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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Saturday, 31 January 2009
Thursday, 29 January 2009
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.
.
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.
.
Wednesday, 28 January 2009
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!
:-) (-:
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!
:-) (-:
Tuesday, 27 January 2009
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.
.
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.
.
Sunday, 25 January 2009
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.
. .
U
.
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.
. .
U
.
Saturday, 24 January 2009
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! :-)
.
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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