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ElementsOfStyle - Elements of Style Modern Algebra Fall...

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Elements of Style Modern Algebra, Fall 2011 Anders O.F. Hendrickson Years of elementary school math taught us incorrectly that the answer to a math problem is just a single number, “the right answer.” It is time to unlearn those lessons: those days are over. From here on out, mathematics is about discovering proofs and writing them clearly and compellingly. The following rules apply whenever you write a proof. I will refer to them, by number, in my comments on your homework and tests. You will find them summarized on the last page. Take them to heart; repeat them to yourself before breakfast; bind them to your forehead; ponder their wisdom with your friends over lunch. 1. The burden of communication lies on you, not on your reader. It is your job to explain your thoughts; it is not your reader’s job to guess them from a few hints. You are trying to convince a skeptical reader who doesn’t believe you, so you need to argue with airtight logic in crystal clear language; otherwise he will continue to doubt. If you didn’t write something on the paper, then (a) you didn’t communicate it, (b) the reader didn’t learn it, and (c) the grader has to assume you didn’t know it in the first place. 2. Tell the reader what you’re proving. The reader doesn’t necessarily know or remember what “Problem 5c” is. Even a professor grading a stack of papers might lose track from time to time. Therefore the statement you are proving should be on the same page as the beginning of your proof. For an exam this won’t be a problem, of course, but on your homework, recopy the claim you are proving. This has the additional advantage that when you study for tests by reviewing your homework, you won’t have to flip back in the textbook to know what you were proving. 3. Use English words. Although there will usually be equations or mathematical state- ments in your proofs, use English sentences to connect them and display their logical relationships. If you look in your textbook, you’ll see that each proof consists mostly of English words. 4. Use complete sentences. If you wrote a history essay in sentence fragments, the reader would not understand what you meant; likewise in mathematics you must use complete sentences, with verbs, to convey your logical train of thought. Some complete sentences can be written purely in mathematical symbols, such as equations (like a 3 = b - 1 ), inequalities (like o ( a ) < 5), and other relations (like 5 10 or 7 Z ). These statements usually express a relationship between two mathematical objects , like numbers (e.g., 7), group elements (e.g., g 2 h ), or sets (e.g., G and R ). 5. Show the logical connections among your sentences. Use phrases like “There- fore” or “because” or “if. . . then. . . ” or “if and only if” to connect your sentences.
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Elements of Style Modern Algebra, Fall 2011 Page 2 of 6 6. Know the difference between statements and objects. A mathematical object is a thing , a noun, such as a group, an element, a vector space, a number, an ordered pair, etc. Objects either exist or don’t exist.
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