Friday, 25 December 2009
See this math challenge 20 and its answer in the comment.
But a twist of the questioning will and can make it more challenging without changing the math expression.
Below is one:
Base on math challenge 20, if all unknowns CANNOT be repeated, what are they?
Again they are integers and below 10.
Enjoy the math thrill answering this.
Saturday, 19 December 2009
Monday, 7 December 2009
Percentage problems can be tricky at times when you are careless.
Let's us look at one maths word problem related to it.
There are 3 persons, John, Mary and Jane.
John is richer than Mary by 10%, and Mary is richer than Jane by 20%.
Is John richer than Jane by (10 + 20)% = 30% ?
Most students, upon quick thinking, will acknowledge that 30% is the correct answer.
Is it so?
To verify the answer, let us assume that Jane has $1000.
As such, Mary will have (100+20)% of $1000 = 1.2 x $1000 = $1200.
John is then 1.1 x $1200 = $1320 richer than Jane ==> By 32%.
If 30% is correct, we should get 1.3 X $1000 = $1300.
The latter number (dollar) is not the same as the first worked out solution.
Mistake in understanding what is percentage:
To assume that John is 10% + 20% richer than Jane is incorrect.
This is due to the fact that percentage has to take a common reference for this to be correct.
In the word problem, the percentages of comparison are not to a common reference.
The first one is to Mary, while the next is to Jane.
These made the denominator of the ratio different.
Thus adding the percentage up is a mistake, and an easy one too!
Sunday, 29 November 2009
Everyone knows what a square root is for and what it does to a number.
But have you tried multiple square rooting ?
What do I mean?
Let's take an example.
Start with a number, say, 3.
i) Square root this 3.
ii) You will get a number after step one.
iii) Square root this new number again.
iv) Continue the steps above and look closely at the number.
You will notice that the number after multiple square rootings, will give you closer and closer to or approaching the number "1".
Now let's start with another number. This time, let's choose, 0.4
After doing the same procedures, you will again notice that the answer is getting and approaching the number "1"!
Amazing isn't it?
What other wonders can you find from this Square Root maths operator?
Share with me in the comment section.
Maths is interesting.........
Monday, 23 November 2009
Logarithm operator exists as a mathematical tool to allow us to convert a number to another form in terms of base and power.
Can you make 0.5 in terms of 10y or 5y ?
The above is a maths question that can use logarithm to solve.
The application requires the Power rule that states that logkDm is equivalent to mlogkD.
With that knowledge, to convert the above question of 0.5 to various base, we simply log the 0.5 to its respecive base.
Let's go for the base 10.
Original: 0.5 = 10y
Performing "log" on both sides:
log10 0.5 = log1010y
-0.301 = y log1010
-0.301 = y
or another way to put the outcome is 0.5 = 10-0.301
We have managed to convert the original number of 0.5 to one with base 10 and a power (index) of "-0.301".
We can also similarly do the same for a base of 5. => 0.5 = 5y
Here we just "log" to base 5.
log5 (0.5) = log5 5y = y
-0.431 = y
Thus 0.5 = 5-0.431
From the 2 examples above, you can see the wonders of having logarithm as a conversion tool.
Once you understand this principles, you will appreciate logarithm.
Friday, 20 November 2009
Everyone knows what happen when we add a number to another.
We also know what happen when we do subtraction.
There is also no problem with multplication or division of numbers.
These are all simple mathematical operations that are basic.
What about doing a logarithmic operation in math?
This is a slightly complex but interesting question.
When we do a logarithmic operation on a number, what we actually do want out of the mathematical process is the power or index with reference to a base number.
105 has a base number of 10 and index of 5.
Doing a "log" of the above will reveal an answer of 5.
This is the power after doing a "log" operation.
Thus, when anyone does a logarithmic operation on a number (or expression), he is trying to find the index with respect to a base reference.
log 105 = 5.
Full written log expression is "logkP". When the "k" is left out, it implies that k = 10.
And log 10 = 1.
I hope the above can explain why we do logarithmic operation and its significant.
There must be a reason for each math operation, otherwise we will be learning and doing some insane process on earth!
Maths Is Interesting!
Tuesday, 17 November 2009
There are the logical deduction type whereby you have to visualise and come out with an answer.
Below is one good example that I would like it to be a challenge.
(Don't be frightened by this, it is just for fun.....)
Above you will find a stack of cubic boxes. There are the blue and the yellow cubes.
What is the least quantity of blue boxes must we use in order to hold the yellow boxes in the same place?
You may present your answer and how you arrive at the answer in the comment section.
Thanks for trying. And I am waiting for that interesting mathematical deduction...
Maths Is Interesting!
Monday, 26 October 2009
Reading maths question is a critical skill to learn.
Without understanding the question, you will not be able to solve the challenge correctly.
A typical maths problem uses the word "of" to express ratio.
What really does this simple word means?
To an adult this is the understanding of English.
But to a kid, this is not English but a maths question!
What then is "of" in maths?
Answer: It means MULTIPLY.
half of the time ==> 0.5 x time
2/3 of the class ==> (2/3) x (class total)
If 40% of the apples are rotten, how many are left? ==> 0.4 x apples are rotten
Learn English well to handle maths.
This simple word "of" may make you happy or sad.
The choice is yours.
But remember, "Maths Is Interesting!".
So despair not, enjoy your maths.
Wednesday, 21 October 2009
Geometry in maths can means dealing with angles from a square or a rectangle.
Normally the question is to determine an unknown angle given some shape and angles.
However, mistakes can happen when basic knowledge of relationship between angles and shapes are not proper understood.
Here, I will stress on the square and rectangular matters. This is basic but can pose a tricky problem to the unwarys. Poor thing.....
Let's look at the diagram below.
