Category: Maths

I was thinking of it while speaking to some of my friends and here is a realization I had!

Well just a profound realization that I had recently, its about how important prime numbers are for Mathematics….

For a moment, it occurred to me that all the natural numbers except one and primes can be represented as prime numbers…. Am sure every one knows this…. its called prime factorization….

Just imagine if there were no prime numbers… there will be no numbers other than unity (1)!!!
Surprising isn’t it? Yes but that is true…. It’s really hard to have anything if not for prime numbers and the operation of scaling them. Yes, call it scaling… multiplying one with another to get a composite number.

The analogy that I can draw from this understanding is… We can consider prime numbers to be “Bricks” that occur naturally in nature, we just pile up these naturally occurring bricks to form what we want… These Primes are for sure the pillars on which the whole natural numbers stand and hence our understanding.

These primes have not definite pattern of repetitions, they are scattered across the whole space… occasionally you stumble by them and start using them from then on… They are the pillars that hold the numbers… All the composite numbers are constituents built by just scaling them!!!

Beauty isn’t it…

Well these are my personal view on what role differential equations have got to play…. read on if u think its worth…

As humans we are mostly interested in knowing what will happen to some thing in the future… What will happen to him? what will be the weather in 2 days, or some more thing else… the whole point is simple…
“I want to know what is going to happen…”

So what is it that a person gets to start off on prediction?
He has seen what has happened in the past, he can think what are the dependant and the independent parameters based on this experience…. he continues from here and makes an assumption as to the directly varying or indirectly varying and then comes up with an equation….
Well this was a crude that worked wonderfully with the initial understanding….
but, what the person currently sees is only a small portion of the whole, the smallest part of the system almost negligibly small instance that he considers… so at this particular instant if he has to form an equation it is better he forms a differential eqn, that is a better way to write it down…. so then if he has a differential eqn, and wants to know what happens in the long run, he just has to integrate and find out…

An example:
well now this is how I start my prediction… if the change is always a constant… lets say…
we want to measure some distance…. I see that I have been travelling some 3mt in 2 seconds… for a consistent amount of time…. so now I say this is a constant for every small interval of time say a second… I travel 1.5 mt So now I continue to write the differential eqn…
dx/dt = 1.5

now I integrate to form my equation…. X = 1.5t… I continue this way to form the eqn…

well there is a catch – in writing these dif-eqn… most of these are unsolvable… that’s coz we cant predict to the minutest detail…. true rt!!!

Well I am writing this not coz no one knows but coz it will help some who dont know to understand it.
It is said that 1/0 is infinity. Well actually it only tends to infinity and is not exactly infinity, but can safely assumed to be based on the proof.

The proof is simple:
1/(1/10) = 10
1/ (1/100) = 100 note 1/100 < 1/10 and the result increases from 10 to 100
1/ (1/1000) = 1000 …
so on…

The basic understanding that can be derived from here is that as the denominator gets smaller and smaller the value get larger.
Assuming that zero is the smallest possible value possible, 1/0 has to be infinity….
Simple right!!! [:)]

Hey this is only for better understanding… may be some of those who find it useful can pick it up from here… not my original work, but this is the way I understood it!!

We have had the identity:
(a+b)^2 = a^2 + b^2+ 2ab
Now to understand what this identity means, I used the graphical repsentation,
Now we can assume safely that ‘a^2’ represents a sqare of side ‘a’

So as well should (a+b)^2, but then what does the identity mean?
Well it means that to make a square of side (a+b)^2 just given geometrically to be proven.
ok that should be simple rt!!!
a figure with area ab will have to be a rectangle
so,
To form a sqare of side (a+b)^2 we have to just add up the geometrical figures 2 squares one each of side a and b.
But then we have to have the space that can be filled by rectangles with side a&b… we have to have 2 such.. so there fore we have
(a+b)^2 = a^2 + b^2+ 2ab
ie
sq of side = sq of side + sq of side + 2 rectangles of area
a + b a b a * b

Well this is my understandin of Diffrentiation…. its easy if we understand this.. but the graphical method is undoubtably the best if thought properly.

x^2 when diffrentiated gives 2x.
Ha ha, where the hell do you think this will be used, I can use it more better only if I understand it well na!!
ok the understanding can be got simply by replacing by numbers say I replace 1000 instead of x.
so x^2 should be 1000000
ie
x = 1000 => x^2 = 1000000

Now assume I increase x by 1
so we have x = 1001 and so x^2 = 1002001,
ie
x = 1001 => x^2 = 1002001

so we see that the change is just most significantly 2*x* the change + a miniscle number
The number being more smaller the change becomes very prominent only on the 2x part.
so that is why we have
derivative of x^2 is 2x
Similarly we can get the description for x^3
and so on!!!

You may ask…
x = 1000.5 => x^2 = 1001000.25
which is not 2x!!! well that is why in the first principles of diffrentiation we do have to devide by the inc. and then ignore the very small value when compared to the large value. So that we can approximate the change at any place to be 2x.
ie the change is around 2*x*the inc!!! inc being small, and then normalizing using the small inc (very small but not zero!!!) we realize that the change is just 2 times x …. why normalize?? coz that will make it more generic rt!!! [:)]…said in other words… we have now made the change in the whole values as independent as possible of the minute changes!!!.

Well all this is just gimic.. the actual meaning of a derivative is the rate at which somethign is changing…
Lets understand it better by taking a line say y = x
the derivative is 1….
it means that there is no change of the rate at which the slope is changing.. its constant.
Now if we have a parabola say y = x^2
Lets see this
x 1 2 3 4 5 6 7 8

y 1 4 9 16 25 36 49 64

dX 1 1 1 1 1 1 1

dY 3 5 7 9 11 13 15

dY/Dx 3 5 7 9 11 13 15 — we see that the values are all odd integers and can be represented by a straight line so the derivative of a parablola is a line..
It is at each of these minute intervals if we calulate the slope.. the rate at which the slope of the prabola changes we get that to be a straight line.