Math Challenge 13

Once upon a time, there are three numbers. They like to play with each other mathematically.

One day, they decided to add themselves up to see how big they can become.
They kept the answer for future reference.

Another day when they met, they decided this time to multiply themselves.
They got a huge surprise. The answer remained the same as when they added up.

Question: What are the 3 numbers?

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Complex Number | Common Mistake (1)

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.

Message: "Touch me not" i said.

Mastery takes place when we do not repeat mistakes.

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Using "i" Imaginary | 2 key functions

Complex number consists of both real and imaginary terms.

The imaginary term utilised the letter"i" as an operator.

This "i" is a special element in mathematics.

What does it do?

It has 2 key functions:

1) It can change a real term or value to an imaginary term, and vice versa.

Example:
Given Z = 2. If you perform an "i" multiplication on this Z, you will get i x 2 = i2.

The real number, 2, became an imaginary term, i2 !

Likewise, an imaginary number i4 multiplied by "i", gives you a real number, i4 x i = -4 !

2) It can rotate a target by 90 degree anti-clockwise.

Example:
Given Z = 2 (lying on the horizontal axis at 0 degree). By multiplying an "i" to it, the number changes to i2, which means lying on the vertical axis at 90 degree from the original.

The "i" works as a rotating operator on its target.

Interesting?

The cheeky, little "i" can make number change direction as well as characteristics; a real number into an imaginary one!

:-) (-:

Complex Number Explained (2)

Complex number consists of real and imaginary terms to cover the concern that not all real number can satisfy any maths question.

An example was provided in this post (click the link for information).

What is then the different between this Real and Imaginary term?

In graphical form, the Real term is presented in the horizontal axis whereas the Imaginary term is along the vertical axis.

This is purposely so to has no impact of the Imaginary term on the Real term.
(Think in term of the cosine aspect of a pure Imaginary axis).
The overlapping portion of the Imaginary vertical part is ZERO on the Real axis.

See the diagram below for understanding.

The concept of this diagram is to allow learners visually see that the Real and Imaginary terms are unique in themselves and have no link to each other.

But another issue appears.

What is it?

Looking at the diagram, you will see that the complex number defined as Z = a + ib,
where "a" is the real term and "ib" is the imaginary,
will create directional value.

Some solution to specific maths problem requires the complete a + ib format.

Thus the introduction of complex number to offset the impossibility of solving equation using only real numbers, forces the angular dimension into the answer.

With this angular dimension coming into the mathematically picture, the principles of quadrant, as in the trigonometry studies, will be utilized to identify the various answers.

Complex number is then made "complex" mainly due to this directional information added to the normal Real numbers.

Do not be frighten off by this new addition, as, if you know very well how it comes about, you will welcome it. This imaginary term helps you solve many interesting maths equation that normal working cannot.

Love this complex number.

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Complex Number Explained

Mathematician believes that all expressions in maths have solution.

But there are equations that seems to be out of sort.

No solution looks fitting.

One example is shown below.

Example:
x - (x + 1)² = 2

Here, by pure comparison without touching on mathematics, you can deduce that x is definitely smaller than ( x + 1).

Furthermore, what if the (x + 1) term is squared!

x is surely smaller than (x + 1)² by this logically deduction.

Now the question, can x - (x + 1)² be a POSITIVE number?

You will fully agree that it is impossible.
A smaller number minus a bigger number will give us a NEGATIVE outcome.

Then how do you get the answer to the above expression?
What is the "x" value that produces a positive "2"?

There is no way for any REAL number to satisfy this!

To solve this type of "impossible" equation, you need to venture into the "Complex Number" concept. Since real number cannot meet the criteria to resolve the maths question, you need to imagine a number to meet this task.

"Complex Number" consists of number formed by a REAL term and an IMAGINARY term.

It is this imaginary term that will give you an answer to the challenging question.

With the understanding of "imaginary number", you will be in a better position to appreciate the usefulness of solving any maths problem with complex number.

Maths is interesting, right?

When you cannot get an answer in the normal sense, you imagine a number!
What a way to get an answer.

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U

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Simple Factor Multiplication

Multiplying is simple.

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

Simple?
Sure it is.

How about y(y - 1)?
Answer is y² - y.

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).

The correct answer is then:
(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.

Cheers! :-)
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Indices | Interesting Mistakes (3)

If you are not careful with the base and power in a mathematics index expression, simple mistakes will occur. These simple mistakes will expose the weakness in your basic understanding of indices concepts and principles.

But, although, mistakes do occur, rest assure that learning from them is a good thing to have in the learning process.

What is the common mistake student normally make when doing indices?

Here is an example:

y = 9^{x + 1}

y = (3²)^{x + 1}

y = 3^{2x + 1}

You see the error here?

Yes, the power to the base 3 is wrongly done.

It should be 2(x + 1) = 2x + 2. The last term was left out of the multiplication.

A good way to prevent this mistake, or slip-of-the-mind error, is to use parentheses.