This is so since the corner where angle A lies is 90 degree divided EQUALLY by half due to the diagonal lines reaching to the opposite side. (symmetrical sides).
However, if the side M and N are not equal in length, then angle A WILL NOT be 45 degree. It will depends on the ratio of side M and N.
Note this message and unnecessary mistake can be avoided.
Sometime it is to test the logical thinkng through maths, by not telling you angle A is 45 degree but stating that the box is a square.
This type of maths problem will require you to calculate another angle but using angle A which is not given.
It is tricky but good to have. Your brain will be stretched to make it "flexible" for future use.
Maths is good in this sense as it twists our mind and makes our life interesting!
Work hard as well as smart.
For more examples on avoiding unnecessary mistakes, visit this time calculation post.
Saturday, 17 October 2009
Maths is interesting!
There are many exciting concepts and techniques one can apply.
Read on ...
There are many ways to solve a maths question.
Normally it involves taking many steps with related sequence.
However, there are also simple ways to handle a maths problem.
One such solving method is simply through visual steps.
This has no working at all.
What do I mean?
Let's look at an example.
Determine the angle B from the diagram below. Angle A is 40 degree.
There is no working at all. Just the visual determination.
Concept of this trigonometrical question in geometry:
When 2 straight lines cross and meet at an angle to each other, the angle opposite to any one is the same.
As such, angle C is also equal to angle D.
This is visual maths.
Wednesday, 14 October 2009
From the logarithm equation below, can you determine what is the expression for y?
log y = x + log x
You may give your answer in the comment section and also help explain how you get it (for the sake of sharing).
Tuesday, 6 October 2009
There are the hours, the minutes and the seconds to handle.
We can add or subtract them.
We do not operate in the hundreds, or ones, like any simple arithmetric.
We are calculating in terms of sixties.
1 hour = 60 minutes
1 minute = 60 seconds
Hence when given a maths question on subtracting two times, how do we go about to avoid mistakes?
Let's take an example to illustrate.
John start his journey to the market at 09:15am. If he arrived at the market at 10:05am, how long did he travelled?
We can use the conventional method of carrying back 60 to the minutes, since the ending minute is smaller than the starting minute. And reducing the 10 to 9 (hour).
Next, we can then subtract with the new numbers.
That is 9:65 - 9:15 = 50 minutes.
This answer is fine.
Change all the times to mintes.
10:05 ==> (10 x 60) + 5 = 605
09:15 ==> ( 9 x 60) + 15 = 555
Subtracting the two new numbers gives 50 minutes directly.
And that is the answer!
Comparing approach 1 with approach 2, you would notice that the latter seems to be simpler.
This is because we have simplified the mixture of hours and minutes to only one dimension, that is, the minutes.
Thus, subtracting the newly created numbers involved only the minutes, avoiding the distraction of handling the hours.
To do maths, clear the mind of the unnecessary.
In the example above, we have removed the "hours" factor to focus purely on the "minutes".
Maths is interesting in that it is up to us to "play" with the techniques.
We can work with it or go against it.
The choice is up to us to select.
Hope you pick up some tips to make maths interesting.
Wednesday, 30 September 2009
Just a simple slip-of-the-mind type of error can cause havoc.
Once a step is incorrect, the following steps will make use of the wrong numbers to "accumulate" the errors.
If the maths teacher is merciful, she will look for the application of concepts instead of hard numbers or the final outcome.
But maths is maths.
Numbers are always there.
Techniques and systematic approach are almost a must in handling maths questions.
So can we avoid making maths errors if every step is important?
Yes, if you try hard. But note, we are human. Thus to completely eliminate errors everytime is calling for being an angel!
One simple way to reduce maths mistakes is to stay focus.
Being focus and understanding the maths probem is one of the easy method to solve making careless mistakes.
Pay attention to ever steps you write and know the purpose of each working.
Every expression must have a meaning. Otherwise what for write it down as a working.
Simply focus and do not be distracted by the surrounding. This is one of the main cause of making errors.
If you are working in front of the television, switch it off or re-locate to another pleasant place.
If there is too many people around and talking aloud, move away if you do not have any ear plugs.
Just stay away from areas or surroundings that are dysfunctional to your maths learning (and in fact to any learning).
Apply what I wrote and you will find a different.
Happy maths learning and do not forget that "Maths Is Interesting!"
Sunday, 27 September 2009
Is maths interesting and is easy to learn or teach?
There are always argument over how to really capture maths students' attention and make them understand the principles and concepts in maths.
One school of thought is to approach the real life case studies.
Here actual applications are taught to make aware the usefulness of maths.
Problem-based case studies are planned into the curriculum to allow the learners to learn maths.
Another approach that is conventional is to pump in concepts and techniques using symbols and hard formulae.
Here students learn maths consists of symbols and their true meaning in the workings.
No real life indication is mentioned or just skimmed through. The focus is purely the use of maths tools to solve questions.
What are the advantages and disadvantages?
1) For real life approach, there are chances that the learners may couple their maths learning to only that particular application.
If train speed is mentioned or used, the students may only understand that the maths tools apply to train and not areoplane. The scope of aplication is a factor and serve to be a disadvantage.
The advantage is that the learners can relate to the usefulness of maths and will pay more attention and feel more fulfilled.
2) The advantage of pure symbolic approach in the conventional teaching method is that scope of the maths tools are wide. No tying down of the maths techniques to any specific area enables fredom of use.
The disadvantage, of course, is that the students may take a longer time to see that actual useulness of the maths tools and principles.
So what is the best method to learn maths?
I suppose the answer lies with the type of students and the topics to be taught.
No one way is best.
It has to be customised to suit the students or majority of them
Flexibility is thus the best methods and getting a good maths teacher who can read the minds of the students is the better choice.
