45-List of Formulas - 03

Click here for the previous list of formulas.

01) The number of elements in a set: 
            The Number of distinct elements in a finite set A is denoted by n (A).
02) Some Important Results:

Cartesian Product of two sets:
Relation:
                  If A and B are any two non-empty sets, then any subset of A x B is called a relation from A to B. If R is the relation from A to B and (x, y) belongs to R, then we say that xRy. Here Set A is the Domain and set B is the Co-Domain. Y is called the image of x under relation R. Similarly,  x is called the pre-image of y under relation R.

Range:
                  A set of all the elements of Co-Domain B associated with the elements of Domain A is called the Range of Relation R.

Function:
                  A and B are two non-empty sets. The Function f from A to B is said to be a function if every element of set A is associated with the unique element of set B and is denoted as f: A -> B. and we say that y = f(x). Here y is the image of x under function f.
                  Here we say that A is the Domain and B is Co-Domain. 

Range:
                  A set of all the elements of Co-Domain B associated with the elements of Domain A is called the Range of Function f.

 Onto Function:
                 f: A-> B is Onto function if every element of Co-Domain B is associated with some 
                 elements of A under Function f. (Note: Range of function "f" is the same as Co-Domain).

Into Function:
                 f: A ->; B is Into function if there exists at least one element extra in Co-Domain B which is not associated with any element of Domain A under Function f. (Note: Range of function "f" is not same as Co-Domain).

Even Function:
                 f : A -> B is said to be Even  Function if f (-x) = f (x).
                 Example: f (x) = cos x    
                 Example: f (x) = x²

Odd Function:
                 f : A -> B is said to be Even  Function if f (-x) = - f (x).
                 Example: f (x) = sin x    
                 Example: f (x) = x³

Composite Function: 

              If f: A ->; B, g : B-> C, then gof : A -> C where gof (x) = g[f(x)]
Inverse Function:
              If f: A -> B is one-one onto Function, then g: B ->A is known as an inverse function of f and is denoted as f^-1: B -> A.

Click here for the next list of formulas.

Anil Satpute

44-List of Formulas - 02

Click here for the previous list of formulas.

Key points formula:

Today we will see the list of Formula (Formulae) of Set Relations and functions.

These are important key points.

Click here for the next list of formulas.

Anil Satpute

43-List of Formulas - 01

Today we will see the list of Trigonometry Formulae (Formulas). 

 

In Hindi, the very beautiful prayer of the goddess "Saraswati" ( the goddess of knowledge and arts ) is there. Please see the following lines (in Hindi)

सरस्वती के भंडार की बढी अपुरब बात 

जो जो खर्चे बढी जात है बीन खर्चे घटी जात . 

The English script of the above Prayer is as follows:

Saraswati ke Bhandar ki Badhi Apurab Bat

Jo Jo Kharche Badhi Jat Hai, Bin Kharche Ghati Jat.

English meaning of the above prayer:

It is the fact of the store of knowledge and arts (Goddess "Saraswati") that if we spread our knowledge among others, it will increase else, it will decrease.

So I request all my dear Students to tell about the above method to your friends and relatives.

Click here for the next list of formulas.

Anil Satpute

42-Magic Square-4 (different cells and addition part-1)

Click here for the next blog on the magic square.

Today, we will some Properties of the magic square. we have already seen how to prepare the Magic Square Using Magical Sketch in blog-001 " Numbers are the Stars, We must play with them " 

Please observe the following Magic square.

Here the sum of the numbers in all columns, rows, both diagonals is 34. Now let us check four houses above Magic Square. See the following diagram for choosing the required 4 houses. The addition of numbers in these 4 houses is also 34. This is the most important property of the magic square.

See one more example of this property of Magic Square.

Like this, you can also find so many beautiful ideas which will definitely improve your thinking level. Actually, your thinking level itself is very high but even then it will boost once again to improve your skills.

Click here for the next blog on the magic square.

ANIL SATPUTE

41-Easy Method to check your calculations

Today we will see an easy method to check your calculations.

