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common mistakes in maths...

Blazing goIITian

Posted on
17 Mar 2009 11:44:50 IST
Posts: 5564
17 Mar 2009 11:44:50 IST
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common mistakes in maths...
Engineering Entrance , JEE Main , JEE Advanced , Mathematics , Mathematical Logic

see.. i hv not typed all this ...i he copied it frm sumwhere else..

but atleast i got it here!!

i just want u to hv a luk ...maybe one of these might be ur error..

common mistakes in maths...

1.Have you seen many mistakes like the below?

Question 1:
3x = 27

Solution:
3x = 34 ===> x = 4

Question 2:
9x = 33

Solution:
33x = 33 ===> 3x = 3 ===> x = 1

Why the error?

A simple explanation is that the maths learner is not familiar with the basic multiplication of repeated numbers.

3 x 3 = 9
4 x 4 = 16
5 x 5 = 25
6 x 6 = 36
7 x 7 = 49

2 x 2 x 2 = 8
3 x 3 x 3 = 27

2 x 2 x 2 x 2 = 16

etc.....

Once you have mastered this basic repeated multiplications, you can rest assure that indices question will not be there to haunt you.

How about solving "x" in this 9x + 1 + 2(3x) - 3 = 0 ?

I bet that if you understood the above criteria of learning indices, the equation can be easily solved for x (using quadratic formula as a hint).

All complex things start off with simple things.
Do you agree this applies to maths?

2.For question regarding trigonometry, quadrant is one of the key parameter to obtain correct answers.

A complete cycle (360 degree) is divided into 4 quarters.
They are zones defined for specific trigonometric functions.

The first quarant (0 to 90 degree) gives positive sign for ALL trigonometric functions.
The second quarant (90 to 180 degree) allows only "sine" to have positive number.
For the thrid quarant (180 to 270 degree), "tangent" has positive number only.
Lastly, the fourth quarant (270 to 360 degree), "cosine" gives positive number only.

So, you can see that given a sign of a trigonometrical operation, the specific quadrant can be found or identified.

Example:

sin X = - 0.5 ===> Identifies quadrant as 3rd and 4th.

tan X = 0.2 ===> Identifies the 1st and 3rd quadrant.

This is simple, right?

However, do note the below example.
It causes a mistake that is common!

Example of potential error:

sin 2X = -0.5 ====> which quadrants ?

The answer is not that direct!

Why?

Now the math question is not on "X", but on "2X".

To identify the quadrant, you need to start off from the "2X", working as per normal.
But, after identifying the 2 quadrants, you have to compute the "2X" reference angle.
Using the reference angle, you have to obtain the 2 angles.
After which, you need to divide the angles obtained by 2.
The divided angles is then the final angles lying within the quadrants.

Confused? Never mind. See the numerical solution below.....

Solution:
2X = sin-1 (0.5) = 300
This is the reference angle used to compute the actual answers.

and (Quad 4) = 360- 30 = 3300.

Common mistake is to obtain reference "2X" angle and straight away divide it by 2.
Using this newly found "X", you proceed to identify the angles of the quadrant identified using the "2X". THIS IS INCORRECT!

Do not confuse double angle with single angle.
When the problem is "2X", solve all the way using the "2X" first until reaching the end.
After which, you then divide the angles by 2 to get to the final answers.

Maths is simple if you follow the rules accordingly.
If you mess up double angle with single angle while solving, you just literally mess up the workings.

Maths forces you to follow rules set out. It punishes only if you do not obey orders.

3.There are times when simple algebraic operations are confused by introducing trigonometric functions or logarithmic terms.

Example:

13 = 7 - 3x

This can be easily computed to be
13 - 7 = - 3x
==> 6 = - 3x
==> x = -2

But how about 13 = 7 - 3 tan X ?

Solution: 13 = 4 tan X ==> tan X = 13 / 4 , ..... and got into hot soup!

Why?

A careless mistake has been made.

When tan X was substituted into the original equation, the eyes refused to acknowledge this "complicated" tan X.
The eyes can only see the simpler "7 - 3" and thus compute it to be (7 - 3) = 4!

This caused the 7 - 3 tan X to be 4 tan X, which is WRONG.

The correct mathematical process of solving should maintain.

Therefore,

13 = 7 - 3 tan X
==> 13 - 7 = - 3 tan X
==> 6 = - 3 tan X
==> tan X = -2
.......

Maths is not that complicated when you follow the rules closely, even when the terms have changed into a seemingly complex expression / term.

By following what you have known with simple expression / term, any challenging equation can be easily solved.

This is the power of learning maths properly.
Being discipline in the way you handle maths is the key.