If the classroom teacher is not up to expectation or has some constraints, it will be good to look for private tutors to brush up the maths learning. Note, classroom teaching does not cater for individual needs and this is a fact.
Happy maths learning.
Maths is interesting!
Don't forget it.
Monday, 21 September 2009
Maths forms interesting patterns if you care to look around.
One such example is highlighted below:
12 11 10 09 08 07 06 <== row 1 (decreasing by 1)
01 02 03 04 05 06 07 <== row 2 (increasing by 1)
If you add the numbers in row 1 to that in row 2, you will get, surprisingly, the same answer throughout!
13 13 13 13 13 13 13
Why is it so?
It is a simple trick to the unwaries, actually.
Maths is not that difficult, to start off.
The answers to the above additions seem to be accidental in having the same number.
It is actually not accidental if you think abit.
The answers were intended to be 13 to begin with.
13 = 12 + 1
13 = 11 + 2
13 = 10 + 3 .... and so on.
Though the additions seems magical with one row in ascending mode while the other row in descending mode, it is the visual maths trick that corrupts and confuses the mind.
If your principles of mathematics is good and strong, you will break through this simple trick in no time.
To excite young minds, this is a good one to try on them.
Happy playing with maths.
Maths is interesting!
Wednesday, 16 September 2009
What is the purpose of graph?
This may be the question every learners first ask when they were exposed to this maths topic.
When do we use graph as opposed to using, for example, Argand diagram or vectors sketch?
Graph by nature is a graphical presentation of data that collectively form into information that reflects the trend of some parameters.
It shows the past, current and possibly the future (prediction).
Graph is a relative as well as an absolute maths tool for people using it.
An example of graph application is that in stock market data prediction.
Using past records, people tends to forecast the future through looking at the graph.
Another example is in engineering work.
Collecting data of a certain electrical system behaviour, engineers can predict the failure or potential life of its operation.
A simple graph is plotted with normally 2 parameters.
But this is not always true.
Graph may come in 3 dimensional. The x, y and z direction.
Knowing graph is an alternative problem solving skill or prediction skill.
It allows users to see an overview of the relation between specific targets.
Graph is wonderful if you let it be.
Sunday, 13 September 2009
In graph plotting, something we need to know the length of a segment of the line plotted.
This may be for the distance to be travelled (like in a field trip).
Or it may be for checking the material to be used in building a slanted pole / support.
Let's take an example to illustrate.
From the plot, if we are to calculate the length of the line between the two red crosses, we can use the well-known Pythagoras' Theorem.
However, we need to know the co0ordinates for the crosses or markres first, to check their positions.
For the lower cross, we will have x1 = 2, and y1 = 3.
For the upper cross, x2 = 6 and y2 = 5.
This allows us to determine that the length in the x-axis direction is 6 - 2 = 4 units.
The length in the y-axis direction will be 5 - 3 = 2 units up.
Using then Pythagoras' Theorem, lenght of targetted line segment will be given as sqrt(42 + 22) = 4.472 units.
From graph and its application with other maths theorem, you can find answers easily.
It is the choosing of the appropriate maths tools that is is key to having a solution in a proper way.
Many a times, you may find answers or solutions through different techniques and methods. But the number of steps are more. But it is still correct.
It is through practice and gaining experience in maths problem-solving that helps you reach a level that let you handle maths with mental ease and confidence.
Everyone can achieve that. It is the attitude. Do not fear maths. It is just a tools to solve problems.
Maths is interesting! Love maths !
Sunday, 30 August 2009
If a ratio has its numerator less than the denominator ==> The numerator is relatively smaller in size than the total.
If the numerator is larger than the denominator ==> The numerator is bigger in size than the total.
An example can illustrate the concept.
If a costume is now priced at 90% of its original, $100, it means that the price is not only $90.
It is lesser than the original, since the ratio is 90 / 100 or 0.9.
If the costume is newly priced at 120% of its original, $100, it means that the price is now at $120!
A price value more than the original.
It is a practical real-life number.
- Percentage can be more than 100%.
- The numerator can be more than the denominator.
Interesting? You bet.
:) Happy maths learning.
Wednesday, 26 August 2009
In maths learning, every learners will come across the terms "percent" or "percentage".
Percentage is a ratio.
It is a ratio between two numbers.
The denominator is normally the total of an item.
The unit used is %.
The above may be common knowledge for anyone doing percent maths problems.
However, what is the presentation of the solution that is appropriate?
Let's us do an example.
Class A has 30 students. If 40% of them are girls, how many are boys?
Number of boys in the class = 30 - 40%
Now, is this "30 - 40%" correct?
The idea may be there, but someting is amiss.
What is this "40%"?
Can we just write 40% as it is?
The answer is NO!
Percentage is a ratio of the total (the class size of 30).
We cannot simply write 40% if we want to know the actual number.
The correct way is to present the step or working as: (40/100) x 30 = 12 students
We cannot simply write : 30 - 40%.
The correct way has to be 30 - (40/100) x 30 = 30 - 12 = 28 students.
2 mistakes, in concept, were made:
1) Number cannot subtract a percentage. Their units are different.
2) Percentage is a ratio, an indication of the proportion of a piece to the total. It is a relative term, not an absolute number. Thus, an absolute number cannot operate with a relative term.
Understanding this concept of "percent" and "percentage" will be handy and avoid the unnecessary trouble and anxiety of doing maths.
Thinks of a slice of cake when doing "Percent". It has meaning only when compared to the whole cake.
Remember: Maths Is Interesting! And it WILL be get more interesting.
Sunday, 23 August 2009
When 3 tests has an average score of 80 marks. All the tests are 80, 80 and 80 marks each.
But nte that the three 80 marks each may not be true.
