# Possible to Represent Large Binary Data As An Equation?

Page 1 of 1

## 11 Replies - 4796 Views - Last Post: 21 March 2011 - 11:33 AM

### #1 DillonSalsman Reputation: 13
• Posts: 144
• Joined: 30-October 07

# Possible to Represent Large Binary Data As An Equation?

Posted 13 January 2011 - 04:25 PM

Question has been answered by lordofduct:

Quote

The area of study is called "Algorithmic Information Theory". It requires an extensive understanding of discrete mathematics.

Basically, is it possible read the binary data as an incredibly large decimal number then find a relatively simple way to represent this number in an equation?
I'm just kind of rambling but it was an idea I had in my CS class and I was wondering if such a concept already exists what it is called.

I'm thinking that it would be dramatically easier to transfer these "magic numbers" than the entire binary file, and then simply use these numbers to recreate the decimal number and write it as binary.

Also, I realize that memory would be an issue here. However, Mathematica seems to handle incredibly large numbers on machines with only a gig or two of memory so I am lead to believe that I should worry more about whether or not these magic numbers would actually take less data to represent than the final result?
Kind of like how you can max out a calculators memory relatively quickly by simply using powers and multiplication.

About base 16 to base 10 conversion.. my basic understanding of process for base conversion is you sum the value of: each digit D (Ones place D=1, tens D=2, etc) with a value M from {1..N} you multiply M by base B to the D-1.
For example, the hexadecimal number 2AF3 is equal, in decimal, to (2 × 16^3) + (10 × 16^2) + (15 × 16^1) + (3 × 16^0) , or 10,995.

Assuming there's 2 hexadecimal per byte (2^8= 256 = 16*16) there would be 2*P hexadecimal digits with P equal to the number of bytes in a binary file I'm thinking the amount of decimal digits required to represent a hexadecimal number grows exponentially..

Of course all of this means that the decimal number will take up much more room than the original. But this is okay because the idea is data portability. To the point where you can print out a list of numbers, type them into your computer, run a program which recreates absurdly large decimal number and saves it as binary.

Justification: Just by using a few numbers you can create absurdly large numbers. All that large amounts of binary data is is a specific absurdly binary number. See attachment for the amount of digits calculated using 2^(2^(3141592)): nearly 10 to the 10 to the 10 ^ 6 digits. Divide this by 2 and you would have the number of bytes required to represent that number. Of course you would then have to offset this value by a number x which could be generated using the same method.

```decimal value = readBinaryAsDecimal(myFile);
decimal generatedTotal = 0;
decimal generated = 0;
decimal currentExponent = 0;
decimal[] recipe = decimal[];

while(generatedTotal != value)
while(generatedTotal + generated < value)
{
generated = 2^2^(currentExponent);
}
generatedTotal += 2^2^(--currentExponent)
currentExponent = 0;
}
```

Its also seems logical that there would be "hotspots" in which files could be represented by values relatively close to an easily generated number. There would also be "deadspots" which there were no nearby easily generated numbers. The key here is that for each generated number (GN) added to the total generated (TGN) number, it must increase the number of matching digits in the TGN to the Target Number (TN).

#### Attached image(s)

• This post has been edited by DillonSalsman: 14 January 2011 - 01:46 PM

Is This A Good Question/Topic? 2

## Replies To: Possible to Represent Large Binary Data As An Equation?

### #2 BenignDesign • • holy shitin shishkebobs
•     Reputation: 8120
• Posts: 12,503
• Joined: 28-September 07

## Re: Possible to Represent Large Binary Data As An Equation?

Posted 13 January 2011 - 04:27 PM

Moving this to the Corner Cubicle in hopes of a serious response or two.

### #3 Tom9729 Reputation: 181
• Posts: 2,642
• Joined: 30-December 07

## Re: Possible to Represent Large Binary Data As An Equation?

Posted 13 January 2011 - 05:38 PM

It sounds like you're talking about data compression.

RLE compression for example replaces sequences of repeated data with a number indicating how many times a byte/pixel/whatever was repeated.

JPEG compression does something with sine waves IIRC.

It's really a very interesting subject and there is a lot of information out there on it... ### #4 DillonSalsman Reputation: 13
• Posts: 144
• Joined: 30-October 07

## Re: Possible to Represent Large Binary Data As An Equation?

Posted 13 January 2011 - 06:06 PM

Yes compression, but not using more "binary", rather using formulae to generate binary.
My final essay for ENGL111 (The "can you write essays" class) was about the kind of data compression you are talking about.

