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COMMON MISTAKES
This section lists common mistakes that students
can sometimes make. For ease of reference the errors are divided into
four main categories. The Root and Exponent
Errors
section deals with
errors involving exponents and roots. The Algebraic
Manipulation Errors section includes errors which may occur when
dealing with manipulating expressions. The Trigonometric Errors section deals
with common errors
involving trigonometry. Finally, the Vectors and Matrices Errors section
deal with common errors in vector and matrix operations.
Note that these are listed by ERROR so the formula that appears above
the explanation is INCORRECT. The correct formula (if one exists)
appears
at the
end of the explanation.
- Let us assume that this is true.
Let a=1 and b=4. Then we have:
Since the square root of 5 is not equal to three we can see that you
cannot split the root in such a way.
- Let us assume that the equation is
true.
Let a=-1 and b=-4. Then we have:
This example made use of complex
numbers. It can be shown that with
the use of complex numbers either a or b can be negative but not
both.
- Let us assume that the equation is
true.
Let a=-4. Then we have:
This example made use of complex
numbers. So keep in mind the negative sign cannot be removed from
the square root.
- (cx)n = cxn where c is
constant.
- Let us assume that the equation is
true.
Let c=2, x=3, and n=2. Then we have:
(cx)n = cxn
((2)(3))2 = 2(32)
62 = 2(9)
36=18
It is quite clear that 36 is not equal to 18 so the assumption that
this is true is wrong. The correct formula for removing brackets
in this case is
(cx)n = cnxn.
- Let us assume that the equation is
true.
Let c=8, x=8, and n=3. Then we have:
As 4 is not equal to 16 this formula is flawed. The correct
formula is .
- (x+y)2 = x2+y2
- Let us assume that the equation is
true.
Let x=2 and y=4. Then we have:
(x+y)2 = x2+y2
(2+4)2 = 22+42
62 = 4+16
36 = 20
Since 20 is not equal to 36 (36 being the correct answer) this formula
is wrong. The correct formula is (x+y)2 = x2+2xy+y2
because (x+y)2 means (x+y)(x+y).
- (-x)2 = -x2
- Let us assume that the equation is
true.
Let x=2. Then we have:
(-x)2 = -x2
(-2)2 = -22
4 = -(22)
4 = -4
This example shows that the negative sign cannot be factored from the
brackets when raised to the power 2. The correct way to solve this is
(-x)2 = x2. Note that this property holds
for any even power but if the exponent is odd the negative sign can be
factored as in the example.
-
- Let us assume that the equation is
true.
Let x = -1. Then we have:
The correct way to simplify this is:
- x-(a-b) = x-a-b
- Let us assume that this is true. Let
x=2, a=3, and b=4. Then we have:
x-(a-b) = x-a-b
2-(3-4) = 2-3-4
2-(-1)= -5
2+1 = -5
3 = -5
As can be seen -5 is not equal to 3 so the formula is incorrect.
The error was that the negative sign must be multiplied into both terms
in the bracket leaving x-(a-b) = x-a+b.
- (ax2+b) / x = ax+b
- Let us assume that this is true. Let
x=2, a=3, and b=4. Then we have:
(ax2+b) / x = ax+b
(3(22)+4) / 2 = 3(2)+4
(3(4)+4) / 2 = 6+4
16 / 2 = 10
8 = 10
Since 10 does not equal 8 this formula is incorrect. The error
was believing that in dividing by x you only needed to remove x from
one term in the numerator. In order to do this EVERY term in the
numerator must have an x in it.
- ax2 = x solved by dividing both
sides by x. I.E. solve ax = 1
- Let us assume that this is true.
Let a=2. Then we have:
ax2 = x
ax = 1
2x = 1
x = 1/2
This seems plausible since we get a value for x and when substitute
into the equation it works. However, if we solve it the right way (by
subtracting x from both sides so the equation is equal to zero) we will
see why dividing by x is not the way to solve the equation.
ax2 = x
ax2-x = 0
x(ax-1) = 0
x(2x-1) = 0
so x = 0 or 2x-1 = 0
x = 0 or 2x-1 = 0
x = 0 or 2x = 1
x = 0 or x = 1/2
Therefore this method gives x = 1/2 as an answer as well but it also
gives x = 0 as another possible value for x. This means that the first
method is incorrect and the correct way to solve the equation is to
subtract x from both sides.
- 1 / (x+y) = 1/x + 1/y
- Let us assume that this is true. Let
x=2 and y=3. Then we have:
1 / (x+y) = 1/x + 1/y
1 / (2+3) = 1/2 + 1/3
1/5 = 3/6 + 2/6
1/5 = 5/6
Which is obviously false. It is often better to leave terms of this
type in their original form as splitting them up is usually more
difficult than evaluating directly.
- sin(x) = sin(x) (note that the symbol beside the
second x is degree not power 0)
- In calculus and algebra sin(x) means the sine of x
radians not x degrees. To show an example that these are
different, let x=2. Then we have:
sin(x) = sin(2)
0.03489950
sin(x) = sin(2)
0.90929743
Even though the value of 2 was used in both equations, the difference
between 2 degrees and 2 radians made the answers different. Remember
that in calculus and algebra the value will almost always be in RADIANS.
But read each problem carefully to be sure you don't make the mistake of
assuming the wrong value.
This applies to all trigonometric terms, not just sin(x).
- sin(x+y) = sin(x) + sin(y)
- Let us assume that this is true. Let x= and y=/6.
Then we have:
sin(x+y) = sin(x) + sin(y)
sin(+/6) =
sin() + sin(/6)
sin(7/6) = 0+1/2
-1/2 = 1/2
This shows that the assumed formula is incorrect (since 1/2 is not
equal to -1/2). The correct formula in this situation is
- sin(x+y)
= sin(x)cos(y)+cos(x)sin(y). Formula for the other trigonometric
functions can be found in the Trigonometric Formulas
section.
Note that this applies to any trigonometric function and also for
'-' instead of '+'.
- cos-1(x) = 1 / cos(x)
- Let us assume that this is true. Let x=0.
Then we have:
cos-1(x) = 1 / cos(x)
cos-1(0) = 1 / cos(0)
/2 = 1 / 1
/2 = 1
Since /2 is not equal to 1 so the
assumption is wrong. This is a common mistake since people often
think that because cos2(x)
= (cos(x))2 then cos-1(x) should equal (cos(x))-1.
However, cos-1(x) denotes the inverse cosine function, also
called arccosine, not the cosine function raised to the power -1. See
the Inverse
Trigonometric Functions section for more details.
Note that this observation applies for any trigonometric function.
- u • v = [u1v1, u2v2, ...]
- The dot product gives us a scalar, not another vector. The products are added together, not
put into vector components.
- Matrix multiplication involves taking dot products of rows from A with columns from B to
get the respective row,column entry in AB
- 1/A, where A is a matrix
- There is no such thing as dividing a scalar by a matrix. Division by a matrix
does not exist, and what you're probably thinking of/looking for is the inverse of the
matrix, A-1.
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