Triangles and Congruency

A
Triangle
is a closed
figure formed
by three intersecting
lines. Many shapes around
us are triangular signals and
signs. A triangle will need three
to four measurements to be determined,
Try to draw it where just two angles or two
lines are defined. It is the first closed geometry
having an odd number of sides, Equality of the area 

of a circle and a triangle is impossible to decide. The shape
is based on visual and measurement referring to size, Congruent
triangles are identical both in shape and size. Congruency is a
method for the comparison of triangles. Corresponding parts are equal
in congruent triangles, Five congruent rules prove the congruency of the
triangle, and Side-Side-Side correspondence is it is initial. Other congruency rules
are SAS, ASA, AAS, and RHS, It is not sufficient to prove by SSA or ASS. 🔺🔻🔺

Central tendency

Center of the mass of an object,
A point having the hypothetical prospect.

Uniform density refer it at the centroid,
In another case, it may be outside.

We analyze data by its central vacancy,
Mean is the primitive central tendency.

Mean involves every measurement,
Median and mode are its replacement.

Zero is the sum of deviation from the mean,
Its validity for other measures is unseen.

Symmetric data have unique code,
Equal mean, median, and mode.

Constant time data implies constant time mean,
AM and GM are tied with Φ(phi)of your screen.


A song on this lyrics is available on https://youtu.be/nVmAac8NTXM

Cryptography

Hard to enjoy the secrecy of secret practice,
Philosophy of God lying behind this.

Everyone knows that God knows everything,
Cryptology protects the right to privacy of human beings.

Alice and Bob are fictions in its literature,
Secrecy depends on the various parameter.

How to lock your secrets?
Digitization removing its grievances.

Mathematics is the strength of cryptography,
DLP and integer factorization are its analogy.

World wars had promoted progress in cryptography,

Enigma is crucial in Turing's biography.

Claude Shannon's mathematical theory of cryptography
Made him the father of mathematical cryptography.

A cipher hides all information perfectly,
Iff message size is equal to the size of the key.

Simplex Method

LPP has target to optimize its objective,
Its requirements are some linear relationship.

Linear relationship are in unbalanced condition,
Slacked and surplus variables balanced the equation.

System of linear inequality defines a polytope,
Moving along its edge  is our hope.

Algorithm is to move from start vertex to optimum one,
Process is finished when isn't possible to improve the solution

Simplex method is for solving problems in Linear programming,

Its inventor name is George Dantzing.

Dantzing solved two open problems in statistical theory
Motivated from Prof. Neyman's inquiry.

Carmichael Number

Primes are those which pass AKS primality test,

For pseudo-primes Carmichael raised the zest.

Integer factorization have some fundamental association,

Any prime factor of a Carmichael number have no repetition.

The search for pseudo-primes have various algorithms,

Because of RSA, Carmichael numbers obtain spatial attention.

It has been proved that all Carmichael numbers are odd,

The congruence relation makes this picture broad.

Alford et al. proved they are infinitely many ones,

The finding of smallest such number was a milestone.

At least three primes are always their factors,

The third Carmichael number is Hardy-Ramanujan number.

Their intimacy with prime is wondering Cryptographers,

Unlocking their secrets can open new prime chapters.

Circle

Circle attracts because of its symmetrical joint,

It is equidistant from a fixed point.

Euclid's Element was the finest creation of geometry,

A line passing through the centre is called the line of symmetry.

The width of the circle is called its diameter,

Half of the diameter is the radius, which is a useful parameter.

A circle divides its plane into three parts,

Drawing it from three points is an interesting start.

Try to fix the ratio of the area of a circle to a radius square,

The ratio of perimeter and diameter is also our ancient desire.

A tangent touches the circle at a unique point,

The radius and tangent perpendicularly meet at this unique joint.

Tangents drawn from an external point of the circle are equal,

They are adjacent sides of a cyclic quadrilateral.

To fit a circle in a square is an ancient quadrature problem,

It shows π irrationality and is useful in many theorems.

The chord and curve of the circle cover the segment area,

The radius and curve of a circle decide the sector's criteria.

Real Number

Division of integers is an early math problem,

We can solve it by Euclid division algorithm.

Euclid division algorithm computes highest common factor,

In this we apply Euclid division lemma till we get no remainder.

Factorization of integer remain a challenge for mathematicians,

A composite number has always a unique factorization.

Fundamental Theorem of Arithmetic is crucial for integers,

In Euclid's Elements, it was recorded earlier.

Prime factorization is useful to find LCM and HCF ,

HCF times LCM of two numbers is the product of numbers themselves.

Locating irrationals on number line show beautiful geometry,

For a prime p, It is interesting to prove square root p irrationality.

Irrational numbers characterize by its decimal expansion,

Decimal expansion of rational terminates or non-terminating repetition.

Cantor worked on the enumeration of real and rational numbers,

Combinatorics has so many reward-able conjectures.

Heron’s Formula

Area of a triangle is half of the base multiplied by height,
Calculating height in every triangular shape is difficult to fight.


Heron's was a famous mathematician from Egypt,
Many results of mensuration written in his manuscript.


He formulated sides of the triangle with its area,
Verify this from the Pythagoras criteria.


Archimedes might know this over two centuries earlier,
In some cases, Brahmagupta and Bretschneider raised the portiere.


Heron gave approximation rule of the square root of a number,
Babylonian kept this method with another nomenclature.