Its also seems logical that there would be "hotspots" in which files could be represented by values relatively close to an easily generated number. There would also be "deadspots" which there were no nearby easily generated numbers. The key here is that for each generated number (GN) added to the total generated (TGN) number, it must increase the number of matching digits in the TGN to the Target Number (TN).

This post has been edited by DillonSalsman: 13 January 2011 - 06:30 PM

### #5 Tom9729 Reputation: 181
• Posts: 2,642
• Joined: 30-December 07

## Re: Possible to Represent Large Binary Data As An Equation?

Posted 13 January 2011 - 06:20 PM

foo.mp3 -> binary data -> large number

Find a function f and a magic number x such that f(x) = large number

This is basically what you are saying right?

### #6 DillonSalsman Reputation: 13
• Posts: 144
• Joined: 30-October 07

## Re: Possible to Represent Large Binary Data As An Equation?

Posted 13 January 2011 - 06:31 PM

Yes. Or possibly a relatively short list of numbers.

### #7 NeoTifa • • NeoTifa Codebreaker, the Scourge of Devtester
•      Reputation: 4688
• Posts: 19,536
• Joined: 24-September 08

## Re: Possible to Represent Large Binary Data As An Equation?

Posted 14 January 2011 - 09:00 AM

You mean like boolean algebra?

### #8 lordofduct • • I'm a cheeseburger
•   Reputation: 2668
• Posts: 4,786
• Joined: 24-September 10

## Re: Possible to Represent Large Binary Data As An Equation?

Posted 14 January 2011 - 10:00 AM

So what do you want to know? All the different available methods of compression? That's a bit out of the scope of this forum. There's an entire areas of study based on it, there's tons of books written about it.

The area of study is called "Algorithmic Information Theory". It requires an extensive understanding of discrete mathematics. And a large majority of it actually isn't dependent on binary (though is usually applied to binary data, big difference between being dependent on the binary number system, and being applied to the binary number system).

This post has been edited by lordofduct: 14 January 2011 - 10:01 AM

### #9 DillonSalsman Reputation: 13
• Posts: 144
• Joined: 30-October 07

## Re: Possible to Represent Large Binary Data As An Equation?

Posted 14 January 2011 - 01:26 PM

No you just gave me exactly what I needed to know lordofduct Thank you very much. I realize that it has nothing to do specifically with binary, or any number system. Its just math. AFAIK generic operations have no dependent base.

### #10 lordofduct • • I'm a cheeseburger
•   Reputation: 2668
• Posts: 4,786
• Joined: 24-September 10

## Re: Possible to Represent Large Binary Data As An Equation?

Posted 14 January 2011 - 02:34 PM

The only time the number system pertains to the compression algorithm is if the algorithm explicitly states that it is.

Consider the IEEE standard double floating point number. This is technically a form of simple compression. It's a lossy compression method that allows us to store very large and very small numbers in the finite space of 64-bits. Say the value 2^124 would take 125 bits as an integer, but only 64 as a double.

In this algorithm though it's fairly specific about base. It breaks the 64-bits into 3 parts (sign, exponent, mantissa). This segmentation of the 64-bit cluster is really only extractable in binary form... the arithmetic must treat the value as binary to accurately translate the compression.

Speaking of the double. It's also an example of some forms of compression and storage designs that this area of study covers. It's not exactly the kind of compression you were thinking of (that being the compression of large files), but the subject will show you there's a lot to the topic with complexity spreading in several directions. For instance though the double is lossy, arithmetic can actually be performed on the value with out decompressing it... hence the existence of the floating point arithmetic processor.

### #11 AdamSpeight2008 Reputation: 2298
• Posts: 9,535
• Joined: 29-May 08

## Re: Possible to Represent Large Binary Data As An Equation?

Posted 14 January 2011 - 03:07 PM

This post has been edited by AdamSpeight2008: 14 January 2011 - 03:09 PM

### #12 DillonSalsman Reputation: 13
• Posts: 144
• Joined: 30-October 07

## Re: Possible to Represent Large Binary Data As An Equation?

Posted 21 March 2011 - 11:33 AM

So much interesting reading you guys have given me :]
What brought all this about was this: if I were given a computer who's only inputs were keyboard and mouse, how would I recreate data found elsewhere on it without writing every bit myself.

Page 1 of 1

 .related ul { list-style-type: circle; font-size: 12px; font-weight: bold; } .related li { margin-bottom: 5px; background-position: left 7px !important; margin-left: -35px; } .related h2 { font-size: 18px; font-weight: bold; } .related a { color: blue; }