With a discipline mind, maths becomes fun , .. and interesting.

4.A common mistake occurs normally during simplification to a single logarithm term.

Question like,

" Simplify log X - log Y + log Z into a single term "

catches many students who are careless.

What is the error or mistake made?
- Doing the solving at one go when not familiar with the logarithmic rules

"log Z" is commonly taken to follow the previous log term, which is, "- log Y ".
Since "- log Y " causes the "Y" to be a denominator, "Z" is also taken to be a denominator too!

This is a mis-cue. A mental slip, mathematically.

Look at the sign carefully before jumping to conclusion.
Go slow in the combination to a single log term.

Remember the idiom: "Slow and steady wins the race"

You can apply this to log simplification when you are new to it.

5.Multiplication of complex numbers remains the same as done for normal algebraic operation.

However, due to complex number having 2 terms, namely, real and imaginary terms, care has to be taken for the "i"unit.

This is specially so when multiplication of conjugate is involved.

A popular mistake made while doing this form of multiplication is:

(3 + i2)(3 - i2) = 32 + (i2)2

What is wrong?

The concept of conjugate and its multiplication states that:
(a + ib)(a - ib) = a2 + b2

The "i" symbol is NOT reflection in the final outcome!

Only the "a" and the "b", the numerical part, are extracted out for computation.

Taking the "i" into account will cause the sign of the last term (i2) to be incorrect.
This is because i2 = -1.

Therefore, regardless of the sign in the multiplicands, just pull out the numerical part in the complex number and use them for calculation, that is, the 3 and 2 in the example above.

The correct answer, thus, is (3 + i2)(3 - i2) = 32 + 22.

Looking carefully at the application of the formula, you will notice that this is a simple and easy technique to do conjugate multiplication.

6.Multiplying is simple.

What is 4 x 3?
Answer is 4 x 3 = 12.

Simple?
Sure it is.

Again simple? Sure.

But how about (y + 1)(2y + 3)?
Many of you may find this simple and basic.

But you may still come across some who did not grasp this factor multiplication.
Mistake still occur for this maths operation involving factors.

What is the mistake commonly seen?

(y + 1)(2y + 3) is given as (y)(2y) + (1)(3).
First term multiply by first term, second one multiply with the second one. That's all.
This is incorrect mathematically.

This is a misconception of what multiplication does.

Let me explain.
(y + 1)(2y + 3) can be interpreted as (y)(2y + 3) plus (1)(2y + 3).
This is key to this form of maths operation.

The second term (2y + 3) is multiplied by the first term "y" of the first factor (y + 1).
(2y + 3) is next multiplied by the second term "1" of the first factor.
The result of these two operations are then added up, since it is y add 1 (as reflected in the first factor).

(y + 1)(2y + 3)
= (y)(2y) + (y)(3) + (1)(2y) + (1)(3)
= 2y2 + 3y + 2y + 3
= 2y2 + 5y + 3

Learn from the mistake, and do not repeat it.
This is the basic concept in learning from mistakes. They are our teacher.

Remember, maths is interesting!
A twist can be destructive or constructive.
That is where maths is special and challenging.

7.For maths problem related to indices, you do not only look at the power. You have to take care of the base too.

The common mistake is to ignore the sign of the base when doing computation involving index.

A simple example illustrates the error that is very common when learning maths.

Example:
Solve 3x2 - 4x + 1 = 0 using the quadratic formula.

Solution:
In solving, we need to extract out the a = 3, b= -4 and c = 1 to fill into the quadratic formula

However, the quadratic formula requires the utilisation of the b2 - 4ac expression.

Here the mistake is to fill in b2 as -42 = - 16!

This shouldn't be the case. It should be (-4)2 = 16.

It is a difference of the sign (positive versus negative).

Thus to handle question on indices, you will have to be extra careful on the base and its sign.

8.I came across an amazing mistake on the index conversion which made me thinking.

One student wrote: a-1/2 = a2

What is the going in his brain?

He is not completely wrong. He had applied something related to the properties of indices. But had confused the application through improper usage.

This may be the result of trying to memorise the working instead of fully understanding the mathematics principles.

I suspected he may be using the idea of a-m = 1/am.

However, instead of changing or converting the numerator as a whole, he converted the index only.

Though the mistake made was minor, it created something of a surprise. Many interpretation came out of the simple index law.

Although formula is given at times to aid the solving of mathematics questions, the correct interpretation and understanding of the concepts and writing of the terms has to be digested with clarity.

8.In maths, the many expressions and numbers confuse the working mind when you are not alert.

This is so especially when you do subtraction.

