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# MATH REASONING MATH 310

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This 5 page Class Notes was uploaded by Addison Beer on Wednesday September 9, 2015. The Class Notes belongs to MATH 310 at University of Washington taught by Staff in Fall. Since its upload, it has received 13 views. For similar materials see /class/192091/math-310-university-of-washington in Mathematics (M) at University of Washington.

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Date Created: 09/09/15

Math 310 Introduction to Mathematical Reasoning Spring 2006 Handout 7 Notes about Run On Sentences Mathematical writing seems to be particularly prone to runon sentence errors Perhaps this is because of the complexity of the ideas that need to be expressed although complexity alone does not make a sentence runon see below Perhaps it s just because people who are writing mathematics often tend to focus more on the mathematical content than the mechanics of communication In any case poor mechanics will impede effective communication so it is worth learning what runon sentences are and how to avoid them COMBINING CLAUSES INTO SENTENCES A clause is a part of a sentence that has its own subject and verb and could stand on its own grammatically as a complete sentence There are basically only two legitimate ways to combine clauses into a sentence 1 With a conjunction There are two main types 0 Coordinating conjunctions and or but so for meaning because yet meaning but nor joining two negative clauses If you like acronyms remember FANBOYS These connect clauses of equal status called independent clauses which could be broken into two separate sentences without substantially changing the meaning Usually but not always there will be a comma preceding the conjunction The numberx is nonzero so its square ispositive lt RIGHT The numberx is nonzero and its square ispositive lt RIGHT o Subordinating conjunctions after although as because before if since though when whenever where whereas wherever while unless and until are some of the most common ones These are used to introduce clauses whose meaning depends on the rest of the sentence called dependent clauses The conjunction and the rest of the sentence are essential to understanding what the clause means Subordinating conjunctions are sometimes accompanied by commas The square ofx is positive because x is nonzero lt RIGHT The square ofx is positive although x is negative lt RIGHT A dependent clause with its subordinating conjunction can also come before the main clause In such cases the dependent clause is always set off by a comma Because x is nonzero its square is positive lt RIGHT Although x is negative its square is positive lt RIGHT 2 With a semicolon Two independent clauses can be connected with a semicolon alone In this case no conjunction is used The numberx is nonzero its square is positive lt RIGHT RUN ON SENTENCES Any other way of joining two clauses together yields a runon sentence Note that whether a sentence is runon or not has nothing to do with its length or complexity it depends only on its structure Here are some common types of runon sentences A The straight runon The simplest kind of runon sentence though probably the least common is just two independent clauses smooshed together with no conjunction or punctuation intervening The numberx is nonzero its square is positive lt WRONG Here The number x is nonzero and its square is positive are independent clauses because each can stand on its own as a complete sentence without changing the meaning Most readers recognize immediately that something is missing in this example B The comma splice A more common mistake is using a comma alone without a conjunction between two clauses Letx be a nonzero real number its square ispositive lt WRONG Since the two phrases being joined are complete clauses they can t be joined by a comma alone C The fake conjunction This is a more subtle error and therefore much more common Certain wordsisuch as also besides consequently finally for example furthermore hence in fact however indeed moreover nevertheless otherwise then therefore and thusilook and act in many ways like conjunctions but in fact they re just adverbs Technically they re called conjunctive adverbs If one of these is used between two clauses it must be preceded by a semicolon because there is no conjunction Alternatively a conjunctive adverb can be used to begin a new sentence The setA is empty howeverB is not lt WRONG The setA is empty however B is not lt WRONG The setA is empty however B is not lt RIGHT The setA is empty However B is not lt RIGHT Letx be a nonzero real number then its square is positive lt WRONG Letx be a nonzero real number then its square is positive lt WRONG Letx be a nonzero real number then its square ispositive lt RIGHT Letx be a nonzero real number Then its square is positive lt RIGHT This mistake with then is especially common in mathematical writing One reason is probably because people see an analogy with the following Ifx is a nonzero real number then its square is positive lt RIGHT This is not a runon sentence The difference is that the phrase If X is a positive real number is a dependent clause 39 by the 39 J if It can t