The Sequence of Chapters In Content Of Any Book.

 Every book have a definite content. And every book have a definite aim behind the content. The writers of the books want to express their way of thinking about that subject or topic. May this happen that your way of thinking of reading that  book is different from though of  writer. But if you wanted to understand the concept behind the sequence of chapters and what is the aim of a book then  You have to read book according to sequence of chapters. This topic came in my mind when my teacher start teaching me with chapter number three. I am really confused that why not from the chapter 1st. But , Now I got the answer. Teacher wanted teach easy thing first. In mathematics, nothing is hard and nothing is easy but it is depend upon your concepts. In Mathematics, it is very important to follow the sequence of content at least one time because every chapter is connected with each other. It may possible that Book is divided in sections but trust me these sections also have some important connections and this way of reading books is justice with the writer of that book, justice with the aim of book. Here, I am taking my first example for explaining my thought more clearly. For this example I used the content of NCERT  Mathematics book for the class 11th. The sequence given in this book is as follows:

1) Sets 

2) Relations and functions

3) Trigonometric functions

4) Principle of Mathematical Induction

5) Complex Numbers and Quadratic Equation

6) Linear inequalities

7) Permutations ad combinations

8) Binomial Theorem

9) Sequence and series

10) Straight line 

12) Introduction to three dimensional Geometry 

13) Limits and Derivatives

14) Mathematical Reasoning

15)Statistics

16) Probability 

Appendix:

 Infinite series:

                * Introduction

                * Binomial theorem for any Index

               *  Infinite Geometric series

                * Exponential series Logarithmic series

Mathematical Modeling:

            * Introduction 

            * Preliminaries 

           * What is Mathematical Modeling 

You can also see the photo of the content:

When I saw this content at this stage, I clearly saw that writer want develop a basic understanding of  Mathematics and wanted that the student become familiar with Mathematics in higher educations and one more thing that the sequence of the chapters is essential because Sets are used in chapter Relations and functions and with this concepts students are ready to read a special type of function named trigonometric functions and similarly other chapter are also connected with each other . Even though Appendix is very interesting and relatable with my point of view.

Now I am taking a my second example of NCERT book of history for class 10th.The name of this book is India and Contemporary world II. sequence of chapters in this book is as follows:

1) The Rise of Nationalism in Europe 

2) Nationalism in India

3) The making of a Global world 

4) The age of Industrialization

5) Print culture and the modern world.

This book also have many purpose. One thing is to give a understanding that any incident happens in the world may also have an impact on India. Another concept is to study comparatively and many more concepts that this book wanted to teach us. Now it depend on us that what we thinking about that book while reading. What is our aim to study that book. And students who understand this then it become a bless for them  and a teacher need to explain these concepts and teach students with this concepts and if teacher found a new concept related to that book then teacher need to explain that concept.

"Take a good book to bed with you - books do not snore"                 ---George R.R. Martin

Thank you.

S.K. Ojha 

Did You Know These Facts About Average Of Numbers

 

In our classes teacher teaches us that average of n number is given as

And this is only about average. But this is not fare with average concept. Here, I give you some basic understanding and some concept related to numbers.

First thing I cleared here that average is a tool which help us to see a broader picture of many problem or solution .Some key words that you must here in your daily life that average internet connection speed, Average income of a family, etc.

Wikipedia said that an average is a single number taken as representative of a list of numbers.

This is a different view of seeing average beside of its formulae. With the help of average you can say many things about its entries which are used in average. I know these things are not in your books but you have to think these things to yourself. And here I give you a way how to think?

Take a example, if average of 10 numbers is 1 then what you can say about these ten numbers. What that 10 number be? For this come on the mathematical form.

         

=   a1+a2+a3+………..+a10 = 10

From above it is clear that the sum of that 10 numbers is 10, if average is 1. If entry is whole number then a1, a2, a3 ………, a10 ≤ 10. Which means that entries lies between 1 to 10. Not only this, but you can say more about these entries. If a single number is equals to 10 then all other are necessarily equals 0. Similarly, if a entry is equals to 5 then other all entries are less then or equals to 5. Similarly, for other possibilities.

Now the last but not the least, What is the basic difference between Average and Arithmetic mean. This question arises here because average of two numbers is 

 and Arithmetic mean is same. Then what is difference between them?

Arithmetic mean is only defined for special kind of progression called Arithmetic progression in which difference between any two consecutive entries is same.

