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Holi, the festival of colors, was celebrated this year on 27th and 28th of March all over India. I decided to move to my own village, Kasturwa and then to Surya’s Village, Shiv Patti, on this occasion. Here are some images from the events taken with my Nokia device, which I thought were worth sharing.
ENJOY READING! Err… Watching.
The greatest number theorist in mathematical universe, Leonhard Euler had discovered some formulas and relations in number theory, which were based on practices and were correct to limited extent. The prime generating equation by Euler is a binomial which is actually very specific and yields more primes than any other relations out there in number theory. Euler told that the equation yields many prime numbers with the values of x being input from x=0 to x=k-2; k being a prime.
Let’s see how many primes we can get by using different values of k and x:
|Serial Number||Value of k (prime)||Value of x (from x=0 to x=k-2)||Value of f(x)=(x^2+x)+k||Not a Prime?|
The above table yields many prime numbers, which again can be put at the place of k and so on the table can be progressed.
According to Euler, 41 was the most appropriate value of k yielding more prime numbers than any other k. In the list below, each value of f(x) is a prime for k=41:
So, the Euler’s Prime Generating Equation can be written as
; where x is an integer ranging from 0 to 39.
Wait. What if we increase the value of x beyond the limit of 39? What will we get?
The next values of f(x) in this series would be 1681, 1763, 1847, 1933, 2021, 2111, 2203, 2297, 2393, … .
Are all these prime numbers too? The answer is no. 1681 is not a prime number, neither are 1763 and 2021. Though all others are prime numbers.
“Irrational numbers are those real numbers which are not rational numbers!”
Def.1: Rational Number
A rational number is a real number which can be expressed in the form of where and are both integers relatively prime to each other and being non-zero.
Following two statements are equivalent to the definition 1.
1. is rational if and only if and are integers relatively prime to each other and does not equal to zero.
If we consider the integral , it is once seen to be an infinite and improper integral. This integral is infinite because the upper limit of integration is infinite and it is improper because is a point of infinite discontinuity of the integrand, if , where is either real number or real part of a complex number. This integral is known as Euler’s Integral. This is of a great importance in mathematical analysis and calculus. The result, i.e., integral, is defined as a new function of real number , as .
[This post is under review.]
If you are aware of elementary facts of geometry, then you might know that the area of a disk with radius is .
The radius is actually the measure(length) of a line joining the center of disk and any point on the circumference of the disk or any other circular lamina. Radius for a disk is always same, irrespective of the location of point at circumference to which you are joining the center of disk. The area of disk is defined as the ‘measure of surface‘ surrounded by the round edge (circumference) of the disk.
The area of a disk can be derived by breaking it into a number of identical parts of disk as units — calculating their areas and summing them up till disk is reformed. There are many ways to imagine a unit of disk. We can imagine the disk to be made up of several concentric very thin rings increasing in radius from zero to the radius of disc. In this method we can take an arbitrary ring, calculate its area and then in similar manner, induce areas of other rings -sum them till whole disk is obtained. (more…)
Triangle inequality has its name on a geometrical fact that the length of one side of a triangle can never be greater than the sum of the lengths of other two sides of the triangle. If , and be the three sides of a triangle, then neither can be greater than , nor can be greater than , nor can be than .
Consider the triangle in the image, side shall be equal to the sum of other two sides and , only if the triangle behaves like a straight line. Thinking practically, one can say that one side is formed by joining the end points of two other sides.
In modulus form, represents the side if represents side and represents side . A modulus is nothing, but the distance of a point on the number line from point zero.
For example, the distance of and from on the initial line is . So we may write that .
Triangle inequalities are not only valid for real numbers but also for complex numbers, vectors and in Euclidean spaces. In this article, I shall discuss them separately. (more…)