“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…)