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The Easiest Math Formula

Published by Kenneth Kho in Math on June 21st, 2012 | No Comments »

We know Infinities are very tricky things and also have perplexed mathematicians and philosophers for thousands of years. Sometimes the sum of a never ending list of numbers will become infinitely large and sometimes it will get closer and closer to a definite number. And the other hand it will defy having any type of definite sum at all. A little while ago my friend was giving a talk about infinity that included a look at the simple geometric series below :

Square = ½ + ¼ + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + … and so on ….

Forever, every term in the sum exactly half the size of it’s predecessor. The sum of this kind of series is actually equal to 1, but someone in the audience who was not a mathematician wanted to know if there was any way to show him that this sum is true.

$the two math formula$ Fortunately there is a simple demonstration that just uses a picture. Just draw a square of size 1 x 1, so its area is 1. Now let’s divide the square in half by dividing it from top to bottom into two rectangles. Each of them must have an equal area to ½ . Now divide one of these rectangles in two to make two smaller rectangles, each with area equal to ¼ .Now divide one of these smaller rectangles in half again to make two more rectangles, again each of area equal to 1/8. Keep on going like this, making a rectangle of half the area of the previous one, and look at the picture. The original square has just had its whole area sub-divided into a never ending sequence of regions that fill it completely. The total area of the square is equal to the sum of the areas of the pieces that my friend have left intact at each stage of the cutting process, and the areas of these pieces is just equal to our formula “Square” – above. So the sum of the formula “Square” must be equal to ONE (1) –> the total area of the square.

In-fact when we encounter a formula like “Square” for the first time, we work out its sum in another way then we notice that each successive term is one half of the previous one and then multiply the whole formula by ½ so we have :

½ x Square = ¼ + 1/8 + 1/16 + 1/32 + 1/64 + 1/128 + 1/256 + … and so on ….

But we notice that the formula on the right is just the original formula, Square, minus the first term which is ½ . Finally we have that ½ x Square = Square – ½ , and Square = 1 again.

Is it easy ? think again.

Tags: math formula

The Counting System Named Eights

Published by Kenneth Kho in Math on June 11th, 2012 | No Comments »

Every of us know how to count in tens. Ten ones sure make ten and ten tens will make a hundred, ten hundreds make a thousand and etc. This counting system is what we called a decimal system. There is no limit to its scope if you have enough labels to call the results. For this we have words like million, billion and also trillion for some of the large numbers, but not for everyone that you might need to write down. Instead of that we have a handy notation that writes 10ⁿto denote the number which is 1 then followed by n noughts that why a thousand is 10³.

The origin of all these tens in this counting system is not hard to find. Actually it is already at our fingertips. Most ancient human cultures used their fingers in some way for counting and as a result you find counting systems based on groups of five(one hand fingers), ten (two hand fingers), groups of twenty (all your fingers plus your toes), or mixtures of all or some of these systems. Our own counting system betrays a complicated history in which different counting systems merged to form the new ones by the presence of old words that reflect the previous base. So we have a word like dozen for twelve, or score for twenty, with its interesting triple meaning of twenty, to make a mark or to keep count. All this three meanings reflect the time when tallies were kept on pieces of wood by marking or scoring them in groups of twenty.

$counting system image$ Despite the ubiquity of the base ten counting system in early culture, there is one unusual case where a Central American Indian society used a base eight (8) counting system. Can you think why this might be used ? Normally I used to ask mathematicians if they could think of some good reason and they usually responded by saying that eight was a good number, to use because eight has lots of factors like it divides exactly by two and four, so you can divide portions into quarters without creating a new type of quantity that all of us call a fraction. The only time I got the right answer though was when I asked a large group of eight year old children and one girl immediately produced the answer : they were counting the gaps between their fingers. If you hold things between your fingers, strings or pieces of material, this is a natural way to count. This base eighters were finger counters too.

Tags: counting system, eights, math, ten

Math Calculation For Doubling Your Money

Published by Kenneth Kho in Math on May 20th, 2012 | No Comments »

Everybody know about this : The value of your investments can go down as well as go up.