Do you look at expression as a block or isolated terms?

Let's take an example.

If Z = 2x + 1, and A = x - 1, perform Z - A.

Do you directly work the subtraction out, like this :
2x + 1 - x - 1, or

Do you treat the A as a piece or block, like:
2x + 1 - (x - 1)

Looking at expression or numbers, requires "seeing" skill. You need to see with your mathematical mind.

Always understand that expressions and numbers alike are to be operated as a whole.
The use of parentheses is a good habit.

Parentheses can be used to group the expression or target, and make it visually clear to the mind that you are working on a piece of information.

From the above 2 ways of seeing the Z - A, you will notice that the first work-out will give a mistake that is very popular among math learners. It is always repeated even after tons of corrections.

The true mistake lies in the way you look at numbers or maths expressions.

If you can't and always make careless mistake, apply the parentheses ( or bracket) to the desired target.

Know your strength and weakness while doing maths. It will at least help reduce some careless mistakes along the way.

9.Look at the difference in algebraic operation for the 2 math examples below:

Example A:
2x + 4y - 3z = 3
after re-shuffling, becomes
-3z +2x + 4y - 3 = 0

Example B:
3x - 4y + z = 2
becomes, after re-shuffling,
-z -4y -3z -2 = 0

Example A can be seen to be correct mathematically, whereas, Example B isn't.

Why so?

Example B, after having the terms re-shuffled, has the signs of those terms changed!

The explanation to this sign change is that since, the terms were moved from left to right, and right to left, the sign must change. A shocking mistake has been made!

This is a mis-understanding and also a mis-conception.

What was missed out here is that the movement of left to right (or vice versa) has to cross over the "equal" symbol.

4x -5y = 1

0 = 1 - 4x + 5y <== This is correct sign change after re-shuffling across the "=" symbol.
-5y + 4x = 1 <== This is correct re-shuffled terms with no change in sign.

Message:
As long as the terms remain on the same side of the "=" symbol, the terms will not have their signs changed, even though their positions may have shifted.

The sign change results only when the term moves across the "equal" symbol, crossing over the opposite side.

Thus, do not confuse re-shuffling of terms within the same side to crossing the "equal" symbol.
This simple mistake can produce a big mistake through wrong understanding of math principles.

Math make us think properly and logically with reasoning to every steps taken. It is a good subject that aids mankind. Treasure the learning.

10.It is human to err.

But to err repeatedly is wrong.

Knowing the scope of a mathematical tools or concepts is a necessary part to handling math well.

In trigonometry, you need to understand the principles of using these trigonometrical functions.

What is the confine of their usage?
What is the factors before proper usage?

Cited below is a common mistake in using trigonometric operation improperly.

It is the solution of parameter for specific triangle.

Example:
Determine the length of "a" given angle A, angle B and length "b".

Incorrect working to calculate length "a":-

sin A = a / b ==> a = (b) times (sin A). ===> Incorrect!

Why?
To understand sine operation, you need to know its condition of usage.

Trigonometric function sine, cosine and tangent is defined using "right-angled" triangle.

To solve for the above problem, you need to know that angle B is not right-angled.
Therefore, you cannot simply use sin A = a/b.

Angle B has to be of "right-angle" or 900 for that to be correct.

The proper method is to apply the "Law of Sine" for this particular example.

"Law of Sine" : (a / sin A) = (b / sin B) = (c / sin C)

Applying this Sine Law does not require the angles to be at right-angle.

However, do note that you need to know more parameters in the triangle to use the "Law of Sine". Example is the Angle B.

Message: You need to understand the scope of trigonometric operations to apply them correctly.

Application of math principles and concepts requires mental preparation of selecting and strategising correct technique.
And this is what make maths learning interesting.

11.In algebra, a very common mistake you can see learner making is the below:

52 - 22 = (5 - 2)2 = 32 = 9

This is very interesting.

It seems to be correct. That is the problem with this form of mathematical operation.

If you are aware that A2 - B2 = (A + B)(A - B), then this mistake will not occur.

It is this slip-of-the-mind type of human error.
It occurs when you are not alert or too tired after too many assignment quesions.

The correct answer is 52 - 22 = 25 - 4 = 21.
Or (5 + 2)(5 - 2) = 7 x 3 = 21.
Simple?

This is why maths is interesting. It catches you when you are not alert!

12.We do encounter question like,

"Find the angle of A in cos (A + 45) = 0.42 ".

What do you do?