In Number system, every number is formed by digits. For our method, we require to get the digit addition (that is Addition of all the digits in the number). 

Now we will see some example of digit addition.

Example-01:  Let the given Number be 795832. The digit addition of this number means to get an addition of all the digit till single digit. 
      
           Digit Addition = 7 + 9 + 5 + 8 + 3 + 2
                                    = 34
                                    = 3 + 4
                                    = 7
 So here Digit addition of the Number 795832 is 7 (which is in the lowest form)

Now we will see still an easy method to do this digit addition. See the following steps to do the fast digit addition.

Step-01: Take out all 9's from the given number if any. In the given number 795832, one 9 is observed, so take it away from the number, So now we have the number for digit addition will be 75832. (Note: The removal of all 9's is only for getting digit addition. For original calculation we must take the complete number).

Step-02: Take out all the digits whose addition comes out to be 9. in our resulted number in step-01, 75832, if we observe very carefully, we will notice that the digits 7 and 2 gives addition as 9 so remove these two digits, so now we have the number for digit addition will be 583.

Step-03: Now we need to take the addition of all the remaining digits. Here we have 
           Digit Addition = (5 + 8) + 3
                                    = 16
                                    = 1 + 6
                                    =  7
Which is as the addition of all the digits calculated by adding all the digits and then reducing it till single digit.

Now we will see all the above 3 steps for getting addition.

           Digit Addition = 7 + 9 + 5 + 8 + 3 + 2
                                    = (5 + 8) + 3
                                    = 13 + 3
                                    = 16
                                    = 1 + 6
                                    = 7
Example-02: Find the digit addition of the Number 543249815
           Digit Addition = 5 + 4 + 3 + 2 + 4 + 9 + 8 + 1 + 5
                                    = (5 + 4) + (3 + 2 + 4) + (9) + (8 + 1) + 5    
                                        (All terms in the brackets gives addition as 9)
                                    =  5

So the digit addition of 543249817 is 5.

Now we will see the procedure for checking our calculations.

Example-01: Add the three numbers 9432, 7859 and 8147

                         9432
                         7859
                     +  8147
                 -----------------
                       25438

Now we check our calculations by using digit addition method.

A) Digit addition of the Numbers = (9 + 4 + 3 + 2) + (7 + 8 + 5 + 9) + (8 + 1 + 4 + 7)
                                                         = (4 + 3 + 2) + (7 + 8 + 5) + (4 + 7)
                                                         = (9) + (20) + (11)
                                                         = (2 + 0) + (1 + 1)
                                                         = (2) + (2)
                                                         =  4
B) Digit addition of our result      = (2 + 5 + 4 + 3 + 8)
                                                         = (2 + 3 + 8)
                                                         = (5 + 8)
                                                         = (13)
                                                         = (1 + 3)
                                                         =  4
So our answer of addition is correct.
Example-02: Find the Multiplication of 279 and 563 
(Please Click on  "Simple Method of Multiplication" to learn a new method of multiplication, We will take the same example here to check the result)

A) Digit addition of the Numbers 
            = (2 + 7 + 9) x (5 +  6 + 3) 
            = (9) x (5)
            = (45)
            = (4 + 5)
            = (9)
B) Digit addition of our result
            = (1 + 5 + 7 + 0 + 7 + 7)
            = (6 + 21)
            = (6 + 3)
            = (9)

So we can say our multiplication is right.

For Subtraction also we need to transfer the negative term to another side only to check our result. 

For Example, If we would like to check the calculations for the subtraction A - B = C, then for checking purpose we need to take this subtraction as 
A = B + C .

The same thing is to be done for the division. (In the division, we need to transfer the division into the product form)

For Example, If we got the result as X / Y = Z, then we need to check our calculations for 
X = YZ.

Please study the above method & try your self by doing some more operations with the help of your parents. You can also ask me your query if any by filling the Entry form available at the top left the tab of this blog.

Enjoy Learning Mathematics by creating such type of interesting things.