Recently you will have discovered that the value of your investments can plummet as well as go down. So if you want to play safe and place your cash in a very straightforward savings account with a fixed or slowly, changing rate of interest. How long do you think will it to double your money? Although nothing in this world is so certain as death and taxes (sometimes the version of the latter that goes with the former), let’s forget about them both and work out a handy rule of thumb for the time needed to double all your money.

Start out by putting an amount A in a savings account with an annual fractional rate of interest r (so 5% interest corresponds to r = 0.05), then it will have grown to A x (1+r) after 1 year, to A x (1+r)² after 2 years, to A x (1+r)³after 3 years and so on. After say n years your savings will have become an amount equal to A x (1+r)ⁿ. This will be equal to twice your original investment, that is 2A, when (1+r)ⁿ = 2. If we take natural logarithms of this formula and note that 1n(2) = 0.69 approximately, and 1n(1+r) is approximately equal to r when r is much less than 1 (which it always is – typically r is about 0.05 to 0.06 at present in the United Kingdom), then the number of years needed for your investment to double is given by the simple formula n = 0.69/r. Let’s round 0.69 off to 0.7 and think of r as R per cent, so R = 100r, then we have the handy rule that n = 70/R

This shows for example that when the rate R is 7% we need about 10 years to double our money, but if interest rates fall to 3.5% we will need 20 years.

Tags: double your money, math calculation

Verbal Conjuring

Published by Kenneth Kho in Math on April 15th, 2012 | No Comments »

Skilful conjuring, viewed close up, can be very perplexing, and it becomes astonishing if the conjuror shows you how it was done. How easily he misled you, how blind you were to what was going on right under your nose, how simple it was.

You soon realize how incompetent a judge you would be in the face of claims of spoon bending or levitation. Scientists are the easiest to fool, they are not used to Nature conspiring to trick them. They believe almost everything they see is true. Magicians believe nothing.

In this spirit, I want to relate to you a little mathematical story, adapted from a version by Frank Morgan, that is an example of verbal conjuring. You keep track of everything in the story but something seems to disappear in the telling : “a sum of money no less” and you have to figure out where it went, or indeed, whether it was ever there.

There are 3 travelers arrive at a cheap hotel late at night, each with just $10 in his wallet. They decide to share one large room and the hotel charges them $30 for a one night stay, so they each put in $10. After they have gone up to their room with the bell boy carrying their bags, the hotel receptionist receives an email from the head office of the hotel chain saying that they are running a special offer and the room rate is reduced to $25 for guests staying tonight. The receptionist, being scrupulously honest about such things, immediately sends her bell boy back to the room of the 3 new guests with a $5 note for their rebate. The bell boy is less scrupulous. He had not received a tip for carrying their bags and he can’t see how to split $5 into 3, so he decides to keep $2 for himself as a “tip” and give the 3 guests a rebate of $1 each. Each of the 3 guests has therefore spent $9 on the room and the bell boy has $2 in his pocket. That makes a total of $29. But they paid $30 – what happened to the other $1 ?

Tags: Verbal Conjuring

Junior High School Geometry Online Course

Published by Kenneth Kho in Math on March 27th, 2012 | 2 Comments »

Nowadays, we would find our life easier because so many online courses like geometry online course that could teach you geometry without required us to physically attend geometry class at school. Actually, What we need is just a desktop with the fast internet connection then we could always jump to learn the online geometry course. It is not like few years back that we have to physically attend geometry class (imagine the time that we have to spend for attending the class and comeback home).

Now with this geometry online class, we have a chance to learn geometry lessons without leaving our home.

Below is the statement that I try to solve one of the geometry problem, even this is not one of the real online geometry courses, but my expectation is to help others to understand the blue print of the simple online geometry course.

Before we jump to the complex of the geometry problem, I would like to play one piano song for you and let your mind relax and it would be easier for you to absorb the geometry lessons as below :

Let’s Start :

There is a triangle with <A, <B, <C. Assume the “<” is angle. ¼<C more 12^o than (2x<A)-2^o. ½<A less 5^o than (1:<B)x2^o. 2<B more 6^o than <C-10^o. Then I ask you how much <A ? (The answer may use the fraction)

The answer:

First Equation:

¼<C=2<A-2^o

Instant result:

<C=8<A-8^o

Second Equation:

½<A = (1:<B)x2^o-5^o

Instant result:

<A=<B-2½^o

Third Equation:

2<B=<C-10^o+6^o

Instant result:

<B=½<C-2^o

Make substitution <C to make it <A!

