Math 213 Supplement (3rd ed rev) (Fall 2011)pp1 to 9

Math 213 Supplement (3rd ed rev) (Fall 2011)pp1 to 9 - Math...

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Math 213 Supplement Jack Calcut Elizabeth Haverstick Karen McLaren

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2 DEDUCTIVE REASONING Deductive reasoning is the cornerstone of mathematics. Ever since Euclid’s time, mathematicians have sought to build mathematical knowledge by following a methodical process: identify a small list of assumptions, and then use logic to prove many other facts using only these assumptions. In the ideal math world, all true facts would be derivable from some short list of assumptions. Unfortunately, this has been proved impossible—but it doesn’t keep researchers from looking for new patterns or trying to prove patterns that have already been identified as probably true. The Importance of Deductive Reasoning Why bother to prove a pattern or rule that has been recognized in many examples—can’t we just assume from experience that it is “very likely” to be true? After all, this is how we are accustomed to making decisions in other areas of our life. In mathematics, however, proof is recognized as the “guarantee” that a fact must be true. It is the secure foundation on which other facts can be built. It guards us against becoming sloppy and accepting “facts” too quickly that may end up contradicting each other. Frankly, this is often a difficult process. Many times, very intelligent mathematicians have thought that they had a proof of some statement only to realize later (or worse, have someone else point out) that their reasoning was flawed somehow. Perhaps they didn’t recognize a hidden assumption or had a gap in their logic. It may take years and collaboration among several mathematicians to produce a valid proof in some cases. However, this perseverance is necessary for the certainty mathematicians seek. A couple of examples may help to illustrate the importance of insisting on proofs: Example #1: One child says to another, “I have magical powers. Think of a number. Now multiply it by two. Add eight. Divide by two. Subtract the number you started with. I can tell you what you’ve got left . . .four!” Should the other child be impressed? Why or why not? Example #2: Look at the design below. How many square units of area are contained in it? Now copy the design onto a piece of graph paper. Cut around the outside and across the diagonal drawn. Rearrange the four pieces you get into a rectangle. What is the area of this rectangle? Rearrange the pieces again to make a square. What is the area of this square? What seems odd about your results in this exercise? What is happening?
3 These examples illustrate a couple of the important roles that proofs fulfill: they can tell us when a certain outcome is assured; they can keep us from accepting “facts” that seem to be true when we cut out paper and move it around, but which are not really true. Thus, deductive reasoning allows mathematicians to build a collection of mathematical facts with confidence. Beginning a Deductive System

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This note was uploaded on 03/12/2012 for the course MATH 0102 taught by Professor Mclaren during the Spring '12 term at Maryland.

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Math 213 Supplement (3rd ed rev) (Fall 2011)pp1 to 9 - Math...

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