Limit:A Bud for Calculus

Calculus is one of the most fascinating, rigorous and importance subject in Mathematics. The two independent founders of calculus Sir Issac Newton and Gottfried Leibniz have done the great contribution for mathematics and the whole world by founding this amusing subject. Newtonian mechanics stands on the base of calculus and goes along with calculus. The pillar on which calculus stands is the concept of limit. The limit concept not only is the base for calculus but also the savior point when other subjects in calculus get puzzled. Limit is the base and ultimate savior. I say limit the mother of calculus.

There are many things in Mathematics that can't be represented in concrete form or can not be perceived at some glances. To provide the meaning and intuition to good degree of approximation, limits are used. What will happen if you built a tower of millions of millions of meters of height? I don't know what other things may happen but I am quite sure you will take help of limits to know what happens. Limit is approximation and the approximation tends to exact reality.

The mathematical definition of limit is mathematically exquisite. But we don't have math sense organs; so how beautiful the definition of limits may be we need some kind of intuition and illustration behind it.

 Let's understand limit a bit more practically. Limit is the result that we get if we go near and near to the situation or condition.

Let me draw a circle and make a inscribed regular polygon of n sides as shown in figure.

Here, area of the regular polygon < area of circle

How will it be if the polygon has  infinitely many sides?It seems puzzled at first hearing but ask limit to help and you will get clear view of the situation. In addition you will too learn what limit actually means.If the regular polygon has three sides it seems inscribed triangle. If it has four sides it seems inscribed square and so on.If we join every vertices of n-sided polygon to center of circle we get 'n' number of triangles. And the sum of all the 'n' triangles give total area of the polygon.

What happens if we make n infinitely large? Does the area goes to infinity? Or does the area vanishes? Lets put n = infinity and see what happens!

Oh dear! What is it? Looking at the first term I feel that I get huge enormous area. But second term says the area vanishes. It is like infinity*0. I don't know what result it gives. This is really confusing expression we have got. And using simple algebra it takes us nowhere. But don't worry. We have this awesome tool called limit and we make use of it here.

Yes the task is solved. The infinity sided polygon tends to be a circle as we get 'A' as area of the circle. Pretty interesting, isn't it?This is the limit concept and how it is beneficial to all we crazy explorers. This is its vigor.

In the next article I use limit in world's population case. How will the population and world go after thousands of years? Will all the available resources vanish? Will the mankind end due to lack of resources? We will find out in next article of limit.

Thank you.

Feeling of Relief at Vacation

Being at the state of no mind and doing the stuffs, thinking the ideas haphazardly has been the ecstasy for me. When the mind mediates with its emptiness, then it leads to spiritual journey. In the meantime random thoughts arise and you are pushed to do some activity. The activities you do which are of immense interest, insatiable hunger and unflinching passion make you feel content with your life and your doings. The journey of walking takes the route of targeting nowhere. You need not to fix the goal. Following the target is boring for me. Enjoying with the stuffs I see on my travel is pleasing.

The intermediate level that made me close with some of the subjects like Physics, Mathematics and Chemistry had carved the pattern of those subjects in my my kind deeply. My mind all filled up with the carved pattern of Physics and Mathematics started to seek whole universe from the eyes of Physics and Mathematics. Being optimistic I hope my mind was nature friendly with those subjects too. Trying to seek logic and reasonable explanation beyond every phenomenon and tending to seek mathematical model for every process, my mind was searching for adventurous journey of creation and universe truths. Though the journey may not be which I had intended to go before but exploring the path which otherwise I would not have traveled certainly makes me learn many things that I could have unlearned had I taken the straight route to my destination. The road dilemma and insufficient wisdom made me to swerve my original path not my intend to learn many things did which I stated above.

Chautari
Pic courtesy: http://www.everestuncensored.org

Sometimes exhausted and torn off of the journey, I lie on the shades of trees in Chautari(resting place at the foot of the trees where one finds shelter from the sun) imagining my own dreams and going into that emptiness. While I may contradict myself saying emptiness means something, but it certainly has a meaning. The fresh and sophomore years of engineering have passed. In the vacation the university provides after each semester, I have nothing to do and it is the feeling of ultimate happiness for me. Though I have got nothing to do for anyone, I have my mind engaged more and empty in this period. At the daytime, I wake my passionate thoughts, I grow my emotional love, I raise my spiritual soul and I continue my pleasing journey and at the night I am empty. I go into the bare world of meditation. Living in the day and dying at the night creates spiritual journey of my life and it is mind-soothing, body resting and soul appeasing.

The days when I should wake up all night and bustle all day are over for a certain period and I breadth the moment of life. Is there any point of doing things which I don't know either why I am doing or why it is to be done? I feel not. Studying and engaging in engineering in the way I am doing is meaningless rather sweeping the floor of my bedroom, sleeping early with meditation and waking early with chirping of birds in swinging of the branches is meaningful(for me). And the living of life with meaningful moments is satisfying for me. Every second I feel contented and those all the seconds being meaningful make my life to be meaningful. It is not that reaching a target only is meaningful. Nevertheless living the moments of an engineer lively is meaningful.