Because<C=8<A-8^o, so the result is:

½<C-2^o=4<A-2^o(<B=½<C-2^o=4<A-2^o)

Now, we have three equation, just combine it! We already knew the triangle formula is the three angle, if we combine, the result will always 180^o, same like our <A,<B,<C, the formula is :

<A+<B+<C=180^o

Now, we come to counting again…

<A=<A , <B=4<A-2^o , <C=8<A-8^o

Combine them with the statement the result must 180^o

<A+4<A-2^o+8<A-8^o = 180^o

13<A-10^o = 180^o

<A = 14 8/13^o

Hopefully this simple geometry online course could provide any one to learn with a better understanding for the concepts of online geometry class. Through reading, geometry online lessons and the independent exploration would make the learner leave the online class with more complete understanding of geometry and be able to think in a geometrical world.

Good Luck.

Tags: geometry class online, geometry online course, geometry online lessons, online geometry class, online geometry course

Makes Your Live Easier By Learning Calculus

Published by Kenneth Kho in Math on February 25th, 2012 | No Comments »

The difference between an amateur and a professional is that as an amateur one is at liberty to study only those things one likes, but as a professional, you must also study what one does not like. Consequently there are parts of a mathematical education that will seem laborious to a student just as all those hours of winter running in the cold and rain will be unattractive, but essential to the aspiring Olympic athlete. If students asked why they needed to learn some of the more intricate and unexciting parts of calculus, I used to tell them this story, one that the Russian physicist George Gamow tells us in his quirky autobiography, My world line. It is about the remarkable experience of one of Gamow’s friends, a young physicist from Vladivostok called Igor Tamm, who went on to share the Nobel prize for physics in 1958 for his part in discovering and understanding what is now known as the “Cerenkov Effect”.

In the Russian revolutionary period, Tamm was a young professor teaching physics at the University of Odessa in the Ukraine. Food was in short supply in the city and so he made a trip to a nearby village, which was under the apparent control of the communists, in an attempt to trade some silver spoons for something more edible like chickens. Suddenly the village was captured by an anti communist bandit leader and his militia, armed with rifles and explosives. The bandits were suspicious of Tamm, who was dressed in city clothes, and took him to their leader, who demanded to know who he was and what he did. Tamm tried to explain that he was merely a university professor looking for food.

“what kind of professor ?” the bandit leader asked.

“I teach mathematics” Tamm replied.

“Mathematics ?” said the bandit. “All right! Then give me an estimate of the error one makes by cutting off Maclaurin’s series at the 9^th term. Do this and you will go free. If you fail, and you will be shot!”

Tamm was not a little astonished. At gunpoint, somewhat nervously, he managed to work out the answer to the problem : a tricky piece of mathematics that students are taught in their first course of calculus in a university degree course of mathematics. He showed it to the bandit leader, who perused it and declared “Correct ! Go home !”.

Tamm never discovered who that strange bandit leader was. He probably ended up in charge of university quality assurance somewhere.

Tags: Calculus, course, george gamow, igor tamm, mathematics, nobel prize for physics, physicist, physicist george, physics, student, village

Have Fun with Small Mind-reading Tricks

Published by Kenneth Kho in Science on February 2nd, 2012 | No Comments »

Sometimes learning math might give you some fun, especially for someone who love numbers. Math also can teach you small tricks like mind-reading.

Try to think of a number between 1 and 9. Then multiply it by number 9 and add the digits of this new number together. Then subtract 4 from your answer and you will be left with a single digit number. Next, try convert this number to a letter. If you number is 1 it becomes A, 2 becomes B, 3 becomes C, 4 becomes D, 5 becomes E, 6 becomes F and so on. Now think of a type of animal that begins with your chosen letter and imagine that animal as strongly as you can. Hold it vividly in the forefront of your mind. It’s an Elephant.

This is a very simple trick, and you ought to be able to work out how I was able to guess the animal of your choice with such a high likelihood of success. There is a little mathematics involved, in that some simple properties of numbers are exploited, but there is also a psychological and even zoological ingredient as well.

