"Expanding Pattern" and "Condensing Pattern"

Consider like this:

There are only three MAJOR NUMBERS in this world. All the other numbers are their added or multiplied replica.

Say it is like this: 0 and infinity are the two extremities of the positive number line: 1 is the midpoint of this number line. There is infinite distance between 0 and 1(just consider) and there is infinite distance between 1 and infinity. Why i am saying this is you can find infinite number of numbers between 0 and 1 like, 0.1, 0.9, 0.9999, 0.8576, 0.00078544, 0.0506640478 like. Also you can find infinite numbers between numbers 1 and infinity like 2, 3, 6589,787965743559207, and so on.

So you have your calculator with you?

Take a number between 1 and 0, say 0.999 and square it continuously, finally you will only get 0. Of course, there is some negligible number. But when you have squared infinite times it would be a perfect ZERO. Similarly when you take a number between 1 and infinity and square it infinite times you will get INFINITY.

From this it is clear that on squaring a number it takes you to the either extremities of the 01infinity number line. It is like expanding, is it not. So I call powering as "THE EXPANDING PATTERN" when the power lies between 1 and infinity. When you square 1 infinite times you get only 1, because it is the neutral midpoint.

So here, for a simple narration lets take squaring and square rooting.

I call Squaring as the Expanding Pattern.

Now when you take any number between 0 to 1 or 1 to infinity and take root of it with the calculator infinite times you will always get 1. So if the power is fractional on simplifying or if the power lies between 0 to 1, on infinitely taking the root, you get "1" always.

So, regardless whether a number lies close to infinity or close to zero on taking the root continuously or infinite times, you get 1 only. So you are always pulled to the number 1. So I call taking root as THE CONDENSING PATTERN.

Squaring numbers in 50's :

Lets take, 52

Square 52 = 2704

Steps :

1. Add the once digit to 25.
2. Attach the square of once digit. If it is a single digit to fill the vacancy of the tenth digit put zero. Here 04 will be attached to 25+2.
3. So the answer is 2704.

Try all  the numbers in the 50's for practice.

Squaring numbers in 41 - 49

Take 46:

The steps would be:

1. Take 15.(this is not the square of 4, but it is one less than 5 so take 15(this is not the actual reason, however please remember like that)). Now add the that 6 to 15, you get 21.
2. Now square the difference between 50 and 46 and attach to 21. Difference is 4, square of 4 is 16.
3. So on attaching, the answer is 2116.

Now try all the numbers from 41 - 49 and practice this method.

Multiplying Two 3-digit numbers which have zero in the middle

Actually this is a two digit multiplication only.

Consider, 304 * 608

Forget about zero now. Now as you would multiply two numbers which are 2-digits. multiply them.

I will represent them in a picture now, so that it is easy.

I think I have made it clear. Since the three parts were only two-digits I have simply attached them. I say, this is simplest thing that one can learn as a shortcut. Squaring a three digit number that has a zero in the middle, multiplying two 3-digit numbers which have zero in the middle. You literally have to remember the parts of the answer and just say them. Ya, I know that you can feel that it is simple.

So now try this operation: 705 * 809 = ?.

First, you try the method. Then verify the answer with my steps. I know that you can do it. Try try.





So the parts would be: 56 103 45. If you have the answer already 5610345 is not the anser. I say agin, these are only the parts. You have got the answer correctly as 570345, give yourself a pat on your back and appreciate yourself. You have mastered this trick. If you have not got the, No probs, you will be master in few more seconds.

You can see that you have three digit number in the middle. So add the hundredth digit "1" to the previous number 56. Then as usual, just attach everything. That's it. You are "Master too.

AND ONE MORE "GOOD NEWS":

You just have to add one to the previous number whenever a 3-digit number arises in the middle. Why I am saying this is, even when you multiply 909*909[the largest possible number to use this method(three digit numbers alone)] you get 81 162 81, is it not? Yaa... Its nice to hear. Really a good news.

Similarly were you to multiply a numbers like this: 7005 * 8009. Even more good news, you have no possibility of adding. You have to simply attach or tack up or whatever. You have to add only the thousandth digit which would never occur. Hmm... great great! Really I am not appreciating myself. I simply love maths just it. That's what making me to write like this.

So we will end this topic with a question: What is 9009^2 ?
I will tell the answer. However you too know. You would have already calculated in your mind.


The answer is 81162081.


One more thing, that zero has come before 81, just to fill the vacancy of the hundredth digit.

Any doubts still ask me.

