Mother is God incarnated.....

Just in case, if you couldn’t meet God anywhere

Why don’t you go to your kitchen

Search for that woman who has dirt everywhere

Shy to come out and always forgotten

Unselfish to love and selfish to be loved

Shed you her body and blood

Fed you for your good

Fled from her home not by her choice

Led by philosophies that are not wise

She needs not your mercy but your prayer

Believe then that God is everywhere

Confirm that God is not a piece of a stone

Affirm that God is flesh that can be seen

Love and Hatred

I love you when you integrate me
But for your partitioning of me finer and finer.
I hate you for differentiating me
Though you wait for my limits.

Sum of the Squares of the Degrees

One of the most fundamental results in graphs is that the sum of the degrees of all the vertices of a graph is twice the number of edges in it.

This is is the famous Handshaking Theorem.
It is called Handshaking Theorem due to the analogy of the involvement of two hands in every handshake with that of the two vertices in every relation (edge).

Hence, the total number of relations of the persons (degrees of  the vertices) in a graph is double the number of relations (edges) in it.

What happens if we add the squares of the degrees of all the vertices in a simple graph?

It is found out that the sum of the squares of all vertices in a graph is the the sum of the degrees of the adjacent vertices of all the vertices the graph it self.

Why is it so? It is because, the square of the degree of a vertex is same as adding the degree of a vertex, the degree number of times.
                                                         n²=n+n+n+...+n (n times)
But one can easily verify that every vertex has exactly its degree number of adjacent vertices. Hence, when we sum the degrees of  the adjacent vertices of all the vertices of a graph, every degree is added that degree number of times. This proves the result.

We can also observe that if we add all the degrees and degree squares of all the vertices of a graph, then it is same as summing up all the degrees of all vertices in the closed neighbourhood of all the vertices of that graph.

The First Twenty Five Primes

Every prime number is a positive integer that is divisible by 1 and itself only. There are infinitely many prime numbers. Prime numbers are peculiar numbers which do not have a common predictable property other than that of the indivisibility. Because they are indivisible, the fundamental theorem of arithmetic is that they are the building blocks of numbers. That is to say that any positive integer greater than 1 can be written uniquely as the product of prime numbers.

There are 25 primes below 100. This is a method to remember all the first 25 primes.
We begin with 25 itself.

The first and third primes are 2 and 5 respectively. They are prime numbers but no other prime number would have 2 and 5 as its end digits
Now, 5+2 and 5-2 give the other two single digits primes. i.e., 7 and 3.
Hence, we have all the single digit primes as 2, 3, 5 and 7.

All primes with at least two digits would end only in 1, 3, 7 or 9.

We list all the two digit primes that end in 1.

Wen use 2 and 5 once again for that.
Splitting 2, we get the first two digit prime, 11.
Then, 1+2=3, 1+3=4, 1+5=6 and 1+6=7 are the first digits of the remaining primes.
Hence, the two digit primes are 11, 31, 41, 61 and 71. Thus we get all primes that end in 1.

A total of 9 primes, we have found out so far.

Now, we need to list all the primes that end in 3.

Adding the digits of the primes that end in 1, we get 1+1=2, 3+1=4, 4+1=5, 6+1=7 and 7+1=8.
Appending 3 to the right of the above numbers, we get the two digit primes that end in 3.
They are 23, 43, 53, 73 and 83. Hence, excluding 3, we got 13 primes. Of course, 13 is also a prime number.

What left are the primes that end in 7 or 9. But, this is very easy.

List all the digits in numbers from 1 to 10.
They are 1, 2, 3, 4, 5, 6, 7, 8, 9, 1 and 3.
Appending 7 or 9 as the case may be(by avoiding the trivial composite numbers we get,
17, 29, 37, 47, 59, 67, 79, 89, 97, 19 and 07.
These are the 11 prime numbers that end in 7 or 9.

Hence all the prime numbers below 100 are:

1. 2 and 5
2. 11, 31, 41, 61 and 71
3. 3, 13, 23, 43, 53, 73 and 83.
4. 7, 17, 37, 47, 67 and 97
5. 19, 29, 59, 79 and 89.

