One of the ways to Understand Mathematics

         I have seen many people doing mathematics without thinking at all. They will do mathematics somehow like a computer which doesn’t know why it is doing.

      I don’t like to do pure Mathematics. What I mean by pure is somewhat not thinking about the Physical meaning. The concepts will look as a vague one.

There is difference between saying 

         one apple + two apples = three apples

from simply saying 1 + 2 = 3

      Human brain always used to understand anything by visualization.

      When you say “y is a function of x” there is no so much Understanding. But when you say “Velocity is the function displacement and time” there is a real understanding because it is understandable from everyday experience. 

          The reason is because the mathematics itself is created to explain physical things.

      When we do any mathematics, we should try to understand each and every line. It is nice to read mathematics in an elaborated manner and it will help you a lot to think. For example instead of studying “dy/dt” you can read it as

Rate of change of y with respect to x

   

   Instead of just saying, ∫ y dx (Integral y dx) you can read it as, summation of function y with respect to the independent variable x.     

   So each time you will remember the meaning of differentiation. And you will have a great clarity in the topic.

     

We should avoid using the formulas blindly.

    

     In any problem, we shouldn’t straightly use formulas but we should always try for the derivation.

     

     First it will take more time to completely derive all the things in a problem. But when you derive it for more than 5 or 10 times, then it will start to come fluently. 

     But after you got practiced, no matter what size the derivation is, it will come in a great flow and you can just do it in your mind. 

Depending on the size of the derivation, you need to practice it more times.

      So, we can learn everything in a perfect way.

Nothing needs to be remembered at all as a mug up thing. But everything can be remembered from the fundamental derivations.

    

    And it will also help you in thinking skills. Each time when you derive the formulas, you may not use the same method. But you will have the chance to derive the formula in a different ways. And you will get a different kind of thinking about the single concept.

   

     Maybe it will lead to great innovations!!    

When you try to analyze a mathematical thing to explain in words, you will get deeper understanding than just looking at the equation.

      Max Born got Nobel Prize only for his famous explanation on Quantum Mechanics.

      Explaining in words is a very good thing.

What My opinion is ,

       Studying mathematics using words is the one of the best way to understand any concept.

      So, you will always understand the physics behind that and may understand the real meaning of the concept.   

      That is why I used to discuss any concept in physics with mostly consisted of words other than just mathematical equations. 

Simple Harmonic Motion

When we study about harmonic motions, first we will just examine special motions in Nature. And you may have seen this kind of motion many times in your real life which is called Simple Harmonic Motion.

What is special about Simple Harmonic motion is that, we can completely explain its motion using basic Physics concepts. We can get the complete solution of Equations of motion.

So, instead of talking we can analyze it.

The simplest example is spring with mass m.

We know that the spring will oscillate back and forth if you give a little force to the mass m. On analyzing carefully we get that when the spring is elongated, the force tries to compress and when it is compressed the force tries to elongate it.

This shows that the force always tries to make the spring in constant length. It doesn’t like to change its length. At certain length, the spring doesn’t do anything i.e. neither compressing nor elongating force. This position is called the Equilibrium Position. And the length of the spring at the equilibrium position is commonly known as the length of the spring.

And the elongation or the compression length with its direction could be written as the displacement vector x which is the displacement made by the mass m from the equilibrium position.  

So, from the Newton’s second law of motion it is known that the force acting on the mass is

m d^2 x/ dt^2

When we need to study about the motion of the spring, we will just care about the force acting on the mass.

When I said force, don’t think about what kind of force would be there because without knowing the force itself we can determine its motion.

We don’t need to care whether the forces are gravitational or electrostatic or the electromagnetic or the combination of all these. For our task we just need the total effect.

And from the output it can be deduced that the force should be proportional to the elongation or compression length and should be directed opposite towards the displacement.

Hence it is written as

      

                   m (d^2 x⃑   / dt^2)              α       – x⃑

                    m (d^2 x⃑   /  dt^2)              =       – k  x⃑

   ⃑    used to denote that it is a vector  

the value “k” is the proportionality constant. We need to analyze it.  

The above equation in words follows as,

In Nature, if any object follows an absolute rule that its second order rate of change of position is proportional to the position of the object itself, then it follows the harmonic motion. Provided its mass is not changing with respect to time.

But you may instantaneously ask the question why it is not proportional to square, cube or any other power of the elongation or compression length.   

         

 The answer is yes. The force does really depends on the other powers of x. But mostly we observe the first power in these motions. And moreover, only for the first power of “x” it is easy to solve for the equations of motion.

The difficulty of solving the equations for higher powers of “x” is too much difficult and it will be no more a simple harmonic motion.

That doesn’t mean we shouldn’t think about other powers. But we will do it by step by step.

