0 + 0 + \text{carry } 1 = 1_2\). Write 1, carry 0.
• \(1 + 1 + \text{carry } 0 = 10_2\). Write 0, carry 1.
• \(1 + 1 + \text{carry } 1 = 11_2\). Write 1, carry 1.

The final result is 11010100.

Verification (Optional)

We can convert the binary numbers to decimal to check our result.

• \(1101111_2 = 1 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 1 \times 2^0\)
  \(= 64 + 32 + 0 + 8 + 4 + 2 + 1 = 111_{10}\)
• \(1100101_2 = 1 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0\)
  \(= 64 + 32 + 0 + 0 + 4 + 0 + 1 = 101_{10}\)

Adding the decimal numbers: \(111_{10} + 101_{10} = 212_{10}\).

Now, let's convert our binary result \(11010100_2\) to decimal:

• \(11010100_2 = 1 \times 2^7 + 1 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0\)
  \(= 128 + 64 + 0 + 16 + 0 + 4 + 0 + 0 = 212_{10}\)

Since the decimal sums match (\(212 = 212\)), our binary addition is correct.

Revision Table: Binary Numbers and Addition

Additional Information: Importance of Binary in Computing

Binary numbers are fundamental in computing because digital circuits operate using two distinct voltage levels, which can represent 0 and 1. All data processed by computers, from text and images to instructions and calculations, is ultimately represented and manipulated using binary code. Understanding binary addition is crucial for comprehending how computers perform basic arithmetic operations.

Other important binary operations include subtraction, multiplication, and division, as well as bitwise operations like AND, OR, NOT, and XOR, which are essential in digital logic and programming.