It is assumed to be to make the average correct.
The actual marks may differ from the 80 marks.
They may be 70, 80 and 90 actually, but their average is 80marks.
You may see the post on averaging if you need more information.
Friday, 21 August 2009
Trigonometrical graphs reveal many exciting properties of their functions.
One such function is the "Sinc" function.
This "Sinc" function is represented by the equation (sin x) / x.
NOTE: There is no typo error for the word "Sinc".
This special maths function is the trigonometric sine of an angle divided by that particular angle.
Looking at the graphs of various multiple of sinc functions, you will notice some unique properties in the cross-over angles (markings).
Looking closely at the multiples of radian pi, 2pi and 3pi, you will see that the amplitudes of the various sinc functions are zero.
This is a special characteristics of "Sinc" function.
If you sample these functions at interval of pi, you will get nothing or zero amplitude.
Interesting trigonometry, right?
Thursday, 6 August 2009
There are many ways to dig into the true character of a person.
One way is through maths.
By being in a classroom of maths learners, which I suppose everyone went through or is in one, you will notice many types of characters and behaviours.
Some are strong and stubborn, the never-say-die doer.
Some are the "can do, then do" type.
Some are the easy surrenders of maths.
Some cannot even be bordered to try!
Those who attempted the maths question, also revealed some weakness also.
The careless type and the long-winded type.
They made all sort of mistakes due poor handwriting or not reading the questions properly. They may miss a few variable or maths operators in an equation.
They may indirectly simplified the problems given through seeing or copying wrongly.
Those long-winded are the "better" lot with the will to stay on track.
They do and do, even when applying the wrong technique. They, however, do get the result through their hardworking attitude.
Some are the intelligent type who spot the trick just by reading the maths question.
They are the "flexible" ones.
They are the ones that seemed to enjoy most of the maths lessons.
Whatever, the type you are, maths is still an important life-long subject.
It is a practical module that serves us till we leave this world.
Love maths, and maths will love you, whatever your character and feeling towards maths.
Maths is interesting! You have a choice for that.
Sunday, 2 August 2009
One question those who detest maths will ask is "Is maths really interesting?".
It is a very subjective question.
Everyone has likes and dislikes.
However, in the case of maths, it is the gain versus the lack.
Maths is a necessary life skill to have.
Knowing it makes a whole lot of different.
It will speed up your solving to some daily questions.
"What is the time needed if I drive at 60 km/h for a distance of 90km?"
"What is the area of the metal sheet needed to cover this pillar?"
We are weak in maths due to many reasons.
If you do not arrest these reasons, or reduces the obstacles to it, you will always fear maths.
This will create a mental block to your maths learning.
Practice and practice to reveal your weakness. Learn through mistakes.
You will feel the confidence of handling maths problems after that phase.
Like what Mark Twain said "Action speaks louder than words".
I would like to tweet it in the context of maths.
Instead of pure saying that you cannot do maths or you hate maths, practice (action) on it.
You will feel the difference.
You will get the hang of doing maths.
You will realise that maths is not that difficult.
You will find that it is your mindset that is the block, not maths!
Practice speaks louder than words.
Maths is interesting.
That will be your final conclusion if you take action and do hands-on practice.
Saturday, 25 July 2009
I managed to see a word problem shown below:
" Mary obtained an average of 80 marks for 2 tests. What marks has she to get for an average of 80 marks for 3 tests?"
If a maths student understand the meaning of "average", she can do this without even working out the mathematical steps.
The first statement in the question stated that Mary scored on both tests an average of 80 marks. This implied that for one test, she obtained 80 marks. Same as for the second test.
For the next test (the third one), to get an average of again 80 marks, it meant that all the test she has to get 80 marks each. This is the power of "average". That is to say, all the test can be concluded as the same score.
The actual differences between the individual marks can be offset from each other to achieve a final "average" of same marks (here, 80 marks).
The maths problem is to test the understanding of the word "average" in maths, and its concept.
Thus, knowing concepts do help in solving maths questions.
It does not mean working out the mechanical computation steps to get answers.
With strong maths concept, anyone can solve simple problems without doing working.
But that is maths also (in the brain, the step-less way).
Interesting. Maths is so. Fear not.
Saturday, 18 July 2009
"I don't understand maths!"
Does this sound familiar?
This statement is not only from young learners, but from adults too.
What actually happened?
Maths is interestingly unique. This is so because it represents a different way of presenting messages.
Maths is a "short form" of written langauge.
It is like the short-hand symbols that many people use to note down minutes of meeting.
What is 1 + 3 = 4?
In the common English language, it simply means "one added to three equals four".
However, visually, the maths equation is encoded with symbols.
What is this "+" and "="?
If you do not understand the meaning and usage of these two symbols, then you are lost.
Thetefore, learning and understanding maths is knowing the unique symbols presenting the "message".
How about 3y = 24. What is "y"?
The question is finding "y" given the relationship above.
The equation in English means "When we multiply "y" by 3, it gives 24".
The solution, thus, it English, is: If I divide this 24 by 3, I will get the answer that is 24/3 = 8.
In maths concept: 3y = 24 ===> y = 24/3 = 8.
It is the interpretation of the maths symbol and its communication with the learners that is key to doing maths.
It is not difficult if you understand the language of maths.
Maths is interesting!
Friday, 10 July 2009
Maths does wonders when presented in graphical form.
This is provided the mathematical equation makes it so.
Using the trigonometric relation, x sin x = y cos y, the artistic effect of this equation is shown below.
You can now see that, though maths can be boring at times, it can reflect its beauty through other means.
Don't you agree?
Maths is Interesting! Watch out for it!
Monday, 6 July 2009
Besides the Boolean AND operation and its application, Boolean OR operation also serves a special application.
