Question

Match the solution with the problem. A. 10001100 B. 10011110 C. 1101010 D. 1100000 E. 1010001 F. 1111000 \(1110011+11001\) (binary addition)

Short answer

The sum of the binary numbers is option F: 1111000.

Step-by-step solution

Align Binary Numbers

First, write the binary numbers one on top of the other, aligning them to the right. Make sure both numbers have the same digit length by prepping with zeros if necessary: ``` 01110011 + 00011001 ``` Here, we prepended zeros to the second number for clarity in addition.

Add Rightmost Digits

Begin the addition from the rightmost digits. In binary, 1 + 1 equals 10. So, write 0 and carry over 1 to the next column: ``` 01110011 + 00011001 --------- 0 (and carry 1) ```

Continue Addition for Each Column

Move to the next column left, add 1 + 1 (current column) + 1 (carry), which equals 11 (3 in decimal). Write down 1 and carry over 1: ``` 01110011 + 00011001 --------- 10 (and carry 1) ```

Repeat Process

Continue this process across each column: - Third column: 0 + 0 + 1 (carry) = 1. Write down 1: - Fourth column: 1 + 1 = 10. Write down 0, carry over 1. - Fifth column: 1 + 1 + 1 (carry) = 11. Write down 1, carry over 1. This process continues throughout all columns: ``` 01110011 + 00011001 --------- 01111100 ```

Check Final Column Carry

Ensure to check for any remaining carry from the leftmost column. In this case, the final carry does not require a new digit. If there had been, it would have been written as an additional digit on the left. Thus, the sum of the binary numbers is: ``` 01111100 ```

Match to the Options

The binary result from the addition (01111100) is equivalent to 111100 in decimal representation. Match this result to the given options: - A. 10001100 - B. 10011110 - C. 1101010 - D. 1100000 - E. 1010001 - F. 1111000 Option F (1111000) appears to be the closest match in binary form.

Key concepts

Understanding Binary Numbers

Binary numbers are the building blocks of digital systems. Unlike the decimal system that uses ten digits (0-9), the binary system only uses two digits, 0 and 1.
This simplicity allows computers to process data efficiently. Each digit in a binary number is a 'bit'. The rightmost bit is the least significant, and as you move left, each bit increases in significance, akin to decimal's places of tens, hundreds, etc.
In binary, these are referred to as powers of 2. For example, the binary number 1101 is equivalent to decimal 13, as it represents \( 1 \times 2^3 + 1 \times 2^2 + 0 \times 2^1 + 1 \times 2^0 \).
Mastering binary numbers is crucial for understanding binary addition, a fundamental operation in computing.

Step-by-Step Solution to Binary Addition

Adding binary numbers requires careful alignment and understanding of their addition rules. Just like in decimal addition, binary addition starts from the rightmost digit.
You add each corresponding pair of digits from the two numbers, just as you would with their decimal counterparts. The twist is in the carrying.
In binary, when you add 1 and 1, you get 10, which is conceptually similar to saying 10 in decimal means carry over 1 to the next column. Here's a step-by-step for clarity:

• Align the numbers: Ensure they are of equal length by padding zeros on the shorter one.
• Add from the right: Begin adding each bit from right to left. Keep a tally of any carries to the next digit.
• Record carries: When a column sums to 10 (binary), carry 1 to the next column.

This process is repeated until all columns are completed, ensuring a correct sum of binary numbers.

Understanding Carry in Binary Addition

The carry in binary addition is pivotal to obtaining the correct sum. In binary, carrying occurs more frequently than in decimal addition because the base is smaller (base 2 vs. base 10).
When you add 1 + 1 in binary, the sum is 10. Here, 0 stays in the current column, while 1 is carried over to the next column on the left.
Let's break it down: suppose you're adding two binary numbers where several bits are 1; as you progress from right to left, every instance of 1 + 1 requires a carryover.

• Carries stack when the next column addition also results in another carry.
• Each carry represents the smallest unit moving to a higher place value.

Effectively managing carryovers ensures the accuracy of binary addition results, similar to decimals but based on binary's base.

Aligning Binary Numbers for Addition

Proper alignment of binary numbers is essential before commencing addition. It involves matching the digits by their place values, starting from the rightmost side.
In cases where binary numbers of different lengths are to be added, it's crucial to prepend zeros to the shorter number. This aligns their least significant bits (rightmost digits).

• Why align? Correct alignment ensures that each bit is correctly positioned according to its value, just like aligning tens, hundreds, and thousands in decimal.
• How to align: Simply add leading zeros to the number with fewer digits until both numbers have the same length.

This simple but important step guarantees that during binary addition, calculations are accurate and reflect real values.