Do you ever think how your imagination can help you remember mathematical formula and concept with the help of mathematical analysis ?

-


 

INTRODUCTION -  It is say that "If you don't  know the basic of any subject , you doesn't know anything about this subject".  As example "If you wish to build a building but the base of the building is not strong, then how the building is strong ?" In mathematics the base is "Analysis", the most fundamental part of mathematics. Many of  my reader may be familiar with "Real Analysis", "Complex Analysis" and many analytical concept we study in mathematics. If you are not familiar with this don't worry and if you someone who first ever listen about "ANALYSIS",  I promised you from start to the end of this blog is a great journey for you and it's gives your imagination a challenge to think. I request everyone who read this blog to read this without any pre-assumption and enjoy the journey. 

 

              ANALYSIS  - Now you might be wonder and curious to know about " What is this analysis ?", you may think since its a mathematical topic it might be something very tough . But I say you are ........ 

                                          I told you my story, when I first saw the book of analysis, I say myself that,"what is this ?" I feel like what kind of characters are written and what is its about, in my school life I only study numerical problems, I rarely found any exceptional problem, sometimes some difficult logical problem comes, I try but I can't understand it, like "what is written is this?" Then my teacher told think it  differently like as, you read a story and just imagine what happened. That is Analysis stands for just think of a scenario just imagine what it's told you. 

                                                I just give you an easy example, " Let consider yourself a coach of  a football team, so what's your duty ? As you have to train them and also you have to tell your players what their plus and what their minus points, and how you do that only by watching previous match, if you watched it all better know eleven players of your team". So technically what you are doing, you just know their nature and used it for betterment. Mathematical Analysis is also same like that, you just take a topic just imagine it, if its a geometrical figure watch it 3-dimensionally then you understand the nature of it, if you read some abstract topic imagine it is a story or a scenario, it's not only help you  look it fundamentally but also help you imagine it clearly.




                Just imagine a geometrical figure "Cylinder", if you imagine the cylinder you see the upper and lower their are same radius circle, what's the use of it, use is you don't need to remember the formulas of cylinder like " lateral area,  base area, surface area and volume",  just imagine the picture,   

now area of the circle is "πr2" and perimeter is "2πr" by only using this and your imagination you know the formula of surface area," First imagine what is surface of a cylinder, It's the circle of same radius put one upon one lets its all h by its length then lateral area= 2πrh (because here we only consider perimeter of this circle and same as base area=2πr2 (because here one upper and one lower base) so by this you get surface area which is sum of this two = (2πr2+2πrh)" you can calculate all formulas like this and you use this trick everywhere that's called Analysis or understand the concept fundamentally.

Now let's imagine some more interesting.....


               Remember I told you in introduction about "Real Analysis" let's look at this, imagine a line which is two direction and never end and no gap between it. Every point on the line called Real Number, Mathematician can distinguish them by Rational and Irrational number, but here we not interested on this, rather we more interested in considering them together. Now when you imagine a Real number line then number of things blows' up in your mind,


  • First it's an unending line from negative infinity to positive infinity.
  • Any additions, multiplication, division, subtraction of any numbers belong this number line.
  • You can distinguish the number by their order.
  • If you want to out a number by a bucket many numbers come with it. That means it's neighbor numbers are always with them.
        And many more thing you imagine by analysis (comment down if imagine something more I not mentioned).


                     Just close your eyes and imagine in the whole world you are the only one person nothing exist without you (only consider you get sufficient oxygen and food for yourself) then what happened if you consider same scenario above , just think ,
  • who are your neighbors?
  • Where your boundary of your house
  • Now if you wanna mathematically think as,  " A set that contain only one point"
Now imagine above points and more of it and comment me what you get.


      CONCLUSION - Analysis is not something tough but the main key is you have think or imagine as it is and try to feel it, as I mentioned last rather consider it as a point consider it as you are there and face the reality... 



Comments

Post a Comment

Popular posts from this blog

Idea of Superposition

-
Image
Let's start with a question, "Can I be in a different position at the same time ?" Obviously - No. Ok, let's ask the question more specifically, "Can I be in a mood of happy and sad at the same time ?"  🤔🤔🤔 Well, we can say many times in life we are not actually happy or sad, if we try to measure on a scale starting 1 with sadness and 10 with happiness, we may be in between 1 and 10. Let at 1 sad,  2 - 4 is more likely sad, 5,6 neither happy nor sad,  7 - 9  is more likely happy and at 10 is happy. Have a look at the following table and graph - (Scores actually represent inner psychological feelings) So for the whole day, since the average of all scores is 7, we can say more likely happy is the experimented mood, but if we see the data locally it does not say so, it can say we are in some ups and down combination of happy and sad (like a wave spread in a whole day). Two clever observations I would like to mention here, 1. As I mentioned earlier, although
Read more

The development of Linear Algebra

-
Image
                   Linear Algebra is the most useful and powerful tool in Mathematics. In the last blog " Linear Algebra - Purpose to Application " we saw how a higher-order system of linear equations has been solved by transforming the equation into the matrix.                  Although it's good to mention that it only works if in the AX=B [A is m*n, X is n*1, B is m*1, so it has m equations and n unknowns ]  our A matrix is invertible, if it is not invertible i.e, in the sense of matrix it is rank of A is less than n and rank of (A;B) = rank of A < n, so the system of linear equation AX=B has infinitely many solutions, we can find the solution by converting the A matrix into row-echelon form, simply you will get the solution from bottom.         Note:-  1. The rank of a matrix - It is defined as the largest order square matrix whose determinant is non-zero.                   2. Row-echelon form - We said a matrix is in a row echelon form if in the matrix fo
Read more

The Axioms - Probability

-
                The word  " Probability " often comes into our day-to-day life referring to, " What is the probability of some random event ? " simply saying what at most chance the event to occur.                 In my last blog " What is the probability you will open this blog ? " I told you, why for a long time,  I have been in a dilemma, about why probability is so useful , my counter-argument was, "Probability tells us about the most probable event to happen based on the available ones, it does not give us the guarantee that the event to be happened for sure, then why it's so overrated ?" and I give you a classical definition of probability.             Although the classical definition easily gives us the probability, it has some ambiguity . 😢😢            1)  It is applicable when the total number of events are finite.                    2) All the events in the event space should be equally likely (they should be symmetrical with eac
Read more