Daily Prep 10A -- MTH 350-01

--- tags: mth350, dailyprep --- # Daily Prep 10A -- MTH 350-01 ## Overview In Module 10, we continue our look at rings by defining **integer multiples and exponents of ring elements**. This is the next logical step in learning to do arithmetic and algebra on these rings. In Module 10A, we'll define these concepts and prove a number of basic theorems about integer multiples and exponents. ## Learning objectives **Basic Learning Objectives:** *Before* our class meeting, use the Resources listed below to learn all of the following. You should be reasonably fluent with all of these tasks prior to our meeting; we will field questions on these, but they will not be retaught. + Develop an intuitive sense of what "integer multiples" and "integer exponents" of ring elements mean. **Advanced Learning Objectives:** *During and after* our class meeting, we will work on learning the following. Fluency with these is not required prior to class. + State the recursive definition (8.2) of what it means to *add multiple ring elements together* and the definition of an *integer multiple* and *integer exponent* of a ring element; use the definition to add specific ring elements together and to multiply and exponentiate them by an integer. + State the definition of what it means to *multiply or exponentiate a ring element by zero or a negative integer*; use the definition to compute zero and negative multiples and exponents of specific ring elements. + Explain line-by-line the portions of the proof of Theorem 8.5 built in Activity 8.6; the proof of Theorem 8.7; and the proof of Lemma 8.8. ## Resources for learning **Reading:** Read through **just Preview Activity 8.1** in Investigation 8. Nothing else! ## Exercises For the exercises this time, you'll need to write things on paper and then scan them, mostly because your work will involve matrices and those are a pain to typeset. You won't be working with a Google Form this time. The exercises come in two parts. **Part 1:** Answer the following questions on your writeup. 1. When we multiply an integer or real number by positive integer, like $4 \cdot 10$ or $5 \cdot \sqrt{2}$, what exactly does the integer multiplication *mean*? If you were to teach this concept to a child, how would you present it? 2. When we multiply an integer or real number by a *negative* integer --- like $(-4) \cdot 2$ or $(-3) \cdot \pi$, what does *that* mean? Again, how would you explain it to a child? 3. When we raise an integer or real number to a positive integer exponent, like $4^3$ or $(\sqrt{7})^4$, what does that mean? How would you explain it to a child? 4. When we raise an integer or real number to a *negative* integer exponent, like $2^{-4}$ or $(2/3)^{-4}$, what does *that* mean? How would you explain it to a child? In all of the above, avoid oversimplified explanations that just tell the child what to write down; focus on the *meaning* of the expression. **Part 2:** Now go and attempt to compute the 21 expressions listed in Preview Activity 8.1. All of the expressions given there are either possible to calculate, or impossible to calculate. For the ones that are possible, calculate them using your intuitive notion of what multiplying or exponentiating by an integer means (see Part 1) and give an explanation. For those that are impossible to calculate, explain. Please note, **you don't have the option to leave these blank and say that you just didn't know what to do**; give each one serious thought and your best idea of how to proceed. **Be mindful of answers that don't make semantic sense!** For example, $\{1,2,3\}^{-3}$ is not equal to $\{1, 1/8, 1/27\}$ (raising each set element to the $-3$) --- that doesn't make sense, because we want the result of $\{1,2,3\}^{-3}$ to be an element of $\mathcal{P}_3$. Likewise do not say that $[5]^{-3} = [1/125]$ because that does make sense either (the integer congruence class of a fraction?). **Note:** Some hints are given later in the section if you want to read ahead. ## Submission and grading Write your responses up in a PDF (either $\LaTeX$, or handwrite and scan) and **email them to the professor** as attachments. A **Pass** mark is given if the Daily Prep is turned in before its deadline and if each item on the Daily Prep has a response that represents a good faith effort to be right. **Mistakes are not penalized**. A **No Pass** is given if an item is left blank (even accidentally), has an answer but it shows insufficient effort (including responses like "I don't know"), or if the Daily Prep is late.