Wednesday, 25 February 2009

Careless Algebraic Mistake

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.

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Sunday, 22 February 2009

Algebra Is Useful

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There are everyday events that requires the use of algebra.

Solving simple math question with unknowns can be done easily with algebra in mind.

Take the example of the math challenge 15 given by clicking this link.

What the challenge requires is the addition of a pair of 2-digit number obtained from a 4-digit number.
The higher 2-digit number is to be added to the lower 2-digit number to obtain the centre 2-digit number.

Example:
1978

Upper 19 is added to lower 78 to produce centre 97.

In that post, you are to come out with more examples of this type of 4-digit numbers.

Use of Algebra can easily solve this cahllenge.

How?

Here it goes...

As in algebra, let's assign "letter" to each digit of this 4-digit number ==> abcd

The upper pair is then 10a + b, and
the lower pair is 10c + d.

Adding them up gives, 10a + b + 10c + d = 10b + c (this is the requirement)
==> 10a + d = 9b - 9c -----(A)

Also a + c + 1 = b ==> a = b - c - 1 -------(B)
and b + d = c + 10 ==> d = c + b + 10 ----(C)

Here, it is necessary to assume b + d >10, since otherwise negative number relation will appear.
(If you find this statement tough, never mind, and read on..)

From the above 3 equations formed, you will then be able to randomly choose numbers that fit them.

You will now appreciate the usefulness of algebra in solving this math challenge.

Enjoy!
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Tuesday, 17 February 2009

Logarithm | Common Mistake

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

Wrong answer given: log X/(YZ)

Correct answer: log XZ/Y

"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.

Cheers! :-D

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Sunday, 15 February 2009

Mathematics Letter S

Using graph and equation, you can create wonders.

Graph does not only mean lines and curves. It can be "letter" too, as seen below.

With creativity and a bit of trying, you can have surprising images formed through graphs.

With mathematics, you are not limited to equation and solving problem. You can have fun and that makes mathematics interesting.

:-)

Friday, 13 February 2009

Math Challenge 16

Jane started off a multiplication process with the number 4.

John followed up by multiplying it by 4 also. This led the result to be 16.

Jane, being her turn now, times the current answer by 4 as required. The answer now is 64.

After a number of alternating multiplication by Jane and John, the result became 4 ,194 ,304.

Guess who did the last x4 operation (without using the calculator or equivalent).

.....

The answer is not as difficult as seems to be. Just a simple stare at the alternate answers will solve the mystery.

Happy thinking.....

;->

Sunday, 8 February 2009

Complex Number | Common Mistakes (2)

Complex number consists of 2 parameters.

They are the modulus (length) and argument (direction).

Argand diagram is the pictoral form of representing this complex number.
In the Argand diagram, quadrants define the position of the "complex" line.

Z = a + ib (click for information) is the general form of writing the complex number.
"a" and "b" will define its polar counterparts, modulus and argument.

Having 4 quadrants in the Argand diagram means having 4 combinations of "a" with "b".

They are:
1) Z = a + ib
2) Z = -a + ib
3) Z = -a - ib
4) Z = a - ib

The first case lies in the first quadrant (Q1).
The second case lies in Q2.
Third case lies in Q3, and
fourth case in Q4.

Therefore knowing the sign of "a" and "b" let you know which quadrant the complex line lies.
And this is where the mistake lies!

The argument is always computed wrongly for the Q2, Q3 and Q4.
Only the positive sign of "a"and "b" is taken to get the value of the angle (argument).

Example of error:
Z = 5 + i5 ==> Argument = +450 (Q1)

However, Z = -5 + i5 ==> Also taken as + 450 forcing it to lie in Q1 (wrong!).
It should lie in Q2 since now the real term is negative.

Values of "a" and "b" are not the only parameters needed to find the angle.
Their signs are equally important!

Think of the Argand diagram representation before the definition of the angle.
This will ensure that the angle is correctly calculate later on since you have an idea which quadrant the complex line should lies then.

Doing the complex number and its conversion from rectangular form to polar form properly will make you happy and like maths. Proper thinking process will path you into a good habit that leads to confidence in maths.

Maths is Interesting! And fun ...

:D, Smile.

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Saturday, 7 February 2009

Geometrical Errors Can Be Exciting

The learning of geometry is to allow anyone to have an idea of correct perspective to objects.

Dimensions are important in this area. Angle of view is equally important.

Drawings of object in the fore-ground and background differs because of geometry.

When the concept of geometry is violated in drawing objects, you will get interesting outcome.

This outcome, however, is apparent only in the virtual sense and cannot be physically produced.

Example of links are quoted below:

http://anythinggorgeous.blogspot.com/2009/01/staircase-of-character.html

http://anythinggorgeous.blogspot.com/2009/01/cunning-overlapping-of-objects.html

http://anythinggorgeous.blogspot.com/2008/12/twisted-rods-or-fishy-rods.html

Geometrically deceptive objects are not easily identified, and have to be closely stared at to reveal their "wonders".

Only through learning maths and its relevant topics, you can then appreciate the importance of having done it. This is very obvious in the above few examples.

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Math Challenge 15

This is an interesting one.

Given a special four-digit number, 1978,
we can get the centre two digit (97) by adding
the left-most 2 digits (19) and the right-most 2 digits (78).

Another of this special number is 1538.

Can you get 2 more of this special numbers?

Brain-raking isn't it?

To be frank, it is not that difficult.
Hint: Algebra can solve this maths thinker.

:-)

Sunday, 1 February 2009

Math Challenge 14

Sometimes when you go for shopping, you may encounter items on offer.

10% discount, 36% discount, ......

So if a \$18 toy is put up at a discount of 25%, and another toy, priced at \$25, is at a discount of 18%, which has a bigger reduction in dollars?

Do not use calculator, though, to answer this challenge.
Otherwise all the toys will be gone!

Be quick.....