Infinities!

"There are more real numbers between 0 and 1 than all the integers"

Ever since I came across the statement that the set of real numbers between 0 and 1 (eg: 0, 0.1, 0.2) is larger than the set of integers (0, 1, 2), I was curious to understand why it is so.

To understand this, I was reading about "Cantor's Diagonal Argument" here and here.

In very simple terms, these articles provide the below explanation.

Every integer is tried to be mapped to every real number with one-to-one correspondence.
Eg:
   1 --> 0.1
   2 --> 0.2
   3 --> 0.3
   and so on... till infinity.

The above gives an impression that we should always be able to easily map an integer to a real number as the left-side (real numbers) is simply right-side multiplied by 10. In other words, multiply real number by 10 and you should get the integer number which it maps to. For 0.11, multiply by 100 and so on. You get one-to-one correspondence. Then how can there be more real numbers between 0 and 1 as described in the first statement?!

This one-to-one correspondence is only true for:
1. real numbers which are not having repeating decimal (eg. 0.1111..., 0.123123123...) or
2. if they are not irrational numbers (eg. pi=3.1415..., e, square root of 2, 0.1234...)
And why is that? It is because of the definition of "Integer", which states that it has to be a complete number. It cannot be incomplete and hence, has to be well-defined. But, real numbers do not have this limitation and they can have an infinite representation even in a single real number. Eg: Real number 0.333... has infinite 3s after the decimal point. (But, it can be also be easily represented as a fraction: 1/3). Similarly, we have pi, which is an irrational number (which cannot be represented by fraction of whole numbers).

So, with our earlier logic of multiplying an integer by 10, we will not be able to map an integer to the real number represented by 0.333... as there is no integer called 333... We have 333 and 3333 and 33333, and this can continue till infinity. But, all these numbers would always be complete in themselves as per the definition of "integer".

What this means is that, we cannot have an integer to represent irrational numbers, repeating decimals and real numbers which having a representation which in itself is infinite (eg: 0.1234...). This breaks the one-to-one correspondence and shows that there should be more real numbers between 0 and 1 than all the integers.

Also, the multiplication logic for mapping may not work to begin with as we will not know what power of 10 to use to multiply as a single real number's representation can be infinite (Eg: 0.123456789101112..., 0.111...).

There can be infinite such real numbers which cannot be mapped to integers. The integers would get exhausted before being mapped to these real numbers.

Hence, "There are more real numbers between 0 and 1 than all the integers".

The same can be extrapolated for:
"There are more real numbers between 0.1 and 0.2 than all the integers"
"There are more real numbers between 0.01 and 0.02 than all the integers"
"There are more real numbers between 0.001 and 0.002 than all the integers"
... and so on.

So, the infinity in integers isn't very big at all. In fact, it is stated that the smallest ordinal infinity is that of the (postive/negative) integers, and any set which has the cardinality of the integers is countably infinite. Read more about Infinity.


Additional points to ponder upon:

Pi
If pi (3.1415..) is irrational, then we can never have a circle which has both diameter and circumference) as a rational number. This is because: pi is defined as ratio of Circumference of a circle to its diameter.
If C is circumference of the circle and d is diameter. Then, pi = C/d.
Now, since pi is proved to be irrational, by definition, it cannot be represented as a fraction of integers. Hence, either C or d has to be irrational.


Logarithm

If, aⁿ = b  -->  a to the power n = b

Then, log_a b = n  -->  log b to the base a = n
   Now, if a=1
then, 1ⁿ = 1, for all n
=> b=1

Hence, log₁ 1 = Any value of n. Hence, undefined.

Conclusion: Logarithm to the base 1 is undefined!

Windows recursive copy and cksum

Option 1  [Recommended – as it gives output in a consistent format of cksum, filesize, filename without path]

USING rcopy to copy all files into a single directory and then cksum

1. Open notepad and copy-paste the below rcopy part into it.
2. Save file as “rcopy_cmd.cmd”
3. To copy all files to a single directory, execute the following command:  rcopy_cmd.cmd [SVN_LATEST_FILE_PATH_ON_LOCAL_MACHINE]
4. Finally, run the command from “D:\TESTCOPY” directory: [CKSUM.EXE_PATH]\cksum.exe * > cksum_off.txt
5. Zip the cksum_off.txt and email it.

       Note:  

         cksum.exe can be from http://gnuwin32.sourceforge.net/packages/coreutils.htm

         Do not forget the dependencies (libintl3.dll, libiconv2.dll) available on same site.

      Standalone recursive copy command execution

rcopy

==START==

@REM  Batch file to recursively get copy files to "D:\TESTCOPY" in Windows

@REM  Usage: rcopy_cmd.cmd [DIR]

@echo off

echo START: %date% %time%

@REM  Go to input DIR

pushd "%1"

@REM  For every file in current dir do (...) [/r = Recursively for every directory]

for /r %%f in (*) do (

  copy "%%f" "D:\TESTCOPY"

  )

@REM  Get back to present working directory

popd

echo END: %date% %time%

@echo on

==END==

cksum.exe "D:\TESTCOPY" > cksum_out.txt

Option 2

      Standalone recursive cksum command execution

cksum_cmd

==START==

@REM  Batch file to recursively get cksum in Windows

@REM  Usage: cksum_cmd.cmd [DIR] > cksum_out.txt

@echo off

echo START: %date% %time%

@REM  Go to input DIR

pushd "%1"

@REM  For every file in current dir do (...) [/r = Recursively for every directory]

for /r %%f in (*) do (

  D:\work\bin\cksum.exe "%%f"

  )

@REM  Get back to present working directory

popd

echo END: %date% %time%

@echo on

==END==

EXTRA

Process to copy all files stated in one list file to one directory.

