MTH 225: Weekly Challenge 1

--- tags: mth225, weekly-challenge --- # MTH 225: Weekly Challenge 1 **Due on Blackboard by 11:59pm Eastern on Saturday September 11.** :::warning **Instructions**: * Your work on Weekly Challenges should consist of **complete solution attempts for all the Application/Extension Problems and complete and thoughtful responses to all the Feedback and Reflection prompts**. Before submitting your work, make sure you've reviewed the [Specifications for Satisfactory Work in MTH 225](/Cy6P0rGZQzuOM3NwZ3ZuMw) document to make sure your work meets the standards to the best of your knowledge. * **Practice Exercises are optional.** You do not need to turn any part of these in. But if you want feedback on any of them, turn those in with the rest of your work and I'll look at it. * You may type up your work or write it by hand on paper, whiteboard, or in a notes app. **Typewritten work is preferred** because it makes revisions easier for you. * If you handwrite your work on paper or a whiteboard, your work needs to be **scanned to a legible, black-and-white PDF**. * All your work is to be submitted as a **single PDF** at the appropriate assignment area on Blackboard in the *Weekly Challenges* folder. Please do not submit multiple PDFs, or files that are not in PDF format. ::: ## Practice Exercises :::info **Practice Exercises are optional and for your benefit only.** You do not need to turn in work or answers unless you want feedback on your work. ::: **Convert each of the following integers from the given base to all three of the other bases we've discussed.** (For example, if the integer is given octal, convert it to decimal, binary, and hexadecimal.) For numbers in base 10, use the conversion algorithm discussed in class ([click here for the video](https://vimeo.com/578187581)). You can check your work [here](https://www.rapidtables.com/convert/number/base-converter.html). | Number | Base | Number | Base | |:------:|:----:|:------:|:----:| | 307 | 10 | 257 | 8 | | 2891 | 10 | 1101011 | 2 | | 34F1 | 16 | 2B7 | 16 | **Perform each of the following binary arithmetic operations.** You can check your work [here](https://www.calculator.net/binary-calculator.html). - $10111 + 10010$ - $10111 - 10010$ - $10111100 + 10001111$ - $10111100 - 10001111$ - $10111100 \times 101$ - $10111100 \times 101$ - $10111100 \div 101$ (that is, $10111100$ divided by $101$) - $111111111111 \div 1010$ ## Application/Extension Problems :::info Your submission should include complete solution attempts at *all* of the problems below. Each attempt should be a good-faith attempt at a correct answer and solution. ::: 1. If $n$ is any positive integer, what is the binary representation for $2^n$? Give an answer to this question that is specific and clear as possible, then explain why your answer is correct. (This was one of the exploration questions from class on September 1.) 2. Take any integer in base 2 form and add it to itself. How does the base 2 representation of the result compare to the base 2 representation of the original integer? Give an answer to this question that is specific and clear as possible, then explain why your answer is correct. (This was one of the exploration questions from class on September 3.) 3. In [the video about the base 10 conversion algorithm](https://vimeo.com/578187581), it was stated that we can use that algorithm to convert a decimal integer into *any* base, not just base 2, 8, or 16. [Go to this website](https://numbergenerator.org/random-6-digit-number-generator) and generate a random 6-digit integer in base 10. Then use the algorithm to convert this integer into **base 20**. See below for a mini-explanation of base 20. Show all your work and explain what you're doing. :::success **Mini-tutorial on base 20**: A number in base 20 is written as a sum of powers of 20. For example, if 371 is in base 20, then in base 10 this is... $$(3 \times 20^2) + (7 \times 20^1) + (1 \times 20^0) = 1200 + 140 + 1 = 1341$$ In base 16 we had to expand our set of digits we could use, adding the letters A-F to the usual digits 0-9. In base 20, we could have place values from 0 to 19, so we need to add more digits. For this, we use the usual 0-9 and the letters A, B, C, D, E, F **and the letters G, H, I, and J**. As in base 16, A represents 10, B represents 11, and so on; now in base 20, G represents 16, H represents 17, I represents 18, and J represents 19. So for example, the integer J6GA in base 20 when converted to base 10 is $$J6GA = (19 \times 20^3) + (6 \times 20^2)+ (16 \times 20^1) + (10 \times 20^0) = 154730.$$ Likewise, the decimal number $2052$ is $52C$ in base 20 (which you can check using the algorithm!). ::: ## Feedback and Reflection :::info Your submission should also include complete responses to *all* of the prompts below. There is no minimum word count, but your responses should be thoughtful and should actually express your thoughts on the prompt. If you'd rather discuss these with me in person, come by drop-in hours or schedule an appointment. ::: Since we are now two weeks into the course and this is your first Weekly Practice, here are some questions about how things are going for you so far. **Please be honest and clear** if you are encountering anything you'd like to see changed. I always take your feedback seriously and will do my best to act quickly on any reasonable suggestions. 1. What aspects of the course so far are helping you learn? 2. What aspects of the course so far are getting in the way of your learning? 3. What's something that we *are not* currently doing in the course (either in class or outside of class) that you'd like to see us *start* doing? 4. What's something that we *are* currently doing in the course (either in class or outside of class) that you'd like to see us *stop* doing, or change the way we're doing it? 5. How is the *structure* of the course working for you so far? For example, do the syllabus and course policies make sense? Are you able to find all the materials easily? Etc.? 6. Finally, ask me one question --- related to the course or not related to the course --- that you'd like to see me answer. (Yes, I will answer everything you throw at me.)