For the AND operation, which is equivalent to the maths multiplication, the use is for it to pull any untied input to a system to zero state.
For the OR operation, the concept is similar, except that now it is the maths addition.
Instead of pulling the untied input to the system to zero, you can set it to the other Boolean state, which is the "1".
In base 2 number system (Boolean), if the input is 000101, and you need the data to be all 1, just OR the input with 111111.
What you will get after this OR (adding) operation is 111111.
This is so due to the fact that 1 + 0 = 1.
Here you will see that option to set the untied input to 0 or 1 can be done using either the AND or OR operation of the Boolean system.
This is maths, if you are aware.
This is the application of maths after understanding the principles of maths operation.
Sunday, 5 July 2009
Boolean means 2-state operation.
OR means any one state that is valid will result in a positive outcome.
Example: When James or Mary come, the show will be start.
In maths, this translates to ADD, except only 2 states can occur (that is, on or off only).
1 = on, 0 = off (or vice versa).
What do I mean?
0 + 0 = 0
0 + 1 = 1
1 + 0 = 1
1 + 1 = 1
From the above addition, you will notice that as long as one state is ON, the result will be ON.
The operation is that of an "add".
Note: Since the operation is Boolean, it cannot go above 1 or 2 and above cannot exist. Only base 2 number (0, 1) can happen.
This field of maths is known as Boolean Algebra, a special maths operation used in digital electronics.
If you are using this Boolean process, make it clear that the number base system is 2, otherwise 1 + 1 = 2!
1 + 1 = 2 in Boolean means that you go over the ceiling!
Another interesting maths concept, right?
Maths is interesting!
Tuesday, 30 June 2009
Boolean, though, of 2 states, can still be useful.
The concepts of multiplication in this AND operation can be used to perform cancellation of unknowns.
Anything times zero gives zero.
This is a common knowledge.
Here we are talking about basic fundamental maths. Nothing difficult.
Cancellation of unknowns in the AND operation means that we are able to mask out the unwanted or redefine the unknowns to a definite state.
Here the unknown become a zero, a known state or condition.
A more specific application is in the masking of unconnected inputs to a processing system. If the inputs are scanned and compared for status updates, the inputs have to be accurate or of a known status. Otherwise, comparison results become meaningless.
Here the Boolean multiplication (AND) forces the unconnected input to a zero. This is then an accurate input status for comparison.
Here, maths is applied to technical application, and its usefulness become apparent.
This is the power of understanding maths , and is exactly what makes maths interesting.
Thursday, 25 June 2009
There is a special field of algebra called the Boolean Algebra.
Here the algebra operates in the base 2 number system.
One special operation it performs is the AND operation.
A simple analogy is " Mary AND John went to the park".
The meaning is that BOTH Mary and John moved together as a whole.
If either one is absent, they did not go to the park!
This is a form of multiplication.
0 x 0 = 0
1 x 0 = 0
0 x 1 = 0
1 x 1 = 1
Only when Both are present , the outcome becomes present.
Here you will notice that maths is applied to real life situation, forming into the English word "AND". But in maths, we call this "AND" as multiplication.
Boolean is utilised when the outcome is of 2 states (on or off, present or absent).
Do you see the interesting part of maths here?
Maths is mingled into daily events and is always around us if you keep an eye for it.
Tuesday, 23 June 2009
Father: How many marks did you get for your math test?
Son: I obtained 100 marks!
Father: Great! You have done me proud. You deserve an ice-cream.
Son: Thanks Dad!
Son to brother: Actually my math test is over a total of 200 marks. I almost failed the test! Luckily dad did not master percentage, otherwise I would not get my free ice-cream.
(To understand percentage, click this link).
Tuesday, 16 June 2009
Many people do understand what percentage is about.
If you see a sales offer with 50% discount, you will know that it is cheaper by half.
If the offer is with 40% less, you will also realise that is a good deal since it is almost half the original price.
But what is this percentage in detail?
A mark of 4 / 5 is reflected as a fraction.
A learned maths person will understand that it is (4 /5) x 100 = 80%
If you tell a student that he achieved 50 marks. Is that enough?
The information will not be enough as the total score is not known (unless using the default told before hand).
Thus a mark of 50 upon 50, and a mark of 50 upon 100 means different story altogether.
I believe you will agree totally!
A percentage will always reflect better information since 80% means 80 / 100.
The base of 100 is taken as the default.
The actual base number is immaterial in this matter.
An information of 50 marks compared to 50% showed the power of using percentage.
For absolute marks, you need to tell the total marks to form a complete message.
Using percentage, the fraction part of the calculation can be ignored.
Percentage and fraction are related. But percentage used a common base number (100) to commonise the value.
For comparison sake, percentage, thus stand a more proper way to tell the result.
Is it easy to tell the closeness between 49 / 56 and 46 / 56,
or is it easier to tell between 78% and 81% ?
The answer is obvious, I hope.
Maths is interesting, and mastering simple concept makes maths learning even more interesting.
Monday, 8 June 2009
To the ignorant, a "-" or minus is a strange symbol.
Is this more so when it is expressed as "-6", "-5 km", "-$3", etc.
What is then the true meaning of this "minus" sign?
Answer: It indicates a reversal of action of intention.
Maths is a useful tool to help explain this concept.
When a car moves forward by 5 km, it moves +5 km. ( By default, no sign means positive)
When it reverses by 5 km, it moves -5 km. The direction of movement reverses!
When a person gain $3, he has +$3 in his pocket.
When he loses $3, he has -$3.
From the 2 examples above, you can see that the "minus" sign is a reversal to the default.
If you "reverse" and then "reverse" again, you find yourself in the positive direction.
