# MY DIGITAL NOTEBOOK

A Personal Blog On Mathematical Sciences and Technology

Home » Articles posted by Gaurav Tiwari (Page 2)

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## Proofs of Irrationality

Tuesday, February 14th, 2012 19:18 / 3 Comments

“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 $\frac{a}{b}$ where $a$ and $b$ are both integers relatively prime to each other and $b$ being non-zero.
Following two statements are equivalent to the definition 1.
1. $x=\frac{a}{b}$ is rational if and only if $a$ and $b$ are integers relatively prime to each other and $b$ does not equal to zero.
2. $x=\frac{a}{b} \in \mathbb{Q} \iff \mathrm{g.c.d.} (a,b) =1, \ a \in \mathbb{Z}, \ b \in \mathbb{Z} \setminus \{0\}$.

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## Gamma Function

If we consider the integral $I =\displaystyle{\int_0^{\infty}} e^{-t} t^{a-1} \mathrm dt$ , 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 $t=0$ is a point of infinite discontinuity of the integrand, if $a<1$, where $a$ 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 $a$, as $\Gamma (a) =\displaystyle{\int_0^{\infty}} e^{-t} t^{a-1} \mathrm dt$ .

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## The Area of a Disk

Friday, January 27th, 2012 16:22 / 1 Comment

[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 $R$ is $\pi R^2$.

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.

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## Triangle Inequality

Friday, January 20th, 2012 10:57 / 5 Comments

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 $a$, $b$ and $c$ be the three sides of a triangle, then neither $a$ can be greater than $b+c$, nor$b$ can be greater than $c+a$, nor $c$ can be than $a+b$.

Triangle

Consider the triangle in the image, side $a$ shall be equal to the sum of other two sides $b$ and $c$, 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, $|x+y|$ represents the side $a$ if $|x|$ represents side $b$ and $|y|$ represents side $c$. A modulus is nothing, but the distance of a point on the number line from point zero.

Visual representation of Triangle inequality

For example, the distance of $5$ and $-5$ from $0$ on the initial line is $5$. So we may write that $|5|=|-5|=5$.

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…)

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## My Five Favs in Math Webcomics

Monday, January 9th, 2012 20:15 / 5 Comments

Cartoons and Comics are very useful in the process of explaining complicated topics, in a very light and humorous way. Like:

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## Progress Report of MY DIGITAL NOTEBOOK in 2011

The WordPress.com stats helper monkeys prepared a 2011 annual report for this blog.

Here’s an excerpt:

The concert hall at the Syndey Opera House holds 2,700 people. This blog was viewed about 53,000 times in 2011. If it were a concert at Sydney Opera House, it would take about 20 sold-out performances for that many people to see it.

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## On Ramanujan’s Nested Radicals

Friday, December 30th, 2011 08:50 / 8 Comments

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