Posts tagged ‘mathematics’

2009-08-3

The 64-bit Difference

I was just reading about the limitations of the WAV audio format.

The WAV format is limited to files that are less than 4 GB in size, because of its use of a 32-bit unsigned integer to record the file size header (some programs limit the file size to 2–4 GB). Although this is equivalent to about 6.8 hours of CD-quality audio (44.1 KHz, 16-bit stereo), it is sometimes necessary to exceed this limit, especially when greater sampling rates or bit resolutions are required. […] Its 64-bit header allows for much longer recording times.

I got to thinking about computer memory and the difference in capacity between N-bit systems. A computer uses an address to access different parts of the memory. The address consists of numbers (internally, ones and zeros), which for a 32-bit system (where the addresses always have a length of 32 bits) would look something like “af34c97b” written using a radix of 16. A 32-bit uses these addresses to look up places in the memory. Each address stands for a certain byte in the memory, so obviously if we only have addresses with 32 bits, we can’t look beyond the address with 32 ones in a row since that is the maximum value 32 bits hold.

Think of when you’re mailing a letter: you can mail the letter to anyone you want using only two numbers for the house or apartment number. You would be able to send it to (0)1-99 Blah Blah St., but not to the guy living at the end of the street at no. 100. Memory addresses work in the same way.

Let’s do some math now. Say your system is working with 32-bit memory addresses. That means the largest value we could have (the farthest down the street we would be able to send the letter) would be 1111 1111 1111 1111 1111 1111 1111 1111, or FFFF FFFF in hexadecimal numbers. Let’s write this figure out in a format that we’re more familiar with, such as Gibibytes (GiB) or as it is more incorrectly known as: Gigabytes (GB). 1 GiB = 1024^3 bytes; 1 GB = 1000^3 bytes.

FFFF FFFF in GiB is 2^32 / 1024^3 = 2^32 / (2^10)^3 = 2^32 / 2^30 = 2^(32-30) = 2^2 = 4 GiB.

You might have heard already that 32-bit systems only can handle 4 GiB of memory, and now you hopefully know why if you didn’t already. Now then, what happens if we double that number, and make it a 64-bit system?

FFFF FFFF FFFF FFFF in GiB would be 2^64 / 2^30 = 2^34 = 17179869184 GiB, or 16 Exbibyte (EiB). A MASSIVE amount of memory. As you can see, with a double increase in address size, we do not get a doubling of the memory space, but rather a number that is the number of bytes in 4 GiB to the power of 2. 4 GiB = 4294967296 Bytes, and 16 EiB = 18446744073709551616 bytes. These numbers are obviously incomprehensible. So I thought it would be easier to demonstrate them with an example, regarding the Wikipedia article quoted at the top of the article.

As the quote says, a 4 GB (actually GiB) WAV file (with file size header of 32 bits) would give us 6.8 hours of music with a sampling rate of 44.1 kHz, a bit depth of 16 bits and 2 channels (stereo).

If we assume the file size is proportional to the playing time of the audio file if the quality specifications remain the same, then we can calculate the playing time of a WAV file with file size header of 64 bits:

17179869184 [GiB] * (6.8 [hours] / 4 [GiB]) [hour-to-filesize ratio] = 29205777612.8 hours of music.

This number is still incomprehensible so let’s walk up the ladder of time units, shall we? Note that when calculating the amount of years, we will use a year length of 365.2425 days, which is the arithmetic mean of amount of days in a year in the Gregorian Calendar, which has a 400-year cycle and 146 097 days: 146097 / 400 = 365.2425 days. This is to take leap years into account. One could also use a day length of 24 hours and 58.3594503 seconds, but that doesn’t feel as nice, somehow.

29205777612.8 hours
= 1216907400.533333333 days
= 173843914.361904762 weeks
= 39981351.585316605 months (average of 30.436875 days/month in one 365.2425-day year)
= 3331779.298776384 years

So we see that just by doubling the address space, we go from 6.8 hours of music — which I guess you could plough through on a really dull and long bus ride — to more than 3.33 million years of music.

That, my friend, is the 64-bit difference.

… hmm? What was that about 128 bits? Shut up. 😦

No but really, to fill a 128-bit hard drive, it would more energy than it would to boil the oceans of the earth. Theoretical breakdown. Enjoy.

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2009-07-18

Open-Source Calculus

I’m (re(re))taking the second course on calculus during the summer at the university. It’s going much better this time around, which is a good thing.

I did have problems with one problem (har), and I couldn’t make sense of it on my messy notebook, and I didn’t feel like doing the entire problem over again. The only thing wrong with my solution is that the answers in the back of the book said term1 – term2, while I kept getting term2 – term1 in my notebook, so I wasn’t too far off but still couldn’t get it right — couldn’t find the erring minus sign. All I really felt like was doing some programming, which I enjoy, but what would I code if I have no software needs?

