Question

Match the solution with the problem. A. 10001100 B. 10011110 C. 1101010 D. 1100000 E. 1010001 F. 1111000 \(1111111+11111\) (binary addition)

Short answer

The correct answer is B: 10011110.

Step-by-step solution

Align the Numbers

Start by aligning the binary numbers one above the other, ensuring that the least significant bits (rightmost bits) are aligned. We have:\[\begin{array}{c}1111111 \+ 11111 \\end{array}\]

Perform Binary Addition from Right to Left

Begin the addition from the rightmost bits. If both bits are 1, write 0 and carry over 1 to the next column. Repeat this process for each pair of bits from right to left. Don't forget to include the carried over bits:\[\begin{array}{ccccccc} & 1 & 1 & 1 & 1 & 1 & 0 \ & & 1 & 1 & 1 & 1 & 1 & 1 \ & + & & 1 & 1 & 1 & 1 & 1 \\hline1 & 0 & 0 & 1 & 1 & 1 & 1 & 0 \\end{array}\]

Double Check Each Column

Ensure each column is correctly added by verifying each pair of bits along with any carried over bits. This helps verify that the carried bits were correctly accounted for.

Verify the Result Against the Options

Compare your result, which is 10011110, with the options provided. Verify each step to ensure accuracy.

Key concepts

binary arithmetic

Binary arithmetic involves operations on binary numbers, which are based on a base-2 numeral system. Instead of using ten digits (0-9) as in the decimal system, binary uses only two digits: 0 and 1. This system makes it the foundation for digital electronics and computer systems since these two states can easily be represented by electrical signals.
In binary addition, similar to decimal addition, you line up numbers by their rightmost digits and proceed from right to left. However, due to the base-2 system, the rules for carrying over differ. Understanding binary arithmetic is essential in computing because it is the basic arithmetic used by all modern digital computers. Without this understanding, it's easy to get lost when working with any kind of digital data processing.

carry over method

When performing binary addition, the carry over method becomes important. This is akin to carrying over numbers in decimal addition when the sum of two digits exceeds the base-10. However, in binary addition, any time two bits (1s) are added, resulting in 10 (which is 2 in decimal), you carry a 1 over to the next higher bit position. For example, if you add three 1s in one column, you will record a 1 in the result position (2 in binary is represented as 10). You then carry a 1 over to the next column. This carry over is essential for maintaining accuracy in binary sums.
Keep in mind the rule:

• 0 + 0 = 0
• 0 + 1 = 1
• 1 + 1 = 10 (write 0 and carry 1)
• 1 + 1 + 1 = 11 (write 1 and carry 1)

The carry over method ensures that each column reflects the correct sum, taking into account any carried values from the previous column.

verification steps

Verification steps in binary addition are a vital process to ensure that every column has been calculated correctly. After performing the addition, go back through your work to double-check each calculation.
It's particularly important to verify that every carry over was correctly added and accounted for in the subsequent higher-order bit. This helps to prevent errors that could cause the entire result to be incorrect.

• Check columns from right to left.
• Ensure the carried over bits were added accurately to the next column.
• Consider each digit’s influence on the next higher bit.

By performing these verification steps, you can be more confident in the accuracy of your binary addition result. This process is similar to checking your work in traditional arithmetic by re-doing calculations or estimating to confirm the results make sense.

alignment of bits

The alignment of bits is a crucial initial step when adding binary numbers. Just like decimal addition, where you align by the least significant digits (rightmost ones), binary numbers need to be aligned in the same way. This ensures that each bit is correctly added to its corresponding bit in the other number.
In the given example, we align the numbers (e.g., 1111111 and 11111), ensuring the least significant bits, or the rightmost bits, correspond to each other. Aligned numbers make the addition process systematic and straightforward, reducing the chance for errors. When bits are correctly aligned:

• Write smaller numbers with leading zeros for visual matching, if necessary.
• Focus on aligned columns for clarity when performing additions.
• Alignment ensures easier tracking of carry overs.

Correct alignment provides a solid foundation to correctly execute further arithmetic processes like carrying over or verification, resulting in precise calculations.