Therefore in our first step we are confined only to the first power of “x”.

and the differential equation for that motion is given by,

                m d^2 x⃑ / dt^2  +    k  x⃑    =   0

Since “m” is a constant

                d^2 x⃑ / dt^2       +    (k/m)  x⃑    =   0

or              d^2 x⃑ / dt^2       +    A  x⃑    =   0

where A = k/m

This is just a second order differential equation. Now it is a mathematical problem. 

The Second way of thinking to the Ultimate Answer

The second way of thinking is very different and also a very non intuitive one.  

          In simple words,

“Why should there be an answer?”

Thinking it for the first time, I just amazed. I always used to think in a logical way. But this question asks “what is meant by logic?”

You can’t find answer for this because the question itself is about the question.

So, my friend argued,

“We are living for the Nothing.”

He said there is no Ultimate answer. We are thinking it to be there is an Ultimate answer.     

For example, if there is some person who found the answer for the Ultimate question then what is after that? 

You can say, “You will never reach the answer because it is about infinite”

But I am talking about what if someone achieved the answer.

What is after the Ultimate?

          The final thing is just “the Nothing”.

It is weird. I just couldn’t simply accept this. It is completely against my point of view. I am basically adapted to the scientific way of thinking.

So, I asked “For nothingness, why the Nature created us”.

He asked me again, “Why do you think there must be a reason behind our life?”

Yeah it is a great question. I couldn’t answer. I remained silent. But then again I asked him a question,

         

It is what “You think to be the Nothing”, but it need not to be. It is just a one way of thinking.

What if there is an answer that answers the Nothingness itself. What if Nature wants us to think that it is Nothing.    

He couldn’t answer.

It is simply the reverse.

I got the point‼

What we think to be the Nothingness is also some kind of an answer from something else. He is again trying to answer. He came to the conclusion of “Nothingness” when he thought about “Something” else.

It is a loop. It is a great loop. There is no starting or the end.

It is simply attached in the closed way,

First way is,

“We think to be there is an answer. But it needn’t be true”.

The second way is,

“We think to be there is nothing. But it also needn’t be true”.

 The following may confuse you, but it has some meaning,

Something is meaningful only when there is the Nothing.

And the Nothing is meaningful only when there is Something.

One is meaningless without the other. Neither one of them is not the starting or the ending. Both start when it starts and both end when it ends.

But the problem with the second way is,

It may leave you some kind of boring life. I don’t want to go so much deep.

So, what is my suggestion is,

Go with the first way of thinking, because scientific developments are based on the first way. But don’t fail to enjoy your life.

          It is the only one life you had. So, enjoy each and every moment of your life. Life is the most precious one.

And we are living for others. Don’t forget that.

I can reason my suggestion. But now I leave you to think about this.

If you like, you can discuss with me. 

The first way of thinking to the Ultimate answer

The question which everyone is trying to find an answer is “what for we are living”.

          I always like people with peculiar characters. One of those characters is my friend named Prasanna. He is a very good philosopher. I think he is the first one I have met in my life, from whom I did learn something.
          I was discussing this question with him for a long time and came to the conclusion of great confusion‼

The discussion is about the ultimate question and the answer –  The question is“Why we are?”

If you try to answer this question, you will never achieve. But I want to give the various ways of thinking to the answer.

The first way of thinking is “There is an answer”

This is the side where I want to be. From my childhood I used to ask a lot of questions. All the time my mind used to think in a very logistic way. 

And the logic is, 

“There should be a reason behind everything”.

         
This is the fundamental postulate of Science. We think there is a reason for everything and the reason is just thought to be the answer. This is the human mentality. This principle just made humans as humans. By following this principle, we have achieved a lot. Scientific developments and research and everything is based on this Principle.

        I say to my friends that we are in a game. It looks as if we can’t find the answer, but there is an answer. Our life is just a puzzle. 

        The example I will give is the converging infinite series.

 If you just look at the question to find the answer for this series,

 1+ ½ + ¼ +1/8 +….

  You will not get the solution.
        

         At my First sight, I said there is no answer, because there are infinite terms.

I reasoned my answer by asking, we don’t know how many terms are in total in this series, then how can I add all these up? But when I found that the answer is 2, I simply amazed. 

Yeah, it is a tending value. But at the first thought I didn’t get this in my school level.   

        In our everyday life itself, we have managed to come up with so many questions like this. On the first sight, we will say for sure that there is no possible answer. But when we see the answer, we will get amazed.

Puzzles are the best example where we come to this kind of situation.

        So, I concluded that our life is also a puzzle but which we don’t know the answer yet. We are just feared looking at the question. But there is an answer.  

         

In the simplest way, I will quote it as

         

“We are...
 So, the Answer.”