Two solutions are presented as below:

Solution 1:
cos (A + 45) = 0.42
==> A + 45 = cos-1 0.42
==> A = 65.17 - 45 = 20.17 (Answer)

Solution 2:
cosA + cos45 = 0.42
==> cosA = 0.42 - 0.707 = - 0.287
==> A = cos-1(-0.287)

You can see that the 2 answers are different.
Why? Or is there 2 valid answers?

Looking carefully at the solutions above, you will see two concepts in approaching the solving.

The first working went through the conventional inverse cosine operation using the summed up angle (A + 45) as a piece.

The second solution used the concept of algebraic factorising to split the angles A and 45 before processing them separately.
What is wrong here?

To reveal the answer in advance, the first solution is correct while the second has a common mathematical fault.

cos (A + 45) means an operation of cosine onto the angles (A + 45) as a whole.

"cos" is not a variable to be operated upon.
Therefore, "cos" cannot be factorised!

The step, cos (A + 45), cannot be equal to cosA + cos45.
This is a common mistake that need to be removed from the brain.
Press the "Delete" button.

13.A common mistake in trigonometry is the misunderstanding that cosA can be taken apart.

What is the true meaning of this "cos"?

"cos", or cosine, is actually a trigonometrical operation on an angle producing a ratio or a number.

Here, cosine is taken as a reference for this type of mistake made.
Sine and tangent are the equivalent.

You cannot take the "cos" apart from the angle A. They must exist together as a pair "cosA".

For double angle 2A, any trigonometrical operation on it will be likewise treated.

Cos2A will be an operation of cosine on this double angle 2A.

"cos" cannot be treated as a variable, standing alone.

Thus cos2A is not to be separated into "cos" "2A" or (cos)(2)(A).

With this principles, cos2A is therefore, not equal to 2cosA, since the 2A is being operated with the function "cosine".

You may wish to pump in some numbers for the angle and try for yourself this verification.
Example: cos 2(20) and 2 cos(20).
Are they really equal?

As long as you understand what is operation (or function) and operand (or the variable operated upon), you will not have any serious problem with math.

14.Confusion does happen when you are bombarded with many numbers, exponential and its likes.

After dealing with indices, logarithm and their multiplication and division, the brain will sort of tangle up and produces weird happenings.

Take 2 examples below:

1) x^n / y^n ==> x /y

2) log x^n / log y^n ==> (log x) / (log y)

By looking at the first example, you may find nothing wrong.
Since the power "n" is similar for the numerator and denominator, you can do the normal cancellation as you do for "mx / my" = x/y.

However, something may tell you that something is amiss.

While "mx / my" is truely x / y, this is because mx means m times x.
There are "m" number of x that are ADDED up.

For x^n, it means "x" is multiplied by itself n times. (or x times x times x times x ....)

Thus x^n is not equal to xn.
The truth of "cancellation" is that since a / a = 1, and this "1" is not required to be written, the disappearance seems to be "cancellation".

Let me explain further with an example(A).
ax / ay = (a/a)(x/y) = (1) (x / y) = x / y. The "1" disappeared and seems to be cancelled off.

The mistake made in Example 1 in the beginning, is the assumption that the powers "n" followed the concept of "ax" in example(A).

Correct answer for x^n / y^n = (x/y)^n. ==> The powers of n are not removed.

Now for the Example 2, at the beginning, it showed a similar cancellation of the powers "n".
But this time round, it can be said to be conditionally correct.

Why?

If the idea that similar "letter" of "n" in the power can be removed through cancellation, then the answer, although correct, is theoretically wrong.

However, if you know that using the Power Law of logarithm, log x^n can become "nlogx" and therefore, log y^n can also be "nlog y", the result of (log x) /(log y) can be rightfully considered correct, since the "n" is removed according to the idea that n/n = 1 and disappeared, or qualified for removal.

In summary, mistakes do happen when the concept of power (x^n) and pure multiplication (x times n) is not clearly understood.

Cancellation of "letters" or symbols in math expressions should be highlighted as a shortcut to removal due to being "1" that can be omitted in the written form.

This concept of "cancellation" is easy if you understand that it is because of a/a = 1.

14.Teacher: John, can you give the answer for X in this log X = log 6 question?

John: No problem. The value for X is simply 6.

Teacher: Correct! How did you get the answer?

John: It is easy. Just do it this way
(John wrote on the board). ==> X = (log 6) /log ==>X = 6.

Teacher: ?????

************************* What happened? ***********************

Logarithm or "log", in its abbreviated form, can be easily misunderstood.

What is this "log"?

Logarithm is an operation on a number that is the reverse of that for indexing a number.

LogaX = Y ==> aY = X

From the above relationship, you will notice that "log" itself cannot stand alone.

That means "log" must come with a number or expression.