Anil Satpute

40-Parenting-16 Encourage your Child to Save Papers

Click here for the next blog on parenting.

Now let us discuss Paper saving.

We are all aware of the Environment. It is necessary to maintain Oxygen Cycle. All Human beings (including Animals) require Oxygen. Trees and Plants breathe out Oxygen in the Daytime and the same Oxygen will be utilized by us. So we need to maintain the Oxygen Level. So cutting down Trees needs to be reduced, which will definitely maintain the Oxygen Level.

We know that for the production of paper, we need trees. This shows that for the production of Papers, we are reducing the oxygen level.

Now, it is time to reduce the maximum use of paper. I know that we can't stop using paper completely, but definitely, we can reduce using paper. For some work, definitely,  only paper copies are necessary. But, for some work, saving papers will be more beneficial. All such things should be told to our children so that they will be habitual for this task. Now we will see how paper saving is beneficial for us. Let us take the simple example of Telecom Services. Everyone uses the telephone, Mobile & so many communication services.  Every month we receive a bill for these services. The same bills can be asked to send through email in soft format. All of us are aware of Soft copies which are generally in PDF format and can be viewed on PC/laptop/tablet/mobile. By means of this soft copy, you can maintain the record of all such bills for years together.

In the same way, you can pay this bill online using Internet Banking or through a Credit Card of By Debit Card.  Which will save the paper used for Cheques (checks) and also the paper money say currency? By doing this we are also protecting our valuable time also. Definitely, such a process needs to be taught to our children so that they will definitely maintain the same procedure in their future life. Now if we are saying to do the same thing to our generation of people then there may be chances of getting the reply like this, "Who are you to teach us?" Or someone may say  " We can't use the internet".

But, our new generation will definitely use all such modern or better amenities only thing is that we need to mold them in such habits.

The above is one example given. , we can use this technique for Electricity Bills, Gas connection bills, or TV cable bills.

In the same way, maximum payments can be done through Credit Cards or Debit Cards. Even if purchase any item from Mall or any shops. If we avoid using cash transactions, we save paper currency.

Making a habit of such techniques will definitely improve our standard of living. So try to encourage you, child, to save paper means save our environment.

I want that the entire world needs to improve its thinking skills so that everyone can invent so many beautiful things about human life.

This is possible only with the help of you along with your child.

Please come forward and start this beautiful work for yourself and show the world that we can also build the world with a powerful thinking level as everyone is Proficient.

ANIL SATPUTE

39-Parenting-15 Prepare Your Child for Math Learning

Click here for the next blog on parenting.

Now we can plan for the new academic year for your child. After their examination, during this vacation, whenever your son/daughter gets time ask him/her to list out the important formulae (formulas) from their previous grade math class. Try to find some technique to remember them. I don't want to have the habit of mugging up technique. Which is not good at all to understand the basic concept of the subject. 

Do see, how we can plan our Math study for next academic year. As per as the subject Math is concerned, (You can definitely try this for other subjects also), we will make some folders for math and label them as follows 

1) School work: Ask your child to write everything that his/her Math teacher says. If any concept is not understood then, it should be cleared at the same time. Then systematically plan for the same day to revise what had been done in Math Class for that day. 
2) Question Set Prepared by Me: Ask your son/daughter to prepare 2 to 5 questions based on the topic and write them on a separate note sheet. These sheets should be numbered topic-wise. (Let us see one example of numbering the sheet. Let us see the example of giving the number for the question sheet prepared by the child on the topic Similarity, the number will be " My-Question/Similarity/7th-Grade/20032013 (today's date)/001 " and so on). This habit will improve his/her ability to maintain or manage proper planning for everything in the future.
3) Happy work: We all know that for Math, we need to practice more to understand the concept. In the school, the Math teacher may allow some work for your child to do at home. Generally, it is called "Home Work " I had given the name " Home Work " as "Happy Work "  So we are labeling this folder as " Happy Work ". We must tell our child to perform their Happy Work within the Time frame for it & all such sheets/notebooks should be kept in the folder labeled " Happy Work ". Every time we need to watch the completion of the work allotted to them by their teacher.
4) My achievements: We know that Every Child is Brilliant, so definitely he/she may have some creative ideas. Those ideas must be written on the sheet and must be maintained properly in some folder labeled " My Achievements "
5) My Doubts: Prepare shortlisted doubts on some sheet and remember that these doubts must be cleared out the very next day by their teachers.
6) Math Dictionary: Help your child to list out the important words of Math with their meanings/Definitions and note down these words in a separate notebook.
7) Math Formulae: Help your child to list out the important topic-wise formulae of Math and note down these formulae in a separate notebook.