I might not have felt this meaningless meaning had I not gone through meaningless moments. Credit for you time and fate. You taught to differentiate between meaningful and meaningless moments. When I said this to my friend he said now only two years left for me to live my meaningful life. No no dear. It is not like that. Let life provide me every meaningless moments for every meaningful seconds. Let life provide a tinge of death for every band of life. Life is supposed to be equipoise. Every oddity in life brings imbalance in living.

Love These Two Idiots

The characters, places and plots in this story are not written relating to some individuals and events. All the things in this story are completely hypothetical. One aspect of this article has been written being inspired by Ron Howard directed film A Beautiful Mind. Apology if it coincides with someone's life.

Everyone takes birth in this world in only a way. I haven't seen a single child coming from different place than his mother's uterus. I can't think the zygote can grow into human soul in any other place than mother's womb. All people born in the same way, same place but some naturally unlucky. Why are some people born healthy and why other having a certain defect? Has the child done mistake in his previous birth or has his parents committed sin? I am just confused by this and have deep empathy towards them. If it is the fault that the child has done in his previous birth he should be known how this sort of punishment is given so that he can think of his next birth and try to make it better. God keeping reason of this bias secret does not do any good to those sufferers. Everything should be clear and everyone has right to know why they are suffering.

I have two close friends, Shaun and Raud. All teachers and students do fun by calling us three idiots. Shaun is from North Karnataka, India and Raud is from Tamid Nadu state of southern India. Looking at them I feel bad and rage towards so called supernatural creator "God". The friend who usually sits left to me in class, Shaun is serious chronic migraine patient. My another friend, Raud whom we call imaginary guy and sometime Mr.i is paranoid sufferer. Looking at Mr.i's beheviour and Shaun's migraine trauma, I feel disappointed. I will go to Raud first and come to Shaun later.

The guy who is unnecessarily fearful, the guy who is freakishly sensitive and the guy who is meticulously careful in every matter, he is none other than my dearest friend Raud. Doctor called him to have suffered from Paranoid Personality disorder. And many of my friends call him a psychopath. Whether to agree on my friend's adjective for him or to agree on his fantasies was a difficult choice for me when I came here in Bangalore. The society where I was brought up had taught me to say such type of persons psychopath and even sociopath to extreme. As the curiosity in me about him grew rampant, I went on to study about paranoid. When I got to know about the disorder in full detail, I thought to make him my best friend and did I. Sometime he seems funny and sometimes he provokes anger. Sometime he empathizes me in my pain and sometimes he shows sudden apathy. Waking up in middle of night murmuring someone has knocked the door may have irritated me, compulsing me to listen his imaginary fantasies may have perplexed me, shouting at me in aggressive manner in early morning may have hexed my day but my feeling of love, empathy and compassion has always grown towards him day by day.

The abnormalities in him hurts me and I try to give as much emotional support as I could. Happiness lies in blossoming the flowers of love, harvesting the seeds of compassion and flourishing the light of wisdom. Hating him, mocking him, hurting him, antipathy-ing him, troubling him and making fun of him as some of my friends do makes me sad and I try to search the solutions for his paranoid. People say he is wrong; even my conscious mind says that. But it's not that he selected to behave that way. He may see imaginary fantasies but he believes they are real. If you could help him to get out of this disorder help otherwise don't make mockery of him. He may fear unnecessarily. That's not his fault. That's what his nerve pulses informed to do and he feared. Nothing more than that. Making this abnormality a reason to hurt him and harassing him is inhumane.

As everybody predicts, Shaun got his chronic migraine from his parents. His father and mother had chronic and episodic migraine respectively. My resentments to the doctors who say your migraine is genetic are understandable to doctors. Feeling emotional I think that "why  can't you find total cure Dr.X?" Before I could say a word they would continue to say,"Look you don't have cure for migraine, only it can be controlled..." "It's okay doctor",I would say and think they don't know anything. I feel many times what are they for if they can't cure a disease called migraine? My feeling that doctors can treat any disease was proved to be false. Shaun, I and Raud are roommates. I see Shaun sleeping more than 15 days a month because of migraine pain. It really makes me feel bad. I can't think what to do for him. More than him I was worried not because he has migraine but because why such unfair to him. Yes everyone gets diseases but why someone gets this vulnerable and mentally, emotionally killing migraine which doesn't have cure? Talking about cure to doctor, one doctor said we can treat it for about six months by injecting BOTOX injection and after four injections each after six months it will go. After hearing this I was like excited and calm but my face turned grim again when I knew its one dose costs 500$. I feel like,"Why so much expensive disease for we poor?" I was too not sure whether it completely cures this chronic migraine. I am not against that my friend should always be free from diseases. But what I am trying to point out is why such disease to the innocent children because their parents had? Yes having curable disease is okay but why this disease which can not be cured and whose actual mechanism is also precisely not known? Is it fair to those innocent children? I completely condemn the pain my friend is getting because of his heredity.

My opinion is that at least the sufferers should know what is going on and almighty can't be biased towards some individuals. I want these sort of abnormalities to be completely cured. And for other people who are making the sufferers life worse from bad; if you can cure, cure it by love, affection and compassion. Otherwise help them to cure this. Listen to them, suggest them and help them. If you can't help then stay quite. Don't touch them. Don't show your animalistic and inhumane behavior by hurting them, harassing them and taking advantage from them.