There is another trick of this general sort that involves only the properties of numbers. It uses the number 1089, which you may well already have listed among your favorites. It was the year in which there was an earthquake in England, it is also a perfect square (33 x 33); but its most striking property is the following.

Pick any 3 digit number in which the digits are all different (like 153), make a second number by reversing the order of the 3 digits (become 351). Now take the smaller of the two numbers away from the larger (351 – 153 = 198, if your number has only 2 digits, like 23 then put a 0 (zero) in front, so 023). Now add this to the number you get by writing it backwards (so 198 + 891 = 1089). Whatever number you chose at the outset, you will end up with 1089 after this sequence of operations.

Tags: Auto, digit, Draft, earthquake in england, Fun, Learning, learning math, letter, math, Mind-reading, perfect square, Small, striking property, Tricks

Learning Math is Fun and Easy

Published by Kenneth Kho in Math on January 10th, 2012 | No Comments »

$Math is fun and easy image$

$Math is fun and easy image$ Just come back from holiday after travelling to other cities for refreshing myself and my family. Now it’s come again for me to have a normal life as usually I did, like wake up at 6 AM, some courses that I have to take until 8 PM and having fun with friends.

I have spent 1 hour to post this small math problem to my blog.

Mostly students always feel bored when they are trying to learn math, especially when facing with numbers, degree and operations on math. Usually their mind will become down at first sight before they even try to solve that problem.

So, again I try to solve another math problem like below, hopefully can help for those who might need this :

1²+ 2²+ 3²+ … + 500²= …

How to do this with the simplest way?

Easy, just use this formula…

500 { 500 + 1 } { ( 2 x 500 ) + 1 } = 500 x 501 x 1001 = 250.750.500 = 41.791.750 6 6 6

Explanations :

Assume the last number equal to n.
Take n multiply { n + 1 } multiply { 2n + 1 } then divide it by 6

Should you have any comments, just do not hesitate to write it down below…….

Tags: Fun, fun with friends, having fun with friends, holiday, learn math, Learning, learning math, life, math problem

The Fast Way to Solve Math degree 3

Published by Kenneth Kho in Math on November 30th, 2011 | No Comments »

$math-image$

Hey Friends, Sometimes when we learn math, we feel bored and we think again and again but the number that comes out always 1, 2, 3, etc.

Even when we go to store, we always dealt with those numbers because we need to make the transaction at the cashier.

Now we come to the real math world again ! Try to solve math with my simplest way.

I have a question and fast way to answer this question !

1³ + 2³+ 3³ + … + 50³ = …

For solving this question , please see the following answer :

Easy…, just take last number , degree with 2 , divided by 4 , then multiply with ( Last number + 1 )².

50² : 4 ( 50 + 1 )² = 625 x 51² = 1.625.625.

That’s all I can do now , hope you come back again.

Feel free to post your comments.

Tags: Auto, degree, Draft, Hey, hey friends, math degree, math world, Solve, transaction, x 512

First Junior High School Math Competition : Mid November 2011

Published by Kenneth Kho in Math on November 13th, 2011 | 3 Comments »

Today I struggle with Math Competition. For sure I am flattered for getting this opportunity.

Few number of the questions in this Junior High School Math Competition, I try to figure it out as below :

1 + 2 + … + 99 + 100 + 99 + … + 2 + 1 = …
How to do this with fast way???
Just take the middle number and multiply the same number too.
100 x 100 = 10.000.
(1 – ½ ) ( 1 – ⅓ ) … ( 1 – 1 / 2012 )
How to do this with fast way???
Just take the last number : 1 / 2012.
745²– 255²= …
How to do this with fast way???
( 745 + 255 ) ( 745 – 255 ) = 490.000.

Another questions not in this Junior High School Math Competition that I found the simplest way to solve :

1 + 2 + … + 118 + 119 = …
How to do this with fast way???
{ ( 1 + 119 ) ( 119 : 2 ) } = 120 x 59,5 = 7140.
1 + 2 + … + 179 + 180 = …
How to do this with fast way???
( 180 + 1 ) ( 180 : 2 )
181 x 90 = 16.290.
11 + 22 + 33 + … =…
10 + 20 + 30 + …
How to do this with fast way???
Just take the first number : 11 / 10.