(-1) x (-1) = + 1
If you reverse 5 times, 5 x (-1) = -5 . You are facing the reverse to the default starting direction.
Now take note of this coming information.
If you reverse the car by 5m and another 5m, you reversed in total (-5) + (-5) = -10m.
If you reverse the car by 5m, followed by changing the direction and moving by another 5m, you moved (-5) + (+5) = 0m.
(Reverse direction followed by forward direction).
Does all these "reversing" cause a daze in you?
Do not despair.
Message is "When there is a reversal, put a minus in front of the number". Simply that!
I owe you $5 ==> -$5.
I gained $5 but lost $3 ==> + $5 + (- $3) = $2 (Action followed by another using "+").
Hope this post on minus sign reduces your anxiety about this little maths symbol.
Sunday, 31 May 2009
By remembering them , you will be in line to solve logarithmic problems and, maybe, fast too.
However, what if you forget them?
Does it mean that you are not able to solve the question regardless of speed?
Do not despair.
As long as you are able to manage the 2 basic laws in logarithm, you are safe.
The product law and the quotient law are must for any students.
Why do I say that?
Let us take the Power Law and do a review.
n log x = log X n
Why is it so? What if you forget this law? Any problem?
These are the very queries any new learners exposed to logarithm will ask.
First allow me to go through the product law of logarithm.
log (XY) = log X + log Y
Here you see that the product of "X" and "Y" in logarithmic operation, becomes a "sum"of the individual logarithms.
4 log Y = log Y + log Y + log Y + log Y (adding up 4 of the log Y)
Using product law, you know that these 4 terms can be combined to log (Y x Y x Y x Y).
log (Y x Y x Y x Y) = log Y4Now, you see, through the product law, you are able to equate the 4 log Y into log Y4 ,
meaning, 4 log Y = log Y4.
You see that you did not utilise the Power Law here,and yet is able to form this formula!
Amazing isn't it.
What is the message here?
The message is that, when you have the basic understanding in logarithmic principles, you will be able to twist and turn any given problems to come out a solution.
You had used the basic product law to discover this unique Power Law.
It is similar to other laws and also can be expanded to cover other maths topics too.
Do enjoy maths.
Do discover more exciting twists it presents wwith a bit of thinking.
Happy learning :-)
Saturday, 9 May 2009
Learning thing with disregard for other is alright for the sake of triggering the mind. But does it benefits more if linked to others?
Does indices related to quadratic equation?
Does multiplication relates to addition?
Does complex number relates to algebra?
All the above questions are common in the mind of a math learners.
If you do not have these questions along the learning phase, something is very wrong.
Learning math in isolation is similar to living in an isolated island all by yourself.
You do not know what is happening in the world.
You do not know if there is famine somewhere, or swine flu going round, or a plane crash near you.
Math has to be done with linkage to many other mathematical topics. It cannot be done in isolation.
Math is a tool that solves real-life problems. With mastery of various mathematical concept and relation among them, you will be better prepare to solve more problems.
Many a time, you will come across students who just study topical math without knowing that they can apply what have been taught to them previously.
They start fresh when a new topic is introduced.
Algebra is different from complex number.
The addition in complex number is done differently from that done in algebra. That's what they assumed, since the heading is different!
Interesting learning ways, right?
That is human nature, to be frank. Only when you are told, sometimes, otherwise you will not know it. Adults learn through experience that this assumption is pulling you down.
The ability to link many things together is a very beneficial skill to permanently internalise.
This does not point to math alone. Others apply.
Math can be tough if learned using an improper learning method.
One good technique is the linking technique where you will see yourself happily doing math, being able to apply and solve questions using previous and current taught concepts.
It motivates you.
This is the STARTING point if you are unaware. This is the point where it decides whether you can sustain math learning.
Learn wide and later deep into math. But start with the correct footing. Link as much to previous as possible. It will be a sure way to happy math doing.
:-) I like math!
Saturday, 2 May 2009
Counting up is an easy task for anyone, even when the base is not ten.
You may refer to this link for counting (up) with base 8.
However, counting down may be slightly harder than normal, especially when dealing with another number base other than 10.
Let's try with base 8 for a start.
Here a count down of 3 from 10 is needed.
How do you go about it?
Thinking of the way base 10 subtraction was dealt with, this is similar.
After all maths is the same. It is the technique that is important.
When you encounter a " bring back" from the upper (rightmost) digit to the lower leftmost digit, you will, for base 10, add a 10 to the "ones" digit.
Here, with base 8, you will do likewise, except that now it is adding the number 8 to the leftmost digit.
Thus, 0 + 8 = 8.
And 8 - 3 = 5.
05 (Answer in base 8).
There is nothing difficult if you look at the technique or concept in counting up or down.
Maths is just playing with methods to resolve numerical issues.
The above is a good example. Hope you agree?
Have fun counting in other number base.
Wednesday, 22 April 2009
It will not be acceptable during tests and examinations. This everyone knows!
But human do make mistakes. This is a fact. Therefore, we have to know our weakness and avoid moving towards it (them).
A very common mistake made while doing maths is to use the variable "Z". From the writing, you can see that it is very similar to the number "2".
During examination, we are tensed up and our mind may not see what the eyes received.
2Z may end up as 22 finally. This "22" will then be used for computation and of course results in a BIG shock!
Thus knowing this danger, avoid using variables that are close to number in writing, unless stated by the maths question itself.
Do not choose a similar looking variable and end up with disappointment.
"b" and "6", "l" and "1" or "S" and "5". These are dangerous combination.
Thus look carefully when dealing with these numbers and variables.
Do not cause unnecessary mistakes. Save your effort to deal with better challenging thinking.
Saturday, 18 April 2009
Number system has an important factor attached to it.
It is the base of the number.
The base defines the quantity of item (in this case, the counted number) before the sequence repeats.