In a very short period, we are starting the Basic Concepts of Math by taking some topics. 

I would like to clarify this point with you, I am not a great person. These are my views. You may know better than me. If you suggest me something more, I will definitely improve myself and improve my Blogs.

Click here for the next blog on parenting.

ANIL SATPUTE

38-Parenting-14 Become curious Parents

Click here for the next blog on parenting.

My elder son "Aaswad Anil Satpute " now studying Computer Engineering in Mumbai. I would like to share one of the stories of his childhood with you.

When he was in Sr. KG, He asked his one interesting question to his Mom.  The question was " What is the number between 2 and 3? "

His Mom was surprised!. She was well aware that in Sr. K KG studies, only the integers have been introduced No concept of decimals or fractions are taught.

Out the curiosity, She asked him, " Aaswad, why did you think about the number between 2 and 3. ?" 

On this question, he had given some explanation. He simply told something like this:

" Mom, see, we know that the double of 2 is 4 and the double of 3 is 6. there should be some number between 2 & 3 whose double is 5 as 5 is between 4 and 6."

After listening to his explanation my wife was surprised. She suddenly explained to him so many new concepts about numbers which were useful for his thinking about numbers.

He was very much curious about any type of machine, about the numbers and also about the computer. (Now also he is curious to learn so many new concepts)

In the 4th Grade, he found out so many new concepts about Mathematical Tables.

This story, I shared with you only to think about " How your child may think about all his/her surroundings". Better you give him/her a chance and allow him/her to think and express their views.

I surely say that everyone is equally brilliant. The only thing is that we need to give them some chance to grow & freely fly in the sky think more freely.

Note: I am a very simple and ordinary person. So again and again I say that we all together will build the entire world with high-level thinking. I will be very happy to listen from Students or Parents to improve my BLOG as well as My eBook "Miraculous World of Numbers". 

So come forward to help me in such a way that we all together will improve our beautiful world with powerful thinking levels.

Click here for the next blog on parenting.

ANIL SATPUTE

37-Easy Method For Creating Mathematical Table

Today I will publish my blog on the article of my younger son Master Abhishek Anil Satpute. Now he is studying in the 11th Grade of Science Stream. In 2008, when he was in 7th Grade (Standard), he found one interesting method: Mathematical Tables.  My elder son Master Aaswad Anil Satpute also found out so many interesting things in Mathematics since his 4th Grade (Standard). (Now he is in the last semester of Computer Engineering)

This Article by Abhishek Anil Satpute is written in his own way. I thank him for allowing me to publish his article.

Easy method for creating mathematical tables

Abhishek Anil Satpute

7th Grade (2008)

While studying the mathematical tables, I found one interesting method for creating 
the tables.

  A)  New Definitions:

  1) Small Friends: Two numbers having their addition as 5 are 
      known as Small Friends

  2) Normal Friends: Two numbers having their addition as 10 are
       known as Normal Friends.

  3) Big Friends: Two numbers having their addition as 15 are 
      known as Big Friends.