With warm regards,

Harish Chakrawarthy :-)

"ZERO" The Expander

I have found this unique quality of zero. It expands a number so that we can easily know about it. At this moment, it is very obvious Indians can really be appreciated for the discovery of "ZERO". I can pronounce it as a discovery only, since all numbers existed in this world even before humans came. I say this because nature had everything in it. I men the knowledge. And the only beings are the Humans who can absorb this knowledge from Nature. I really like "Nature".

Consider 34^2.

If you square the number using the formula I gave, you get 1156. from this you cant tell that a^2 was 9, 2ab was 24 and b^2 was 16.

Now we will introduce a "Zero" in between that 3 and 4. You will get 304. Now squaring 304 you get 92416. I can say the answer for square of any number like this i.e., when someone asks a 3-digit number with a zero in the middle, I can tell the answer within a second. I feel very nice when such things happen, you know. When you do something great and you get appreciated for it, "IT IS SIMPLY SUPERB".

So lets come back to the field. The square of 304 was 92416. Now, even before I told all about my feelings, weren't you able spot out that from the answer you could directly get a^2 was 9 and so on. That's the thing. A wonder, really a wonder. I use "ZERO" to research on many shortcuts like this.

So whenever I use this Zero, I will mostly use it as an "Expander" or "extractor".

When there was no zero in that 34, you add all the numbers from the tenth digit of the separate parts of the number to the previous number, isn't it? You used to do like that.

I will better explain, you get that 9 24 16 as three parts. If it was 34, I will tell you to add the numbers except the once digit to the previous number and attach the once number. So you will get 114 16 and the final answer would be 1156. When you put a zero in between 3 and 4, i.e., when you square 304, it is enough that you add all the numbers from the hundredth digit. So if it was 24 in the middle you simply attach it.

When you square, 909,

You will get the parts as 81 162 81, So here, you need to add only the hundredth digit to the previous number since there is "0" in the middle. So you easily get the answer as 826281. So easy, just like that, isn't it?

"BREAKING THE CODE OF MATHEMATICS"

Hi, Guys!
Imagine a day when you are asked the square of 491 i.e.491^2. And you say the answer just in a few seconds without doing the calculations in a pen or a paper.

The people around you would be simply amazed.

And you know? This is simply VERY SIMPLE if you break the code of maths. You will also find that your brain is more efficient than your thoughts.

This code breaking will really help all the people of any range of age.

So now fasten your seat belts to undertake a simply amazing ride of the wonderful world of maths.

 I assure you it wont be a thrillarium ride for those who fear maths.

Multiplication

OK here comes the most interesting part.

Now if we consider a two 4 digit number, the multiplication would be taking place such that every number in the each 4 digit number is taken into consideration.

1-digit:

If we have the numbers 7 and 5:

                  7
                  5

only two numbers are there, so only 7 and 5 only would be involving in multiplication.

2-Digit:

           42
           76
now see the pattern:

This is new for you(maybe)

1. First 4 and 7
2. then 4*6+2*7
3.then 2 and 6

You can notice that somehow directly or indirectly each number comes into association.

If you put a diagram of sets of how the multiplication takes place, it would be:

The numbers rounded in red represents the first set multiplied.
The numbers in green borders represents the second set.
Similarly the blue the third set.

But why the second set involve multiplication?

Here is the number. You can also notice that a number say 7 of 26, comes two times in the multiplication operation. This is for a 2-digit number. It it was a three digit number, 7 would have come three times exactly in the multiplication operations. We can also say that 7 would have occurred in three sets.

Since numbers have been uniform since their invention by Nature, formulas like (a+b)^2 = a^2+2ab+b^2 exist.

Now you can be able to break the code of multiplication.

Now take a number (66)^2[read as 66 square].

If we take the above formula, in that take a=6. then bis also=6. Since a=6=b.But you may think why we should not take a=60. "Just believe me". Don't take care any zeroes for value calculation. I only mentioned not for value calculation. But zeroes are taken into consideration for digit calculation.

First we shall calculate what is 66 square.

1. What is 6 square i.e.a^2? It is 36.
2.You must have understood. Now calculate 2ab. It is 72.
3. Then b^2 is also 36.
4. If you had learnt the tacking up lesson in the addition lessons(If you had not learnt, please learn), Add the tenth digit to the previous number and simply attach the ones digit. then you get the answer 4356.

Try these calculations in your mind itself. You will be amazed to find that you are able to find that you will be able to square a two digit number in your mind.

Now try squaring the number 34.