Using parentheses at the (x + 1) index will visually group up the "target" for multiplication by 2.

The correct answer: y = 3^{2x + 2}

Take note of this simple maths error and you are on the way to a happy maths learning journey.

Cheers! Making maths interesting goes a long way.....

:-)

Indices | Interesting Mistakes (2)

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. Make full use of parentheses, if possible, to group and focus the targeted term or number.

In summary, index or power affects whatever it includes, base and its sign together.

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U

Indices | Interesting Mistakes (1)

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/a^m.

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.

Otherwise outcomes may appear that amuses the teachers.

:) Maths is fun when looked at from some angle.

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How Is Maths Interesting ?

Anyone who do a taxing job will detest the job finally.

Anyone who has trouble learning a subject will lose interest ultimately.

Anyone who find a task difficult will not have interest developed for it.

These are threads of a human being and a fact of life.

When you put in effort to rectify the undesired for a good cause, you will find fulfilment gradually.

This gradual increase of satisfaction will arouse your interest for the subject or task.

The interest will develop, if the process is correct, to a liking.

Learning maths is the same.

Put in the appropriate effort and see interest develop.

Finding maths interesting will lead you to like maths finally.

You will find that maths is not that difficult after all. You will start to challenge yourself when faced with testing questions. You will like the thrill.

If you find that thrill, I can safely say that you have succeeded with maths learning.

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U SMILE

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Seeing Expression As A Block

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.

How do you go about this?

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.

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U
Smile... Maths is interesting!

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Cranes of Number "Four"

Numbers are everywhere.

Some are obvious but some are not. They have to be discovered.

Below is one that is nicely presented.

Can you see the "number" that they form?



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Trigonometric Form of Complex Number (Part 2)

The trigonometric form of the complex number consists of 2 parts:

i) Modulus (length)
ii) Argument (angle)

An interesting question : Can the argument be different?

Example:

Z = 5 (cos 45⁰ + sin 30⁰)

Is Z possible as the trigonometric form ?

If you comprehend the concept of vector, you will understand that having different angles in an expression carries no meaning.

The complex number Z represents a line of specific length pointing in a specific direction.
(Argand diagram can be drawn to reflect this.)

With a specific direction means only ONE angle is possible with reference to the positive Real horizontal axis.

Thus it is not possible to have the argument of different values in the trigonometric form.

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Trigonometric Form of Complex Number

Trigonometric form of the complex number combines algebra with trigonometry.

Z = r(cosA + jsinA)

In Trigonometry:(examples)

cos(45⁰)= 0.707, sin(45⁰) = 0.707, and
cos(-45⁰⁾ = 0.707, sin(-45⁰) = -0.707,

cos(135⁰) = -0.707, sin (135⁰ = 0.707, and
cos(-135⁰) = -0.707, sin (-135⁰) = -0.707

The above characteristics showed that "cosine" has less impact to the overall angular interpretation of the complex number, Z.

What do I mean?

If Z = 3(cos45⁰ + j sin 45⁰), we know the angle to be +45⁰.

However, if Z = 3(cos45⁰ - j sin 45⁰), what is the angle?
"+45⁰" as in the cosine term, or "-45⁰" as for the sine term.

Looking at the trigonometric examples using positive and negative angles, the sign of the computed values having the same angle for the cosine operation remained.

Rather, the imaginary term of sinA has more information regarding the actual value of the angle.
This is so since "sine" can cause sign change to the computed value with positive or negative angles.

Thus to quickly identify the angle from the trigonometric form, we have just to look at the imaginary or sine portion of the complex number.
The cosine part will not reveal the correct answer.

An understanding of trigonometrical principles, especially the quadrant concepts, has to be strong in order to handle complex number studies.

:D Do not be overly fearful of maths. It is interesting, if you follow the ideas behind it.

:) ....

Mistake in Sign of Reshuffled terms

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.

Cheers!

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Wrong Usage of Trigonometric Ratios

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.

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Special Product A^2 - B^2 and A^2 + B^2

The product format A² - B² is a special form of algebraic expression.

It is equal to (A + B)(A - B).

Many other expressions, like the cos2A = cos²A - sin²B,
can also be expressed in the special product form, that is,

cos2A = (cos A + sin A)(cos A - sin A).

But, how about A² + B² ?

Can it be expressed in the (A + B)(A - B) format?

Why not.

However we need to deviate a bit from the norm.

We need to know the imaginary "i" in complex number system.
Click here for a review to it.

Since i² = -1, we can make use of this property for the special product A² + B².

Here it goes....

A² + B² = A² - (i²)(B²)

Using the basic of Indices, it can be modified to,

A²-(iB)²

This becomes, therefore, (A + iB)(A - iB).

In summary,

A² - B² = (A + B)(A - B), and

A² + B² = (A + iB)(A - iB).

The principles of the special product still holds regardless of the addition or subtraction operation between A square and B square.

Student: What an interesting twist is mathematics in this matter. Anything seems to be simple if we know the technique!

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