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Thinking Mathematically by Robert Blitzer

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Thinking Mathematically by Robert Blitzer Powered By Docstoc
					  Thinking
Mathematically
  Problem Solving and Critical
           Thinking
       Inductive Reasoning
“Inductive reasoning” is the process of
arriving at a general conclusion based
on observations of specific examples.
Example
 You purchased textbooks for 4 classes.
 Each book cost more than $50.00.
 Conclusion: College textbooks cost more
 than $50.00.
        Inductive Reasoning
There is no guarantee that the conclusions
reached by “inductive reasoning” are correct
with no exceptions.

Even though all your textbooks this term
cost more than $50.00 it is still possible (and
even probable) that a textbook for some
course will cost less than $50.00.
        Inductive Reasoning
The conclusion is really a conjecture or
   hypothesis and a case where the conclusion
   is not true is a counterexample.
It takes one counterexample to show a
   hypothesis is false.
         Inductive Reasoning
In mathematics “inductive reasoning” is often
used to find patterns.
 Find the pattern in the following sequence of 6
 numbers and use that pattern to decide what the
 next number should be.
            1,4,16,64,256,1024,?
  Each number is obtained from the previous one
  by multiplying by 4. The next number is

                    4096
        Deductive Reasoning
“Deductive reasoning” is the process of
proving a specific conclusion from one or
more general statements. A conclusion that is
proved true by deductive reasoning is called
a theorem. For example:
 The catalog states that all entering freshmen
 must take a mathematics placement test.
 You are an entering freshman.
 Conclusion: You will have to take a
 mathematics placement test.
       Deductive Reasoning
Examples from Mathematics

  Suppose 3x = 12. We conclude x = 4.

  The length of a rectangle is 6 and its width
 is 5. We conclude its area is 30.
Select a number. Multiply it times 10.
  Subtract 6 from the result. Divide what
  you have by 2. Add 3 to this.

Repeat a few times with other numbers and
 make a conclusion [hypothesis].
What type of reasoning did you use.

Show why this works. Relate deduction to
  logic and proof.
Can you think of examples of inductive and
 deductive reasoning in your areas of
 interest? In social sciences, humanities,
 arts, and sciences?

				
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posted:9/24/2012
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