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Project I
Common numerical/logical errors
Instructions (additional general instructions may be given in class) :
Math 323
2010
Refer to the sheet which was handed out in class entitled,
Common errors of post-calculus students.
(Also available on the next pages of this 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.)
I.
II.
The point is NOT to analyze these statements from the point of view of truth tables or the logic of “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.
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.
III.
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.
IV.
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
whereas right-side = a2 + b2 = 12 + 12. = 2.
left-side = (a + b)2 = (2)2 = 4,
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.
Common Errors of Post-Calculus Students
I. Common numerical/logical errors and misconceptions of post-calculus students:
(All variables refer to real numbers unless otherwise specified.)
1. a. If it is true that x < a, then it is not true that x ≤ a.
b. If x ≠ a, then it can’t be true that x ≤ a.
c. If x ≤ 4, then it is not true that x ≤ 2.
2. If one wants to prove that x ≤ a, and one has proven that x < a, one still has to consider the case
x = a.
3. The following statement is not true: For all real numbers x, x 2 > -1.
4. If y/x > a, then it always follows that y > xa.
5.
|(-x)| = x.
6.
√(x2) = x.
7.
x 2 ≥ x.
8. If a < b, then a2 < b2 (and the converse).
9. If x u = x v, then u = v.
10. The only numbers in the interval [ 0, 2 ] are 0, 1, and 2.
(More specifically, if one needs to check something for all x in [0, 2], one needs only to check x = 0, 1, 2.)
11. The set {1/n : n is a positive integer } (i.e., {x : there exists a positive integer n such that x = 1/n } )
is the same as the set (0, 1].
12. “Let m be the smallest number greater than 1.”
13. a. To prove that x is an element of the interval (3, ∞), one must prove both x > 3 and also x < ∞.
b. If x is an element of the interval (3, ∞), then x > 3 and x “goes on forever ”.
14. a. The equation x2 - 4 x = y has no solutions because there are two variables and only one equation.
b. One can’t solve the equation x2 - 4x = y for x because y is unknown.
c. To solve the equation x2 - 4x = 7 for x ,
I begin by factoring the left hand side: x(x - 4) = 7, so x = 7 or x - 4 = 7. …
15. One can prove a statement by assuming that it is true and showing that it leads to a tautology (such as
1 = 1). E.g., if we want to prove the statement p, we can use the following logic: p  q  r  “1=1”.
(After completing the “proof”, one draws a box around the tautology at the end of the argument and writes, “which is true”.
Happy faces are optional.)
16. a. If x2 < 1, then x < ± 1. (Is this an error? How?)
b. The following statement is not true: If x2 < 1, then x < 1.
II. Other occasional errors, not as common as those above:
1. If we know that k is a negative number, then k = -k. OR: If k is a negative number, then k = -a.
2. If x2 = 2, then x = |√2| .
Common Errors, Addendum
Some common errors which were listed on the sheet of errors for Project I:

|(-x)| = x.

√(x2 ) = x.

If a < b, then a2 < b2 (and the converse).

If
y
/x > a, then it always follows that y > xa.
QUESTION: What do these errors have in common? I.e., how could you explain in a general way the error
that one is making, an explanation that (at least partially) explains all the errors? Or, to look at it another way,
what common restriction could you make on the variables so that all these statements are true?