Mistake with A^2 - B^2

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!

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Key Differences Between Cos and Log Operations

It has been mentioned in previous posts that "cos" and "log" are actually mathematical operators.

They cannot exist alone. They need partners.

For "cos", it needs angle.
For "log", it requires a number.

They operate on something.

So what is their difference since they are both operators?

"cos" changes angle information to number (ratio).

"log" operates on number to get another number.

Their applications, thus, differ in the above matter.

NOTE: "cos" here refers to any trigonometrical functions.

"cos" is useful in getting length or angle information.

"log" can be used to find the value of the power needed.
Example: To find 2^x = 5

Knowing the various mathematical tools, we can apply the correct ones to solve specific problems.

Without understanding the underlying usage and concepts, we will be at a lost to which tools to use.

Math is, therefore, wider in nature than just calculation.
Keep learning....

:-)

Math Challenge 12

Although math may seem difficult at times, proper use of its principles will render it simple to handle and use.

Here, a bit of algebraic juice can help in solving the below math challenge.

Following the pattern, you will observe a "trick" that you can make use to solve the challenge later.

2² - 1² = 3

3² - 2² = 5

6² - 5² = 11

11² - 9² = 40

The above equations are done without detailed written working, just simple mental calculation.

Challenge:
16² - 14² = ??

Do it without calculator.

What is the answer to the above? At the comment session, please.


Hmmm........ tick tock tick tock

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Understanding Math Principles Helps

There are many formulae that a student has to know and sometimes remember for his studies, and applications.

How then can he capture all these necessary formulae for usage?

One way is to understand the principles and derive those required later on for usage.

Example:

In trigonometry, you will come across sine of two angles, sin (A + B).
You have memorised sin (A + B) as "sinA cosB + cosA sinB".

However, when you further need to know sin 2A, what then?
If you know and understand the principles of the sin(A + B), you can easily move on to derive the sin2A.

How?

Since you know sin (A + B), you can equate A = B, to allow you to get the sin(A + A).
As sin (A + A) = sin2A, you will then have no problem achieving
sin(A +A) = sinA cosA + cosA sinA ==> sin 2A = 2sinAcosA

There, you have obtained another formula without the need to memorise it.

Thus mastering the sin (A + B) principles or its equivalent, can allow you to expand your knowledge further.

You see the benefits now?

I have shown only the trigonometry part in math learning, but, you can appreciate that it applies to any other topics as well.

Seek, therefore to handle principles of math well as it will serve you good in the long run.

:-) ..... :-) Forever liking mathematics. Maths is interesting!

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Math Challenge 11

What is the simplest way to solve the below equation?

cos X = log X.

You may wish to give your suggestions in the comment space.

Hints:
1) There are 3 answers to the above relation.
2) Sometimes technique not related to logarithm or trigonometry can be useful.

Happy thinking ....... (:-)

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Mistake in Cos(A + B)

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)
==> A = 106.69 (Answer)

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.

With this post, your trigonometry is getting better right?
Cheers!

You may visit this post for more mistakes to be avoided.

;)

Cos2A IS NOT 2CosA

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.

:-)

Mathematical Flower For Festive Season

Since this is the end of the year, with the festive season coming, I have designed a flower through the graphical means using the mathematical expression indicated.

It is done in polar form with the x as the time base and y as the modulus (amplitude).

Happy greetings to my readers.




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Division of x^n with y^n

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.

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Correct Method over Correct Answer

In math, you do not go for correct answer.

Yes, correct answer motivates. It is the ultimate goal for any math learner.

But is that all to math?

Math calls for more than that.

Read this story before moving on......

You have to ensure that the correct answer is gotten with the correct method and concept or principles.

Many a times, students can obtain correct answers to a math question, if it is not properly though over by the teacher.

If the teacher did not do a scrutiny over the working, the answer may be passed off as correct.
This is disaster.
If the student did not master the concept properly, this sort of happening will also results in learning disaster.

Therefore, seek to learn the correct technique or math method instead of aiming for the answer.

It is better to get wrong answer with the correct method than getting correct answer with wrong technique.

This makes learning math interesting!

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Common "log" mistake

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.

Log_aX = Y ==> a^Y = 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 "-".

What mistake did John made?

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.

Correct answer:
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.

Learn well. Maths does not call for correct answer. It is the thinking behind it.

..... :-)

Interesting Facts of Factorisation in Quadratic Equation Solving

Why do you learn factorisation?

What is the importance of factorisation?

You may have these questions when you are taught this.

Factorisation is the making of an expression into the product form, that is, (....)(....)(....).

Factorisation ends with all terms in a math expression being connected with parentheses.

What for?

To solve an equation, say, a quadratic equation, you normally make the expression equal to zero.

There is a meaning to this "equal to zero".

Using factoring method, (....)(....) = 0, implies that either one of the (..) can be zero.

Click this link for an explanation to the above statement.

This is will not be so when the terms are in the form A + B = 0 (sum format).

Factorising causes a quick, simple solving of quadratic equation by having the product form to the expression.

Example :

x² + 3x + 2 = 0

Factorising: (x + 1) (x + 2) = 0

Through factorising the quadratic equation,
you can equate (x + 1) = 0 or (x + 2) = 0.
Thus, x = -1 or x = -2.
This is made possible by factorising.

The demerit of this factorisation method, however, is that it takes time to figure out the numbers within the factors. Not all expressions can be easily changed to the factor form through simple "guessing".

NOTE: This post is talking about numbers and not matrices, which involve another concept of dealing with AB = 0.

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Meaning of AB = 0 and AB = 1

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.

:-)