Here "a" and "b" would be different. "a" would be "3" and "b" would be "4".

Now try to do in mind itself, although I will guide you. If you had practiced, Just check the number. If you find something had gone wrong, check out the steps for correction.

1. 3^2=9
2.2ab=24
3.b^2=16
4. Then tack up to find the answer. You should get 1156.

This is not only to find two digit numbers, but also to find the squares of any number of digits with two numbers having zeroes between them. For example take the number 34 itself, but insert two zeroes between 3 and 4. that would be 3004. The square of 3004 would be 9024016. Now Think how this answer came.

If you had estimated that, "since two zeroes are present in 3004 then one minus two is one, thus one zero is present between 9 and 24 and 16."THIS is wrong. It means you think too complex. Don't think too complex but think correctly. Actually the same number of zeroes as that in 3004 is present in the answer. But since we are tacking up the tenth digit is added to a zero before. Thus there are actually two zeroes. To prove, lets take 3001^2. The square would be 9006001.

Math Games

You can play some games in these links. Enjoy MATHS!

http://ababasoft.com/kids/index_math.html

http://www.gamequarium.com/math.htm

Addition

Addition has always been simple. But imagine if add some seven 5-digit  numbers in your head and tell the answer in just a second.

Lesson 1:


So lets consider a list of one digit numbers:
5
7
6
9
8
3
1

"Now you can add the numbers easily." This is the best tool for mental arithmetic.Feel easy. The world will feel that you are the best.Now don't get irritated.I just wanted to tell, Find out the set of two numbers that add to 10. Then the number of sets would correspond to 10's. That is in this example you can keep sets of 9and1 then 7and3. So there are two sets. So keep 20. Then add the balance numbers and add them to the 20. The answer would be 39 then.

The answer is : 39

Lesson 2: Tacking up

Here comes the main part. Without learning the next part quick multiplication would be difficult.
                                46
                                59
                                94

1.Now add the first column of numbers.
you get 18
2. Keep this in mind. Now add the second column. You get 19.
3. Now add the tenth digit of 19, that is 1 to 18 and then simply attach the 9. You get 199.

"This method of adding the digits with respective to zeroes present and then simply attaching the once digit is called Tacking up."

Now if it was:
                             406
                             509
                             904
 The same sums would come. That is 18 and 19.
 But the sum would be 1819. But since zero is in middle. Then the tenth digit also would be tacked up.
But if it was a three digit then the hundredth digit would be added up and the remaining two digits would be tacked up.

You know why the left to right addition helps? It would be helping you in mental arithmetic When a list of numbers are given or when you are asked to square a number and also in many methods which you will learn.

Now use this method for adding three digit numbers also.

Lesson 3: Three digit addition

                                        576
                                        487
                                        832

 The sum would be 17 18 15.
So if you tack up then the sum would be 1895.

Lesson 4:
Arithmetic progression:

Say if a series of numbers starts With some 21 and ends with 29.
That is:                             21
                                        22
                                        23
                                        24
                                        25
                                        26
                                        27
                                        28
                                        29

 Find the middle number or find out the average. Here it is 24.5and see how many numbers are there. Here it is 10. Multiply 10 with 24.5. The answer is 245 that is the sum of the series.

Percentage

Calculating percentages: Percentages are interesting to calculate. We convert the multiplication into addition. That's it.

Just consider numbers 1, 2, 5, 10, 20, 50,100. Can you see anything? Yes they are the denominations of Indian currency. These are enough to calculate any percentage values directly.

for 1%: divide the given number by 100.
for 10%: divide the given number by 10.
for 100%: It is the actual given number.

for 2%: divide the given number 100 and multiply by 2 or just divide the given number by 5 and then divide by 100.
for 20%: divide the given number 10 and multiply by 2 or just divide the given number by 5 and then divide by 10.
for 200%: It is the actual given number multiplied by 2.

for 5%: divide the given number 100 and multiply by 5 or just divide the given number by 2 and then divide by 100.
for 50%: divide the given number 10 and multiply by 5 or just divide the given number by 2 and then 10.
for 500%: It is the actual given number  multiplied by 5.

For your number: 25.6% of 488

26.5 = 25 + 1 +.5

25% of 488 = 488/4 = 122.[Because 488 * 25/100 => 488*1/4]

1% of 488 = 488/100= 4.88

.5% of 488 = 488/200 = 2.44

Now add all the values.It gives 129.32 which is the answer. If you wanna calculate 3% find 1% and multiply it by 3. If you wanna calculate 7% calculate 5% and add 2%. That's it.