stand on its own as a sentence so it s not an independent clause If you re not sure whether you re looking at a dependent clause introduced by a subordinating j or an 39 39 clause 39 39 39 by a j 39 adverb try this test if the clause can be moved to the beginning of the sentence it s a dependent clause if it can t it s probably an independent clause and needs a conjunction or a semicolon The square ofx is positive although x is negative lt RIGHT Although x is negative its square is positive lt RIGHT The square ofx is positive nevertheless x is negative lt WRONG Nevertheless x is negative its square is positive lt WRONG The square ofx is positive nevertheless x is negative lt RIGHT Math 310 B amp C Introduction to Mathematical Reasoning Autumn 2005 Handout 3 102805 Notes about Run On Sentences COMPOUND SENTENCES An independent clause is a phrase that could stand on its own as a complete sentence When you join two or more independent clauses together into a single sentence you get a compound sentence There are only two legitimate ways to combine independent clauses into a compound sentence 1 With a coordinating conjunction 7 and or but so for meaning because yet meaning but nor joining two negative clauses If you like acronyms remember FANBOYS Usually but not always there will be a comma preceding the conjunction The real number x is positive so it has a square root RIGHT The square root of 2 is a real number and it is positive RIGHT 2 With a semicolon Let x be a positive real number then x has a square root RIGHT RUN ON SENTENCES Any other way of joining two independent clauses together yields a runon sentence Here are some common types of runon sentences A The straight runon The simplest kind of runon sentence but probably the least common is just two complete sentences mooshed together with no words or punctuation intervening Let x be apositive real number it has a square root WRONG Note that Let x be a positive real number and It has a square root are independent clauses because each can stand on its own as a complete sentence Most readers recognize immediately that something is missing in this example B The comma splice A more common mistake is using a comma alone between two independent clauses Let x be a positive real number it has a square root WRONG Since the two phrases beingjoined by a comma are independent clauses they can t be joined by a comma alone C The fake conjunction This is a more subtle error and therefore much more common Certain wordsialso besides consequently nally furthermore however indeed moreover nevertheless otherwise then therefore and thus look and act in many ways like conjunctions but in fact they re just adverbs Technically they re called conjunctive adverbs If one of these is used to join two independent clauses it must be preceded by a semicolon Examples The setA is empty howeverB is not WRONG The setA is empty however B is not WRONG The setA is empty however B is not RIGHT The setA is empty However B is not RIGHT Let x be a positive real number then it has a square root WRONG Let x be a positive real number then it has a square root WRONG Let x be a positive real number then it has a square root RIGHT Let x be a positive real number Then it has a square root RIGHT This mistake with then is especially common in mathematical writing One reason is probably because people see an analogy with the following If x is a positive real number then it has a square root RIGHT This is not a runon sentence The difference is that the phrase If X is a positive real number can t stand on its own as a sentence so it s not an independent clause Math 310 Introduction to Mathematical Reasoning Spring 2006 Handout 6 Solutions to Problem 23i7ii page 56 i Theorem For any nonzero real numbers zy and any integer n zny Proof The case n 2 0 is taken care of by Exercise 5i7il For the case in which n lt 0 let k in so It gt 0 Notice rst that by the de nition of negative exponents we have 1 16 zk 1l It follows that z kzk l l and similarly that y kyk 17 2 Iy kryk 1A 3 By the result of Exercise 5i7i we know that M zkyk Multiplying both sides of this equation by zy kz ky k and using 173 to simplify we obtain 7k 7k 7k I y my Substituting k in we obtain the desired equationl D 71 in ii Theorem For any nonzero real I and any integers m and n rm 7 z n z Proof There are four cases depending on the signs of m and n The case in which m 2 0 and n 2 0 is taken care of by Exercise 5l7iil For the second case assume that m 2 0 and n lt 0 Let k in so what we have to prove is zm k zmz ki 4 Multiplying both sides by 1 16 and simplifying using equation 1 from the preceding proof we obtain On the other hand if m lt k we apply Exercise 5i7ii again to obtain 1k zmzkim Multiplying both sides by z kzm k and simplifying once again yields The third case in which n 2 0 and m lt 0 is handled in exactly the same way with the roles ofm and n reversedl Finally for the last case assume that m lt 0 and n lt 0 Let k 7m and l in so what we have to prove is z k l z kz li 5 We use Exercise 5i7ii one more time to conclude that Multiplying both sides by z k lz kz l and simplifying we obtain 5 as desired D You may turn in a paragraphstyle proof of part for extra credit you wish

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