@REM Create the directory to which the files are to be copied to

mkdir D:\TESTCPY2

@REM Assuming all files are present in one directory called “All_Files”

@REM Read one line at a time from the list file

for /F "tokens=*" %L in (list_of_files.txt) do (

  copy All_Files\%L D:\TESTCPY2

)

Note:  To use the above command in a batch file, change %L to %%L

 

Free Will versus Destiny

DESTINY

Whatever has happened, whatever is happening and whatever will happen has already been decided. So, whatever one does or does not do is no longer a choice and hence, it is destiny.

Let’s say that the past, present and future information is written by a creator in a book called Predictions. But this book can never be accessed by anyone other the creator. Neither is the creator accessible to anyone.

In this case, there is a book which has all actions written. But, since the book is not accessible, no one other than the book’s creator knows about it. Hence, whatever one does, even thought already predicted, is not known and hence, it is free will.

FREE WILL

At any point in time one can choose what to do or what not to do. This is completely up to the person and the circumstances at the time of action or inaction. Hence, it is free will.

Let’s say that all that one has done and one is doing is being written in a book called Predictions by a creator. But, neither the creator not the book is accessible to anyone. The creator also writes future actions and inaction into this book as this book has no bearing on anyone as it is anyways inaccessible.

As the book is always inaccessible, one can say that whatever one does or does not do is already predicted in the book. Hence, it is destiny.

 Is free will the same as destiny?!

Convex Universe

What if the universe is convex on a grand scale?

Would it explain the acceleration of galaxies moving away from each other?

On a small scale, the universe would behave like a normal cube, so any matter would bend the space to some extent causing gravity. But, on a large scale, the convex universe would be pushing things away from each other and the farther you go, the more convex it would be.

A simpler example to understand this would be to consider the planet Earth. When we consider a small area, say 10 square meters, we perceive it to be flat and, in earlier times, that could have been one of the main reasons to believe that the Earth was flat. But, with some techniques and better understanding we now believe that the Earth is spherical (geoid) in shape. When we look at it from a larger perspective (not just the few square meters), we perceive the Earth as a sphere.

Similarly, when we look at the universe on a smaller scale (say, few thousands of light years), we perceive it to be flat... but, it may well be bent as a convex.

Now, back to pondering whether the universe is finite in space (having a boundary) or is infinite (unbound).

Limitations!

We seem to be bound by limitations and there may be nothing that we can do about it!

Some of the limitations are as follows:

1. Maximum Speed = c (Speed of light in vacuum = 299,792,458 m/s)
2. Minimum Temperature = 0 K (Absolute zero = -273.15 °C)
3. Minimum Energy = h (Planck's constant = 6.62606896(33) x 10^-34 m² kg / s)


I will keep updating this list as and when I come across new limitations. If you are aware of any other limitations, please let me know so that I can add it to this list.

Useful Inventions - Yet to be invented!

A list of inventions which may be very useful, but not yet invented... (apart from the cliched Time Travel Machine, Holodeck, Force Fields / Shields)

1. Instant Disc Reader 

Description:
It would be good to have a small camera/gun type equipment for reading the entire content of the disc (CD, DVD, Blu Ray, HD-DVD) at one shot, just by taking snaps of the disc.

Plan: 
The reader machine would emit light/sound waves and depending on the echo received would be able to decipher the binary data on the disc.


2. Click Photo and Translate Text 

Description:
Imagine that you are in a land which has lot of directions, text, messages in a language that you don't understand. Wouldn't it be great if you could take a photo of the text using your mobile phone and get the text translated.

Plan: 
a. Using your mobile phone take photo of the section to be translated
b. Repeat taking of photo, as required, for clarity and correct focus.
c. Send photo to service provider ‘A’ for OCR (Optical Character Recognition)
d. Receive text (to be translated) from service provider ‘A’
e. Send this text to service provider ‘B’ for translation
f. Receive translated text from service provider ‘B’
g. All above steps should be seamless in one go.

Service providers 'A' and 'B' can be the same company if possible. But, I have kept them separate to utilize the best service providers for the technology and functionality required.

And now ... I enter the world of Blogging

This is my first blog and I am not sure as to what a blog should contain. I have not been through a lot of other blogs. So, as I usually do, I will carry out blogging in my own way and hope for it to be a success. Anyway, I feel there's no right or wrong way of posting blogs. So, I will be mostly be posting my thoughts on various stuff that I find interesting.

This posting of blogs reminds me of Star Trek - "Captain's log - Date: 20070103: ...". I am a big fan of Sci-Fi and get a lot of my inspiration from it. Mixing Science with Science Fiction and trying to develop on it is a great hobby of mine. I hope to strike big with some invention in the near future!

That's it for my first blog. Over and Out.