So, if a₁ ,a₂ , a₃ ,…..a_n  are in Arithmetic progression then arithmetic mean of a₁ and a₃ gives a₂ and arithmetic mean is calculated only when the series is  in Arithmetic progression. But Average is calculated for ant number and no need of any special progression.

Many more concepts are hidden inside the formulae and definition.Some of them are written in books and some of them you have to think your self.

Thank you.

S.K. Ojha

Mathematics of Geometry

Many students, incliding me, never talk about geometry in the class of algebra but now I realize that the right way of study of algebra is to connect it with geometry . It is impossible to explain all Algebra with geometry here. So, Today I give you only a way of thinking.One thing that I always quote that Geometry is extremity of every mathematics or every mathematics have their use in Geometry. So, here I give you a list of topics Whose geometry is very Interesting. I think you must have a pen and copy. 

1)Algebra: 

This branch of mathematics have many uses and always have a geometry. If you get a topic in algebra whose geometry is known then its your turn to discover that. We always write many equation like x=0, y=o or x-y=0 ,x^2-y^2=1 etc. 

All these type of equations have definite geometry in Cartesian plan or Euclidean space. Some of above equation are represents a line, some represents curves and and many other eqations are there in Algebra which give you a solid geometry in 3-D.Also we very often use the term "find the root " as:

*what is the root of x^2=1 or x^4=1 ( In real plan and in complex plan). 

*what is the root of the system of equation

                             x-y=1

                            X-2y=2

Try these questions without calculations. ( Hint : Use geometry) 

2) Vector calculus:-

I strictly recommend you that vector calculus is always with geometry becuase this is frequently used in geometry and in engineering, Some examples are as follows: 

 Dot product of two vector:

With the use of  dot product you can find the angle between them since 

                

Dot product has the geometry interpretation as the length of the projection of x onto the unit vector of y when the two vectors are placed so that their tails coincide.

Similarly, there are many terms as cross product, divergence, gradient, curl, etc,. have a geometry Interpretation. 

As we saw two topics whose geometric interpretation is quite interesting. Similary, differential equations are used in geometric problems ,diffrential calculus, trigonometry, integrations, etc have a geometric use. For undergraduate students Groups also have a geometric interpretation and extremity of Groups and rings are called Algebraic geometry. This post may be a trouble for your teachers beacuse now they have to explain more in their classes. But if teacher denies to explain all your question ,never stop yourself and mail me your doubt I will try my best to explain you. 

🙏🙏Thanks🙏🙏

S. K. Ojha

Discovery of a sub atomic particle Pi Masons: Dr. Bibha Chowdhuri

 *This article is dedicated to the unknown or we can say that known by only few of us, Bibha Chowdhuri. And this article is a question for all of us.*

*Why this name is not known to everyone ?*

*Is this great scientist not deserves the noble prize..?*

*This question is for all of us.*.

I am going to start the introduction of this lady scientist by a recent incident : An unsung brilliant Indian scientist,Bibha chowdhuri,was honoured by the International Astronomical Union by naming a white yellow dwarf star after her name. The star ‘Bibha’ is 340 light years away in the constellation of Sextans.

Bibha Chowdhuri worked at Tata Institute of Fundamental Reserch for 8 years. She worked with Homi J. Bhabha and Vikram Sarabhai. And she also worked with Noble prize winner and great physicist P. M. S. Blacket(Google this name for more)  

And now why I am saying this lady deserve a Noble Prize. Bibha Chowdhuri  discovered the pi-meson, a subatomic particle, in Darjeeling before the independence of India. 

And a book is writen on her work, named "A Jewel Unearthed :Bibha Chowdhuri".

I can't introduce this lady but after reading the above lines you can easily think about her greatness in her field. But my question is still same why this lady is unknown to many of us ? And the biggest question why not Noble Prize to her? For this you have to read and research on the history. What are the factors working behind all these stuff ? Beside of all this question, we are also,intellectual class of the nation, was responsible for all these things. why she didn't get her fame despite this great discovery. If we are using the internet in the right way. And if Indian parents support their children to go in higher education then definitely they came to know about names like Vibha Chowdhuri. Here I took only one example of Bibha Chowdhuri but there are many more names of great ones whose name is unknown to us and they have disappeared in the pages of history.I am quoting here that "Talente is hidden till to be viral".

Thanks

Shivam kumar ojha