# Refreshing MY DIGITAL NOTEBOOK

This September, MY DIGITAL NOTEBOOK will be exactly three years old. On this golden occasion, I’ve planned something big for this almost dead blog – in form of some tweaks and updates.  Here is the list of major changes you’ll be noticing in upcoming days: Continue reading

# Photo Archives from ‘Only Gaurav’

Only Gauravwas a photo-blog, I operated parallel to MY DIGITAL NOTEBOOK. Both sites worked well together until February 2012. After February 2012, I got busy in college stuffs and my blogging life took a huge leap of one year. MY DIGITAL NOTEBOOK was able to survive this gap and now it is continuously getting updated. On the other hand, ‘Only Gaurav‘ wasn’t that lucky. It was neglected due to several reasons. Starting from January 2013, I’m regularly using  Flickr for storing and sharing media, so there seems no chance that I’ll be posting on ‘Only Gaurav‘. Who knows if the blog itself will be wiped-out in nearby future. This gallery is a small collection of some memorable shots which I don’t want to lose any way. Continue reading

# Real Sequences

## Sequence of real numbers

A sequence of real numbers (or a real sequence) is defined as a function $f: \mathbb{N} \to \mathbb{R}$ , where $\mathbb{N}$ is the set of natural numbers and $\mathbb{R}$ is the set of real numbers. Thus, $f(n)=r_n, \ n \in \mathbb{N}, \ r_n \in \mathbb{R}$ is a function which produces a sequence of real numbers $r_n$ . It’s customary to write a sequence as form of functions in brackets, e.g.; $\langle f(n) \rangle$ , $\{ f(n) \}$ . We can alternatively represent a sequence as the function with natural numbers as subscripts, e.g., $\langle f_n \rangle$ , $\{ f_n \}$ . This alternate method is a better representation of a sequence as it distinguishes ‘a sequence’ from ‘a function’. We shall use $\langle f_n \rangle$ notation and when writen $\langle f_n \rangle$ , we mean $\langle f_1, f_2, f_3, \ldots, f_n, \ldots \rangle$ a sequence with infinitely many terms. Since all of $\{ f_1, f_2, f_3, \ldots, f_n, \ldots \}$ are real numbers, this kind of sequence is called a sequence of real numbers.

# Smart Fallacies: i squared equals to 1, 1 equals to 2 and 3

This mathematical fallacy is due to a simple simple assumption, that $-1=\dfrac{-1}{1}=\dfrac{1}{-1}$ .

Proceeding with $\dfrac{-1}{1}=\dfrac{1}{-1}$ and taking square-roots of both sides, we get:

# Set Theory, Functions and Real Number System

## SETS

In mathematics, Set is a well defined collection of distinct objects. The theory of Set as a mathematical discipline rose up with George Cantor, German mathematician, when he was working on some problems in Trigonometric series and series of real numbers, after he recognized the importance of some distinct collections and intervals.

# 0.999…=1?

There is a clinical fallacy which proves 0.999… =1.

First we assume the repeating decimal

n=0.9999…                      …(i)

Then, 10n= 9.9999…                      …(ii)

Subtracting equation (i) from equation (ii) we get

# Four way valid expression

People really like to twist the numbers and digits bringing fun into life. For example, someone asks, “how much is two and two?” : the answer should be four according to basic (decimal based) arithmetic. But the same  with base three (in ternary number system) equals to 11. Two and Two also equals to Twenty Two. Similarly there are many ways you can add them and get different results.

# Bedtime Analysis Stories

There is a collection of 101 (actual there is only one) humorous bedtime stories on Analysis at Collin Macdonald‘s website, which Sunshine Dubois and he wrote years ago. The story is really cool, language is easy (relatively less mathematical)  such that even a undergraduate student with basic analysis background can easily understand it. Continue reading

# Interesting Egyptian Fraction Problem

Here is an interesting mathematical puzzle alike problem involving the use of Egyptian fractions, whose solution sufficiently uses the basic algebra.

## Problem

Let a, b, c, d and e be five non-zero complex numbers, and;