Exponential Equations

Equations are the language of nature. When a person is simply forced to mug up an equation he/she tries to mug it up in a detesting mood. But there is no need to mug up. You just have to know why is it so. Exponential Equations are one of the important equations of nature in which its language is so significant. I know that significant word, but exposing in the web leads to criticisms. So just leave it. But to generally look upon the exponential equations, it can be given as:

N = N₀e^±λt

^{Actually here N is the final value. "t" is the time.} 

"λ" is another constant depending on the process.

N_{0 is called as the initial value.}

 _{The main thing here is the "e" called as the exponential constant. It is just a factor that has the value of 2.718. "e" is also called as Euler's constant after the person who gave it i.e. Leonhard Euler.}

_{Now, here what I wanted to tell is the importance of this equation. I will give examples that are of practical life which follow this equation. Population growth attains exponential growth. When you deposit an amount in the bank for Fixed Deposit, after some time that money would have increased in an exponential manner. When you keep a hot cup of coffee to cool down on the table, the speed at which the coffee cools down would be exponential.} 

_{"For this must I know this equation?", if you think like this, just leave that coffee example, but is the fixed deposit example not important to you?}

_{If you had a doubt about} λ, I will tell how it works. The bank will say that that this percent per annum the interest will be calculated no? That percent is that λ. Actually we call the symbol Lambda. If you know about the symbol please do not scold me. Just for others to know.

I think the above content helped you. If you have any doubts about these types of equations or if you have any suggestions also, just mail me at iforindia123@gmail.com. And sorry for the different sizes of text. It may be irritating for some people. Due to addition of symbols and equations into the content such things occurred.Please do not mind it, OK ?!

Thank you

Law of Uniqueness

           Each number whether in a addition or a subtraction or a multiplication or a division process maintains its own individuality.

          In Vedic Division shortcuts what I have observed is that the long form of division is so much reduced in size, that it makes the humans to calculate in a limited workspace. Some may feel that Vedic division is faster. It is because, when one searches for a shortcut, and if this process appears "as" shortcut, they say to themselves, "Oh yes, it is a shortcut. It will help me to do the calculations faster."

          But does it really help to calculate using mind? The real answer is "No". Certain people can do division using Vedic method so quickly because it is due to years of practice. And even some people like Scott Flansburg do the calculations in fractions of second and tell the answer immediately because, it is scientifically observed that, in his brain he has the ability to say the answer by "instinct". The doctor says such abilities may be due to genetic reasons.

         You know, shortcuts must be easier to learn and easier to use. Even the great math Prodigy Shakunthala Devi did not expose her secrets of mental math.

         Let it be. Soon we will find some shortcuts that can be easily practiced and easily used. However division is a different process. But with the help of Law of Uniqueness that numbers retain their individuality in all processes I will try to find some shortcuts in Division.

Thank you 

"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?