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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