Graph Theory

In Graph Theory, a graph is an accumulation of dots that may or may not be joined to  each  other by lines. It doesn't amount how big the dots are, how long the curves are, or whether the curve are straight, curved, or squiggly. The "dots" needs not to be round! 

All that matters is which dots are joined by which lines.

Two dots can alone be affiliated by one line. If two dots are joined by a line, it's not "legal" to draw another line joining them, even if that line stretches far  from the first one
 Graph theory applications: Graph theory  plays an important role in the design, analysis, and testing of computer programs. It is important for the fact that flow of control and flow of data for any program can be expressed in terms of directed graphs.

Geometry constructions

Construction of geometry involves drawing of shapes.In construction, we use compass and scale. 

Geometry construction rotation: Circling is the movement (turning) of an object about a point through a given  number of degrees in a clockwise or an anticlockwise. A two-dimensional object rotates in the adjustment of a center most or position of rotation.

See the link for Geometry Construction Answers.

Example of construction of geometry:

Construction of parallelogram type of quadrilateral is AB = 11 cm, BC = 10 cm and ÐA = 60o

Solution:

First we can draw AB - 11 cm and make  ÐA = 60o

Then cut the line AD - 10 cm ( accordingly AD = BC)With B as center and a radius 10 cm, draw one arc and with D as center and radius11 cm cut that arc at C.

Join the line BC and DC.Then ABCD is the construction of appropriate parallelogram.

Matrices

What are matrices?
    A rectangular array of entries is called a Matrix. The entries may be real, complex or functions. The entries are also called as the elements of the matrix. The rectangular arrangement of entries are enclosed by bracket or by square bracket.

You can also check the link on matrices determinants and determinants of matrices.

There are different types of matrices -  Row Matrix, Column Matrix, Square Matrix, Diagonal Matrix, Scalar Matrix, Identity or Unit Matrix, Null Matrix or Zero Matrix

Operation on matrices:
Equality of Matrices, Addition of Matrices, Matrix Addition is commutative, Matrix addition is associative, Subtraction of Matrices, Multiplication of a matrix by a scalar, Multiplication of Matrices, Properties of Matrix Multiplication, Transpose of a Matrix, Properties of Transpose, Symmetric Matrix, Skew-Symmetric Matrix, Properties of Symmetric and Skew Symmetric Matrices,

Application of matrices:Matrices are applied on Homogeneous Equations (Constant = 0), Non Homogenous Equations.

Mathematical Induction

Mathematical Induction:It is a technique for proving a statement, a theorem, or a formula that is asserted about every natural number.

Before we take an example on mathematical induction I would also like to provide you a link on Mathematical reasoning. 

Example of mathematical induction:

Mathematical induction is used to prove that the following statement holds for all natural numbers n.

Let us call the statement as P(n)

Show that the statement holds for n = 0.
P(0) amounts to the statement:

On the left-hand side of the equation, 0 is the only term, and so the left-hand side is equal to 0.
On the right-hand side of the equation, 0·(0 + 1)/2 = 0.
The two sides are equal, so the statement is true for n = 0. So, it has been shown that P(0) holds.

Inductive step: Prove that  if P(n) holds, then  P(n + 1) also holds.

Imagine P(n) holds (for some unspecific value of n). It must  be proved that P(n + 1) holds, i.e,

According to the induction hypothesis that P(n) holds, we can rewrite  the left-hand side as follows:

Algebraically:

Hence P(n + 1) holds.

We have proved both the  basis and the inductive step, therefore,it has been proved by mathematical induction that P(n) holds for all natural n

Prime number theorem

Statement of the prime number theory:

Let π(x) be the prime - counting function that gives the cardinal of primes which is less than or equal to x, for any absolute cardinal x. For example, π(10) = 4 because 2, 3, 5 and 7 are prime numbers which are less than or equal to 10.

According to prime number theorem,the limit of the quotient of the two functions π(x) and x / ln(x) as x approaches infinity is 1, which is expressed by the formula:

Using asymptotic notation the above formula can be written as

 

The theorem states that x/ln(x) approximates π(x) which means that the relative error of this approximation approaches 0 as x approaches infinity.

See the link for Prime and composite numbers

An example of the problem on prime number theory is given below: 

 

Solution:

 

Parallel lines

Definition of parallel lines: Two lines are said to be parallel if they never meet each other.Two parallel lines are always apart by the same distance.

There is a link provided below where you will learn in detail about parallel lines:parallel perpendicular lines

The diagram of of parallel lines are shown below:

Properties of parallel lines:
1)When another line is drawn through two or more parallel lines, The resulting angles are same.
2)The sum of the two adjacent angles is 180 degree

I am giving a link on Pair of straight lines as well.

Next time we will learn the above properties with the help of diagram by giving detail explanation.      

Binomial theorem

Binomial Theorem:

According to Binomial theorem, it is possible to expand any power of x + y into a sum of the for

where indicates the corresponding binomial coefficient.By using the summation notation ,the above formula can be written as

A variant of the binomial formula is obtained by substituting  1 for x and x for y, so that it involves only a single variable. In this form, the formula reads

See the given link on how to solve problem on binomial theorem.I hope this blog on the binomial theorem will give you the basic knowledge to solve problem on binomial problems