Math, Math, Math

Need math?
Dude, no path.
Fills us with wrath
Swells us with wreath
Oh! Not to grave
But to grow.
Math, math, math,
Mother of truth
Author of faith
Dude, it's the path.
In depth of doubt
Math keeps us straight.
Work math at home
Strike with warm chime.

Determining the Determinant of a 3x3 matrix

Introduction

Determinant is a unique value representing a square matrix. It is useful in determining the inverse of a matrix. The determinant of a 1x1 matrix is the entry in the matrix itself whereas in the case of a 2x2 matrix, it is the upper left into lower right minus the upper right into the lower left. The determinant thus obtained is called the minor of the 2x2 matrix. Conventionally, the determinant of a higher order matrix is calculated by a recursive method with the help of the minors. Let us now see a method of rotations to find the determinant. 

Let us consider the 2x2 matrix given on the left. Its determinant as per the rule given above is AD-BC. 

However, look at the diagram given right. It can be explained as follows.

Rotation Method

Rotate left the second row to one position and then multiply the entries in the respective columns. Subtract the second column product from that of the first.

This process can be extended into higher orders also.  This is done by carefully completing all the rotations depending on the number of rows. Let us see the case of a 3x3 matrix.

At first, rotate left the second row to one position and third row to two positions. The matrix thus obtained is given on the right.

Multiply the values in each column separately and then add the three products.

Thus we get A= (A1.B2.C3)+(A2.B3.C1)+(A3.B1.C2).

Now we rotate left the second row to one more position and the third row to two more positions. The matrix thus obtained is as follows.

Multiply the values in each column separately and then add the three products. Thus we get B=(A1.B3.C2)+(A2.B1.C3)+(A3.B2.C1).

The determinant of the 3x3 matrix is then

A-B = (A1.B2.C3+A2.B3.C1+A3.B1.C2)-( A1.B3.C2+A2.B1.C3+A3.B2.C1).

Cylindrical Rotation Method

This process is better understood if we can represent the matrix cylindrically. Consider a cylinder that has three horizontal sections which can be rotated freely with respect to a central vertical axis. Entries in the matrix are given on the exterior of the cylinder.

After the first set of two types of left rotations, the cylinder looks like what is on the left.

Multiplying the entries in the columns separately we get

A=(A1.B2.C3)+(A2.B3.C1)+(A3.B1.C2).

After the second set of two types of left rotations, the cylinder looks like what is on the right.

Multiplying the entries in the columns separately we get

B = (A1.B3.C2)+(A2.B1.C3)+(A3.B2.C1).

The determinant of the 3x3 matrix is then

A-B = (A1.B2.C3+A2.B3.C1+A3.B1.C2)-( A1.B3.C2+A2.B1.C3+A3.B2.C1).

Palm Method

This can be visualized in yet another way also.

Let us use the three central fingers on the left palm to represent a 3x3 matrix.

Instead of the first set of rotations, multiply the entries from left as indicated by the dark lines, starting from the diagonal to get the value A.

To get the value B, we can multiply the entries from right starting from the antidiagonal. This can be done also by turning the palm and then multiplying the entries from left as given below from the diagonal.

Conclusion

An advantage of the above mentioned process is the elimination of the repeated use of plus(+) and minus(–) which is sometimes disturbing for beginners and non-Mathematics students. Can this method be extended to higher order matrices?

Before I became a Teacher

Before I became a Teacher never I knew

That sleeping in the class disturbs

The teaching of the teacher.

That talking in the class makes

The teacher shouting in the class.

That not scribbling in the class augments

The degrading of the student.

That correcting the answer scripts is tougher

Than attempting all the questions.

That questioning the answers is harder

Than answering the questions.

That coming late for the class is worse

Than not attending the class.

That absenting in the class is worse

Than not joining for the course.

That photocopying is the greatest insult

That a teacher can ever bear.

That helping a fellow student learn is

The second greatest insult to a teacher.

That teaching is the noblest profession

When all your students become teachers.