Then I thought: I need to get a more structured view of this problem. Why not write it up in LaTeX and make it into a nice, good-enough-to-print, PDF solution. That way, I will get some practice writing LaTeX documents (it’s been a few months, sadly), and writing LaTeX is pretty much programming in a way, so I get to practice LaTeX, scratch my coding itch, and maybe find out where that offending minus sign went.

Sure enough, it worked pretty well.

Solution to problem 6.2.9 in Calculus, A Complete Course (Sixth Edition).

I want to point out that this PDF was produced entirely using Free and Open-Source Software (FOSS):

  • texlive LaTeX distribution (tex->PDF compiler: pdflatex/pdftex)
  • GNU Emacs as the LaTeXt editor
  • Ubuntu Linux to run Emacs and tex compiler
  • Totem to play music while writing up the solution
  • Grip + oggenc to rip CDs to Ogg Vorbis
  • Ogg Vorbis media container format
  • libVorbis 1.2.0 used by oggenc (darn you Ubuntu for not updating libVorbis since 2007)
  • and so forth.

It works, folks!

2009-03-9

Project Euler

I’ve become an active member of Project Euler (PE) again. Project Euler (www.projecteuler.net) is a collection of (around 300 or so) mathematical and programmatical problems. I’m not sure when I started the first time, but it’s been a couple of years, I think. Now I’m back again with a bit more knowledge and feeling a bit more comfortable with programming, several programming and math courses later.

When I first joined, I was mostly only comfortable with Java as my programming language of choice, but I’m very interested in learning to program comfortably in C for a few reasons:

  • It is the traditional language in Unix/Linux applications.
  • I like GNU/Linux and Free and Open-Source Software.
  • It is a lot faster than Java… but still has very similar syntax!
  • I like handling pointers for some reason. I feel like I’m much more in contact with the data as I have full control of everything. It’s just so gratifying to get Segmentation Fault after Segmentation Fault and then, after some debugging and messing around with my pointers, finally getting the sought output from the program.

I don’t believe it’s mandatory to join Project Euler to take part in the problems, but I recommend you sign up for an account, because if you do:

  • you can provide answers to the problems and Project Euler will keep track of the problems you’ve solved.
  • you may participate in the forum thread for a particular problem upon solving it, allowing you to discuss your solution with fellow members. (It’s incredible to find solutions written in the J and K programming languages, which afford extremely concise programming.) Note that only relatively recent forum threads are open to new posts, but all threads are open to read (upon solving the corresponding problem, as I already mentioned).

Project Euler is a very fine tool to learn a programming language. If you’re not a programmer by profession and you don’t really have any software needs that aren’t fulfilled, Project Euler provides you with a great environment to experiment with a programming language’s syntax and libraries with problems and tasks to solve and complete.

My goal right now is to solve as many problems as I can using only the standard libraries in the GNU Library (glibc), namely stdlib, stdio and occasionally string (so far I’ve used string only for the strlen function, which I felt was unnecessary to write myself). It’s going pretty well, and it’s kind of fun to have to figure out solutions on your own instead of relying on already-written functions to do the job for you.

Some of the problems at PE can be solved only with the help of pen and paper, but most of them require up to thousands of iterations to find the solution and isn’t feasible without some computational aid. However, according to the site’s FAQ section, which they call “About”, all the problems are designed in a way that a well-constructed algorithm should take no more than 1 minute to output the solution to the problem. I’m very proud of C. Most problems finish in milliseconds, even on the super-slow laptop I use sometimes.

2009-02-10

1 + 1 = 0

I saw something on YouTube today that was interesting. To relieve you of having to watch the extremely slow-moving video, I will just provide the proof right here.

First assume that

i = √(-1)

Next, assume that

√(ab) = √(a) * √(b)

Now, we will derive the proof:

1 + 1 = 1 + √(1)
= 1 + √[(-1)(-1)]
= 1 + √(-1)√(-1) = 1 + (√(-1))²
= 1 + i²
= 1 + (-1)
= 1 - 1
= 0

Does that not blow your mind? Now, obviously this is crazy. So let’s take a look at how the proof is flawed.

The assumption that √(ab) = √(a) * √(b) is only true if at least one of a and b are positive numbers (including zero). If both a and b are negative then √(ab) = -√(a) * √(b). Hence,

1 + 1 = 1 + √(1)
= 1 + √[(-1)(-1)]
= 1 - √(-1)
√(-1) = 1 - (√(-1))²
= 1 - i²
= 1 - (-1)
= 2

And the world makes sense again. Now I feel like my feet are back on the ground!