"Log" is an operator, like the "+" or "-".

John mis-interpreted the "log" to be a variable!

It made him transfer the "log" over the equal symbol as though it is a number (or equivalent).
the "log" is thus, separated from the "X" that it should operate upon.

A common "log" mistake was made.

The answer can be obtained through logically comparison, that is,
when log X = log 6, X is simply = 6.

The question may be simple, but if the learning is improper, the concept behind it may be drastically, wrong, even though the answer can be correct.

15.AB = 0.

The mathematical statement seems simple.

It means the multiplication of variable A and B equals zero.

Though it seems simple and direct, mistake in understanding the implication of the zero exists.

When we say AB = 0, we indirectly (and logically) deduce that A = 0 or B = 0.

This deduction is with taken regardless of what the other variable is.

When we say A = 0, B can be anything since 0 multiply "anything" = 0.
This is true vice versa for B = 0.

But what if AB = 1? or AB = x?

This is where misconception of the "logically" deduction happens.

Many maths learners assumed that since AB = 0 indicated A = 0 or B = 0,
AB=1 indicated A = 1 or B = 1 also!

This is a grave and serious mistake made.

AB= 1 does not imply A = 1 or B = 1 .

If A = "1" is true, then AB = 1 means "1" x B = 1, which is definitely false, as 1 x B = B!

Likewise when B= 1 is assumed.

Therefore the AB = 0 cannot be applied across the board to cover all else with the same deduction.

The equal to Zero has special meaning, and should not be confused with other number equated.

In summary: AB = 0 means A= 0 or B= 0 only if "= 0".

A little accurate understanding goes a long way.... in maths, especially.

16.This is a very interesting maths question, that may trick many students.

2x = -1

What is the value for x?

I got all sort of answers.
Some gave me x = 0 (knowing that 1 = anything to the power of 0).
Some treat the expression to be 2x = -1 ==> x = -1/2.
Some used the calculator, applying "logarithm" operation and getting an "Error" message!

Those who know the answer, congratulation. You can stop reading this post.

For those interested and wanting to know what's the answer, read on...

Knowing the answer, is fine here. But knowing with understanding is better.

Now, let's put some numerical values of x into the expression.

If we choose x = +3, y = 23 = 8 ( a positive number).
If we choose x = -3, y = 2-3 = 1/8 ( a positive number).
If we choose x = 0, y = 1 (again a positive number).

Conclusion: All the x values substituted will get us positive number instead of the desired negative number.

Then how do we get a NEGATIVE number from the exponential expression?

WE can NEVER get a valid numerical value for x for expression in the form ax or equivalent when it is equated to a negative value.

Clear?
Do not fall into this mathematical trap. You will not look good, if you can solve it!

source:mathsisinteresting/blogspot.com

Blazing goIITian

Joined: 11 Mar 2009 20:15:19 IST
Posts: 5564
17 Mar 2009 11:45:52 IST
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Blazing goIITian

Joined: 8 Aug 2008 05:48:49 IST
Posts: 484
2 Jun 2009 19:39:56 IST
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Good one Nugo

Blazing goIITian

Joined: 1 Mar 2008 12:04:44 IST
Posts: 797
10 Jun 2009 12:10:15 IST
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And pretty looooooooooooooooooooooooooooong too.

Blazing goIITian

Joined: 11 Mar 2009 20:15:19 IST
Posts: 5564
10 Jun 2009 16:10:18 IST
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and pretty boring too...yup ..boooooriiiing

Cool goIITian

Joined: 27 Dec 2009 16:10:49 IST
Posts: 80
28 Mar 2010 00:31:49 IST
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gud article man.... very usful....my bst wishes

Blazing goIITian

Joined: 4 Aug 2009 12:11:46 IST
Posts: 314
28 Mar 2010 13:39:16 IST
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gud article....almost everyone do these mistakes....

Blazing goIITian

Joined: 2 Sep 2007 11:45:43 IST
Posts: 383
4 Apr 2010 17:30:14 IST
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nugo rocks!! pretty good one man..

Blazing goIITian

Joined: 2 Sep 2007 11:45:43 IST
Posts: 383
4 Apr 2010 17:31:44 IST
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but these mistakes are not what a high school student would make.. i found many similar mistakes in my younger sister's book.. she's in 5th!that's why i liked this article

Cool goIITian

Joined: 26 Jan 2010 13:53:06 IST
Posts: 51
5 Apr 2010 20:36:23 IST
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good

Hot goIITian

Joined: 2 Apr 2009 11:50:53 IST
Posts: 197
6 Apr 2010 11:39:41 IST
0 people liked this

its really gud n helpful .......

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