How do i solve this question---**Extraction Content:** **Topic:** Ones’ complement addition **Question Stem:** Add 11110111 and 11111010 using ones’ complement arithmetic. **Solution:** Solution **Mathematical Operation / Calculation Steps:** Binary addition of 11110111 and 11111010: ``` 11110111 + 11111010 ---------- ``` Result of the initial addition showing a carry-out: ``` 1 111110001 ``` The leftmost '1' is the carry-out. End-around carry operation: The carry-out '1' is added to the initial sum: ``` 111110001 + 1 ---------- ``` Final result: ``` 111110010 ``` **Chart/Diagram Description:** Type: Mathematical calculation representation. Main Elements: * Two binary numbers (11110111 and 11111010) arranged vertically for addition. * A plus sign (+) to the right of the second number. * A horizontal line separating the numbers being added from the initial sum. * An initial sum (111110001) below the line, where the leftmost '1' is visually separated and highlighted in red. * A green curved arrow pointing from the separated red '1' (carry-out) towards the rightmost position below the sum. * A second horizontal line separating the initial sum (without the carry-out) and the carry-out from the final result. * The red '1' (carry-out) is shown below the first result line, aligned to the right, with a plus sign (+) next to it, indicating it is to be added. * The final result (111110010) below the second line.

视频信息

答案文本 复制

视频字幕 复制

In ones' complement addition, we first add the binary numbers normally. Here, we're adding 11110111 and 11111010. The initial sum gives us 111110001, with a carry-out of 1 at the leftmost position. In ones' complement, we perform an end-around carry operation, which means we add this carry-out back to the least significant bit of our result. After adding the carry-out, our final result is 111110010. Let's examine the binary addition process in detail. We start by adding the rightmost bits: 1 plus 0 equals 1. Moving left, 1 plus 1 equals 10 in binary, so we write 0 and carry 1. For the next column, 1 plus 0 plus the carried 1 equals 10, so we write 0 and carry 1 again. This process continues for each column. After completing the addition, we get 111110001, with a carry-out of 1. In ones' complement, we perform the end-around carry by adding this carry-out to the result, giving us our final answer: 111110010. The end-around carry operation is what makes ones' complement addition unique. After our initial addition, we have a sum of 111110001 with a carry-out of 1. In ones' complement, instead of discarding this carry-out, we 'wrap it around' and add it to the least significant bit of our result. This is why it's called an end-around carry. So we add 1 to 111110001, which gives us our final answer: 111110010. This end-around carry operation ensures that the arithmetic works correctly in the ones' complement number system. Let's compare ones' complement with the more commonly used two's complement system. In ones' complement, we negate a number by inverting all its bits, while in two's complement, we invert all bits and add 1. A key difference is that ones' complement has two representations for zero: positive zero and negative zero, whereas two's complement has only one. For addition, ones' complement requires the end-around carry operation we've been discussing, while two's complement uses standard binary addition without any special handling. While ones' complement was used in some early computers, two's complement is the standard in modern computing systems because of its simpler arithmetic operations. Let's summarize what we've learned about ones' complement addition. Our original problem was to add the binary numbers 11110111 and 11111010 using ones' complement arithmetic. First, we performed standard binary addition, which gave us 111110001 with a carry-out of 1. Then, we performed the end-around carry operation by adding this carry-out back to our result. This gave us our final answer: 111110010. While ones' complement was used in some early computer systems, it has largely been replaced by two's complement in modern computing due to the simpler arithmetic operations. The key feature to remember about ones' complement is the end-around carry operation, which is essential for correct arithmetic in this number system.