Understanding Binary Addition

Binary addition is a fundamental operation in digital electronics and computer systems. Unlike decimal addition which uses base 10, binary addition uses base 2. The rules for binary addition are simple:

• \(0 + 0 = 0\) (with no carry)
• \(0 + 1 = 1\) (with no carry)
• \(1 + 0 = 1\) (with no carry)
• \(1 + 1 = 0\) (with a carry of 1 to the next higher position)
• \(1 + 1 + 1 = 1\) (with a carry of 1 to the next higher position)

When adding two binary numbers, we align them by their rightmost digit and add column by column, propagating any carries to the left, just like in decimal addition.

Let's find the sum of the given binary numbers: 1100100 and 101110.

Step-by-Step Binary Addition of 1100100 and 101110

To add 1100100 and 101110, we can align the numbers and add column by column from right to left.

Let's perform the addition column by column:

• Rightmost column (Position 1): \(0 + 0 = 0\). Carry = 0. Result digit = 0.
• Position 2: \(0 + 1 = 1\). Carry = 0. Result digit = 1.
• Position 3: \(1 + 1 = 0\). Carry = 1. Result digit = 0.
• Position 4: \(0 + 1 + \text{carry (1)} = 1 + 1 = 0\). Carry = 1. Result digit = 0.
• Position 5: \(0 + 0 + \text{carry (1)} = 0 + 1 = 1\). Carry = 0. Result digit = 1.
• Position 6: \(1 + 1 + \text{carry (0)} = 1 + 1 = 0\). Carry = 1. Result digit = 0.
• Position 7: \(1 + 0 + \text{carry (1)} = 1 + 1 = 0\). Carry = 1. Result digit = 0.
• Final Carry: The carry from Position 7 is 1, which becomes the leftmost digit of the sum. Result digit = 1.

Reading the resulting digits from left to right (including the final carry), we get 10010010.

Final Sum of Binary Numbers

The sum of the binary numbers 1100100 and 101110 is 10010010.

This matches one of the provided options.

Revision Table: Key Binary Addition Rules

Additional Information on Binary Numbers

Binary numbers are the foundation of digital computing. A binary digit (bit) can only be 0 or 1. Larger binary numbers represent values using powers of 2, similar to how decimal numbers use powers of 10. Understanding binary arithmetic, including binary addition, subtraction, multiplication, and division, is crucial for studying computer architecture, digital logic design, and low-level programming.

For instance, converting binary numbers to decimal can help verify the addition:

• \(1100100_2 = 1 \times 2^6 + 1 \times 2^5 + 0 \times 2^4 + 0 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 0 \times 2^0\)
• \(1100100_2 = 64 + 32 + 0 + 0 + 4 + 0 + 0 = 100_{10}\)
• \(101110_2 = 1 \times 2^5 + 0 \times 2^4 + 1 \times 2^3 + 1 \times 2^2 + 1 \times 2^1 + 0 \times 2^0\)
• \(101110_2 = 32 + 0 + 8 + 4 + 2 + 0 = 46_{10}\)

Their decimal sum is \(100 + 46 = 146_{10}\).

Let's convert the binary sum 10010010 to decimal:

• \(10010010_2 = 1 \times 2^7 + 0 \times 2^6 + 0 \times 2^5 + 1 \times 2^4 + 0 \times 2^3 + 0 \times 2^2 + 1 \times 2^1 + 0 \times 2^0\)
• \(10010010_2 = 128 + 0 + 0 + 16 + 0 + 0 + 2 + 0 = 146_{10}\)

The decimal sum matches the decimal conversion of the binary sum, confirming the correctness of the binary addition.