Math 323 Project I
Common numerical/logical errors		 2008

Instructions (additional general instructions may be given in class):

Refer to the sheet which was handed out in class entitled,
      Common errors of post-calculus students.
The list is (or will be) available online as a pdf document and as a Word document .

You are supposed to explain what is wrong with the statements in the list of common errors. When doing this, use the following guidelines. (These are points to keep in mind when presenting your answer. Don't number the points in your answer in accordance with the numbers on these guidelines. Just write a sentence or two, or a paragraph or two, to explain the problem, as discussed in these guidelines.)

  1. The point is NOT to analyze these statements from the point of view of truth tables or the logic if “IF ... THEN ... ” statements. The point is to understand what the statements are saying, to understand why they are wrong, and to think about why a student might make this mistake.

  2. Explain so that I know that you understand what is wrong with the given statement, and so that another student (e.g., a good high school student, or a calculus student, or another Math 323 student) would see what was wrong with the statement.
    For the statement in the list which begins “ To solve the equation  x2 - 4x = y ... ”,  you don't have to comment on the “ I don't have a clue ... ”  part of the statement.

  3. If a counterexample is appropriate, give a counterexample.  Also, if there is a simple correct version of the incorrect statement, give a correct version and, perhaps, explain the difference between the correct and the incorrect statement.

  4. Don't include a lot of trivial arithmetic calculations and explanations.
    The examples below contain the maximum amount of arithmetic and 8th grade algebra which should be presented.
Examples. (These are examples, not models or patterns or templates; you may and, in some cases, should adjust the format and the wording to be appropriate to the given statement and to express clearly the issues involved.)

(a2 + b2)1/2 = a + b.
This is not true because, e.g., if  a = 3  and  b = 4,  we get
      left-side = (a2 + b2)1/2 = (32 + 42)1/2 = (25)1/2 = 5,
whereas    right-side = a + b = 3 + 4 = 7.
So it is not always true that  (a2 + b2)1/2 = a + b.
There is no standard way to “simplify”  (a2 + b2)1/2  for arbitrary  a  and  b.
(a + b)2 = a2 + b2.
This is not true because, e.g., if  a = 1  and  b = 1,  we get   left-side = (a + b)2 = (2)2 = 4,
whereas    right-side = a2 + b2 = 12 + 12 = 2.
So it is not always true that  (a + b)2 = a2 + b2.
A correct formula for  (a + b)2  is  (a + b)2 = a2 + 2ab + b2,
which can be obtained, e.g., by multiplying, using the distributive and commutative properties:
(a + b)(a + b) = a2 + ab + ba + b2 = a2 + 2ab + b2.


Last modified Jan 17, 2008

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