Take for example, base 8.
The count is: 0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12 ....., 17, 20, 21, 22, 23,.......
The number "8" does not appear in the base 8 number system.
Once the 7 is reached, the next number increments to 10.
That is, the range is from 0 to 7 only.
How about the addition?
Since 5 + 3 exceed the maximum count of 8, the final added answer is 10, the start of next sequence.
It can be seen from the 2 examples above that the second digit in the addition is added with "1" after the first (right-most) digit shoots over "8".
This counting technique is nothing different from our normal base 10 (decimal) method of addition.
Thus, knowing the meaning of the base in the number system helps in proper counting, and includes addition and subtraction with the respective base.
It is nothing complex and abstract. Just counting with the correct quantity of number.
Monday, 13 April 2009
Counting from 1 to 100 is normal for anyone. Just increment by 1. It's that simple.
But counting time may be another story for young math students.
Why is this so?
There is the seconds, the mintes, and the hours to handle. They differ by the umber 60.
The "carry over" through addition is the number SIX!
1 min 30 sec added by 40 sec gives .......
1 min 70 sec?
Here the 70 sec includes 60 sec + 10 sec. You need to understand that for time, 60 sec means one whole minute.
Thus, 70 sec = 60 sec + 10 sec ===> 70 sec = 1 min + 10 sec.
Therefore, 1 min 30 sec add 40 sec gives ==> 2 min 10 sec.
Compare this to adding 30 by 40. Answer = 70. No more analysis.
Time base on 1 hour = 60 min, 1 min = 60 sec concept.
To test true understanding of addition (and subtraction), time is a good gauge and tool to assess learners.
Test it on young kids today to annoy them... *#^&@!
Saturday, 11 April 2009
There are some maths questions that will be out to catch the careless learner.
A common one is that which require you to calculate the distance or length given the gap of items.
Look at the diagram below for an example.
Here, can you find the length from the left-most pillar to the 8th pillar, given the distance from start to 3rd pillar is 30m?
Solution: (Wrong slip-of-the-mind working)
Since the distance is 30m for 3rd pillar, answer to 8th pillar has to be 80m.
Seems to be right and logical. ==> Careful here!
Look at the step distance in between pillar. It is 30m / 2 = 15m.
As the gap between pillar from start to 8th pillar is only 7 gaps,
the actual correct distance is 7 x 15m = 105m.
Interestingly tricky question, right?
Be careful and alert for this "step" or "gap" maths problem.
Slamp down this carelessness, and mistake will eventually disappear (for this type).
Friday, 10 April 2009
("letters" here may mean symbols like the theta, beta, etc)
Some of these symbols are constants and some represent variables.
Knowing what are constants and variables is crucial for mastery of maths learning.
What are constants?
Constants, as the word literally means, are items that have number that never change.
What are variables?
Variables are symbols that changes in value.
y = 3 x + 2
y and x are variables, and 3 and 2 are obviously constants.
log x = 5y
x and y are variables , and 5 is constant.
y = mx + c ( for straight line equation)
y and x are variables, and m and c are constants.
This may be confusing to some maths students when they start plotting graphs.
Here, the straight line is continuously moving with the value of x and y.
So why is the "m" identified as constant?
You need to know that "m" represents the "GRADIENT" of the line.
The line has the same slope at any value of x and y.
Thus "m" is a constant.
This is a typical concept that commonly goes wrong.
Therefore when you really understand what changes are considered "variables" and those that remain stable are known as "constants", you are in line for good maths study!
Thursday, 2 April 2009
In the study of algebra, symbolic representation of number or unknown is key concept to solving mathematical equations.
The letter "x" or "y" are examples.
Other symbols can also be used as long as the usages are understood.
In the expression, x + 0.5 = 3.
This meant that the unknown "x" added to 0.5 will give a total of 3.
"x" here is nothing other than an unknown item to be solved.
It should be a number that relates to that maths equation. Nothing more, nothing less.
x2 + 2x - 1 = 0
This "x" again is an unknown number to be found out.
Thus this algebraic expression and its "x" are just mathematical item representing a relationship.
Many students learning maths, when faced with this "x" always look puzzled.
With this post, the queries of this "x" (or "y", etc) should be cleared.
With this knowledge of the symbolic representation of unknowns, other areas of maths can be explained easily.
Topics like the trigonometry and logarithm will be expanded from this symbolic concept.
Cos A and log B will thus be finalised to a number, with this "A" and "B" yet to be solved.
cos A + 2 = 2.4
will then be nothing more than to relate this unknown "A" to the expression.
It is also an easier way to explain and express this relationship between the unknown (A) and the other number (2 and 2.4).
log x + log 2 = 3
means that "x" is related to the 2 and 3 according to the given equation.
From the above few examples, the question now of what really is this "letter" doing forms meaning, right?
Maths starts off easy when this concept is clear.
Alot of the maths study involves this simple "trick" of presenting unknowns.
You notice how clever past mathematicians were now?
The use of simple symbol to pass off as number to carry on with maths solving.
Without this algebraic presentation, maths will not be as interesting as now.
Alot of guessing will have to be done and .... guess what? Maths will be HELL then.
Enjoy this symbolic concept in maths.
Enjoy your maths.
Friday, 27 March 2009
I have seen lower primary school kids learning mathematics.
They are exposed to many logic "games" which tested their mathematically analytical skill.
One of them is the math Word problem topic.
Here they are always given a scenario and asked to give an answer.
They are not taught algebra, however.
The expectation is for them to think out logically.
This is good in a way.
But along the journey of learning mathematics, they will sooner or later be told of an exciting area called the "Algebra".
Here, algebra comes in helpful for those who did not do well in the logical word problem questions.