B) Table of friends:

	Small Friends	Normal Friends	Big Friends
1	4	9	14 NA
2	3	8	13 NA
3	2	7	12 NA
4	1	6	11 NA
5	0	5	10 NA
6	-	4	9
7	-	3	8
8	-	2	7
9	-	1	6

c) Procedure for creating unit-placed digits:

1)  Take the number of which the table is to create.                           (A)

2)  Write the unit digit of the double of the unit placed digit of (A).      (B)

3)  Write the friend number of the unit digit of (B) as given below.      (C)

Digit in the units place	Type of friends to be used
1, 5 or 7	Small Friends (05)
2, 4, 6 or 8	Normal Friends (10)
3 or 9	Big Friends (15)

4)   Write the friend number of the unit digit of (A) as given below. (D)

Digit in the units place	Type of friends to be used
1, 3, or 5	Small Friends (05)
2, 4, 6 or 8	Normal Friends (10)
7 or 9	Big Friends (15)

5) Write 5 if (A) is odd & 0 if (A) is even.

6) Write the normal friend number of the unit digit of (D).

7) Write the normal friend number of the unit digit of (C).

8) Write the normal friend number of the unit digit of (B).

9) Write the normal friend number of the unit digit of (A).

10) Write the 10th multiple of (A).

D)     Procedure for creating ten’s/hundred's placed digits:

Now we will write the ten’s placed digits of the table. We will follow a few simple 

Rules as given below.

1)  If ten’s placed digit of (A) is “0”, then while writing the table for (A), for Increasing 

the digit in unit place, increment the ten’s placed digit by “0” else by “1”

2) If ten’s placed digit of (A) is “1”, then while writing the table for (A), for Increasing 

a digit in unit place, increment the ten’s placed digit by “1” else by “2”

3) If ten’s placed digit of (A) is “2”, then while writing the table for (A), for Increasing 

a digit in unit place, increment the ten’s placed digit by “2” else by “3” and so on.

E)     Examples: 

1)  Table of 06 (A) Unit place is 6, so for the 3rd step take Normal Friends, for the 4th step take Normal Friends, and the ten’s placed digit is 0, so increment either by 0 or by 1.

01	Take the number 06 (A)	06
02	Double is 12, so take 2 (B). 2<6 so 0+1=1, so write 12.	12
03	Normal-friend of 2 is 8 (C) so take 8. 8>2 so 1+0=1, so write 18	18
04	Normal-friend of 6 is 4 (D), so take 4. 4<8 so 1+1=2, so write 24	24
05	6 (A) is Even so take 0, 0<4 so 2+1=3 so write 30	30
06	Normal-friend of (D) 4 is 6, so take 6, 6>0 so 3+0=3, so write 36	36
07	Normal-friend of (C) 8 is 2, so take 2, 2<6 so 3+1=4, so write 42	42
08	Normal-friend of (B) 2 is 8, so take 8, 8>6 so 4+0=4, so write 48	48
09	Normal-friend of (A) 6 is 4, so take 4, 4<6 so 4+1=5, so write 54	54
10	Always take 0, 0 < 4, so 5+1=6, so write 60	60

2)      Table of 27 (A): Unit's place is 7, so for the 3rd step take Small Friends, for 

     4th step takes Big Friends, and the ten’s placed digit is 2, so increment either 

       by 2 or by 3.

01	Take the number 27 (A)	27
02	Double is 14, so take 4 (B). 4<7 so 2+3=5, so write 54.	54
03	Small-friend of 4 is 1 (C) so take 1. 1<4 so 5+3=8, so write 81	81
04	Big friend of 7 is 8 (D), so take 8. 8>1 so 8+2=10, so write 108	108
05	7 (A) is Odd so take 5, 5<8 so 10+3=13 so write 135	135
06	Normal friend of 8 (D) is 2, so take 2, 2<5 so 13+3=16, so write 162	162
07	Normal friend of 1 (C) is 9, so take 9, 9>2 so 16+2=18, so write 189	189
08	Normal-friend of 4 (B) is 6, so take 6, 6<9 so 18+3=21, so write 216	216
09	Normal-friend of 7 (A) is 3, so take 3, 3<6 so 21+3=24, so write 243	243
10	Always take 0, 0<3, so 24+3=27, so write 270	270

Once again I thank Master Abhishek for his article.

Anil Satpute