It is because, in algebra, the unknown can be replaced by a symbol, normally a letter.
This solves the poor kid the time to "guess" the answers, with iteration of checking and re-trying at times.
With the use of algebra, the kid can attach the unknown to a letter and proceed with the calculation.
When this algebra concept is not mastered at a later stage while studying math, the learner will face tremendous obstacles along the way. The meaning of the "letter" will be an alien to him, not knowing the power of its usage, and thus the magic of algebra application.
Thus, in conclusion, any math student has to die-die, managed simple algebra in order to have a good time learning math.
Hope this advise and information helps.
Monday, 16 March 2009
Anxiety is caused by not being able to fulfill your desire but strongly wishing for it.
Maths anxiety is likewise. Wanting to master maths but is unable to grasp the concepts.
Don't worry. It is not the end of the world!
Maths is just a tool for you to solve problems easily.
Maths lets you have a systematic approach in solving questions.
But if you do not understand maths, does it mean that you cannot solve problems?
You still can, but maybe only through more guessing (that is, non-systematical).
While learning or doing your maths homework, forgo the idea to quickly master the topic.
Forcing your learning through at a faster pace than you are able to handle does not serve any purpose.
Learning needs time to digest and analyse information. It is just like eating a meal.
Eating too fast will get you indigestion!
Same to learning maths, as well as other subjects.
Just understand that not knowing maths is OK.
Knowing maths is a privilege, a bonus.
With this mindset, you will find maths interesting.
It is a tool only for helping you find answers in an "education" and impressive way.
Life still goes on without you knowing maths.
(In fact, you actually use it, except that you did not know that it is called maths!).
Does this message make you feel better?
Hope it does.
Let maths be your slave.
And not you being the slave to maths.
Thursday, 12 March 2009
Have you seen many mistakes like the below?
3x = 27
3x = 34 ===> x = 4
9x = 33
33x = 33 ===> 3x = 3 ===> x = 1
Why the error?
A simple explanation is that the maths learner is not familiar with the basic multiplication of repeated numbers.
3 x 3 = 9
4 x 4 = 16
5 x 5 = 25
6 x 6 = 36
7 x 7 = 49
2 x 2 x 2 = 8
3 x 3 x 3 = 27
2 x 2 x 2 x 2 = 16
Once you have mastered this basic repeated multiplications, you can rest assure that indices question will not be there to haunt you.
How about solving "x" in this 9x + 1 + 2(3x) - 3 = 0 ?
I bet that if you understood the above criteria of learning indices, the equation can be easily solved for x (using quadratic formula as a hint).
All complex things start off with simple things.
Do you agree this applies to maths?
Tuesday, 10 March 2009
Why do I say that?
How do you gauge whether you have mastered a subject or topic?
Your teacher will award you marks for the assessment done to check your understanding, right?
What is this process? It is maths!
Maths learning is being monitored through itself!
Applying maths to learn maths.
Giving marks is counting, logical thinking and judging. These are related to maths.
Thus you see that maths is interesting in that it checks itself like no subject does.
*** : - ) ****
Friday, 6 March 2009
What is this quadrant about?
A complete cycle (360 degree) is divided into 4 quarters.
They are zones defined for specific trigonometric functions.
The first quarant (0 to 90 degree) gives positive sign for ALL trigonometric functions.
The second quarant (90 to 180 degree) allows only "sine" to have positive number.
For the thrid quarant (180 to 270 degree), "tangent" has positive number only.
Lastly, the fourth quarant (270 to 360 degree), "cosine" gives positive number only.
So, you can see that given a sign of a trigonometrical operation, the specific quadrant can be found or identified.
sin X = - 0.5 ===> Identifies quadrant as 3rd and 4th.
tan X = 0.2 ===> Identifies the 1st and 3rd quadrant.
This is simple, right?
However, do note the below example.
It causes a mistake that is common!
Example of potential error:
sin 2X = -0.5 ====> which quadrants ?
The answer is not that direct!
Now the math question is not on "X", but on "2X".
To identify the quadrant, you need to start off from the "2X", working as per normal.
But, after identifying the 2 quadrants, you have to compute the "2X" reference angle.
Using the reference angle, you have to obtain the 2 angles.
After which, you need to divide the angles obtained by 2.
The divided angles is then the final angles lying within the quadrants.
Confused? Never mind. See the numerical solution below.....
2X = sin-1 (0.5) = 300
This is the reference angle used to compute the actual answers.
Final answers are (Quad 3)= 180 + 30 = 2100
and (Quad 4) = 360- 30 = 3300.
Common mistake is to obtain reference "2X" angle and straight away divide it by 2.
Using this newly found "X", you proceed to identify the angles of the quadrant identified using the "2X". THIS IS INCORRECT!
Do not confuse double angle with single angle.
When the problem is "2X", solve all the way using the "2X" first until reaching the end.
After which, you then divide the angles by 2 to get to the final answers.
Maths is simple if you follow the rules accordingly.
If you mess up double angle with single angle while solving, you just literally mess up the workings.
Maths forces you to follow rules set out. It punishes only if you do not obey orders.
Maths is interesting isn't it? Never expect that maths can police your behaviour while practicing it, right?
Wednesday, 25 February 2009
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!
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.
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.
Sunday, 22 February 2009
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.
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.
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.
Tuesday, 17 February 2009
" 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
- Sign interpretation
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.
Sunday, 15 February 2009
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
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.
Sunday, 8 February 2009
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".
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 ...
Saturday, 7 February 2009
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:
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.
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
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!
Saturday, 31 January 2009
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?
Thursday, 29 January 2009
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
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.
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.
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.
The cheeky, little "i" can make number change direction as well as characteristics; a real number into an imaginary one!
Tuesday, 27 January 2009
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.