View Full Version : 2^*024 Decimal Precision

SyntaXmasteR

08-15-2007, 02:51 AM

DECIMAL TO BASE(N) CONVERTER (http://www.syntax******.info/tools/base_converter.php) << CHECK IT OUT

I wanted to throw out a second tool for you guys. I've been coding this for about *2 hours straight.

This tool will convert any decimal number to any base. This is the first program online that does anything close to this with *00% precision so you never have to worry about loosing data.

Other tools give 2^*2 decimal precision:

42*4*672*6 (WoW!)

My tool gives you 2^*024 decimal precision:

*7*76****4862**5*0772**05**078*0247**6*7*76*78*42*065727*4*008**577*2675805500*6***2708477*224075*60

2**20***87*87*****5765878*7688*44*66224*28474*06**474*24*777678**424865485276*022**60*2460*4***45*08

2*520850057688*8*50682*4246288*47******05408272*7*6**505*06845862*82***47245**847*7*6*048*5*56*2*624

224**72*6

I stopped at 2^*024 so my CPU would not die, but I could pushed it even further. I also created bases up to *5 because I really didnt like only *2!

Any comments or recommendations on this tool?

~SyntaX

Moonbat

08-15-2007, 04:19 PM

It doesn't serve much of a purpose (at least to average people), but it is indeed impressive.

SyntaXmasteR

08-15-2007, 04:51 PM

*) MOAR SOURCE!!** (Though I can't blame you for keeping it to yourself if you do, because it sounds like something I'd do)

I have no problem sharing different parts of my source, but I dont hand it out like candy anymore because I've had users stamp their name on it and redistribute. If you have any questions about specific functions of it, let me know.

2) The base system lacks a little bit of order (such as !@#$&#*7;^&*())

I used the order the characters fall into on the ASCII Chart (http://www.asciitable.com/).

It doesn't serve much of a purpose (at least to average people), but it is indeed impressive.

To most it would seem useless, so it is mainly for those who have a deeper understanding of programming precision restrictions.

Try to enter this number into any calculator program:

*7*76****4862**5*0772**05**078*0247**6*7*76*78*42*065727*4*008**577*2675805500*6***2708477*224075*60 2**20***87*87*****5765878*7688*44*66224*28474*06**474*24*777678**424865485276*022**60*2460*4***45*08 2*520850057688*8*50682*4246288*47******05408272*7*6**505*06845862*82***47245**847*7*6*048*5*56*2*624 224**72*5

If you are on a windows machine you will hear crazy error beeps. I'm not sure how other OS calculators handle the number but the decimal precision is cut wayyyy short most of the time.

To give you a better example. Its like me saying 2+2=5 (Then saying, "well I was close!"). This is how most calculators and computers handle numbers when they get too large. I'm providing EXACT calculations :-)

Another example: Enter **52*2*504606846*76 into your calculator (which is 2^60). Now do the binary conversion and you will probably get something like *.^+****00. Inaccurate! Now enter that same number into my calculator. You will get the exact binary equivalent which is: *000000000000000000000000000000000000000000000000000000000000

~SyntaX

Moonbat

08-15-2007, 08:38 PM

To most it would seem useless, so it is mainly for those who have a deeper understanding of programming precision restrictions.

Try to enter this number into any calculator program:

*7*76****4862**5*0772**05**078*0247**6*7*76*78*42*065727*4*008**577*2675805500*6***2708477*224075*60 2**20***87*87*****5765878*7688*44*66224*28474*06**474*24*777678**424865485276*022**60*2460*4***45*08 2*520850057688*8*50682*4246288*47******05408272*7*6**505*06845862*82***47245**847*7*6*048*5*56*2*624 224**72*5

If you are on a windows machine you will hear crazy error beeps. I'm not sure how other OS calculators handle the number but the decimal precision is cut wayyyy short most of the time.

To give you a better example. Its like me saying 2+2=5 (Then saying, "well I was close!"). This is how most calculators and computers handle numbers when they get too large. I'm providing EXACT calculations :-)

Another example: Enter **52*2*504606846*76 into your calculator (which is 2^60). Now do the binary conversion and you will probably get something like *.^+****00. Inaccurate! Now enter that same number into my calculator. You will get the exact binary equivalent which is: *000000000000000000000000000000000000000000000000000000000000

~SyntaX

Hmm, I never thought of it that way, but it still is useless to me, sorry :p

But one day I might find it useful, so thanks in advance. :D

SyntaXmasteR

08-17-2007, 04:19 PM

SyntaX******, if you could send me some of your source code, I'd love to see it and I might be able to understand it within the next 2 years (I can't read code from other peoples' big projects - let alone my own).

What is your username? I'll ad a module to display source code for your username only. Will you be logging on from the same IP? If so I will double secure it with your username and ip.

What parts of the source code did you want to see? specific math functions? loops?

DECIMAL TO BASE(N) CONVERTER (http://www.syntax******.info/tools/base_converter.php) << CHECK IT OUT

I wanted to throw out a second tool for you guys. I've been coding this for about *2 hours straight.

This tool will convert any decimal number to any base. This is the first program online that does anything close to this with *00&#*7; precision so you never have to worry about loosing data.

Other tools give 2^*2 decimal precision:

42*4*672*6 (WoW!)

My tool gives you 2^*024 decimal precision:

*7*76****4862**5*0772**05**078*0247**6*7*76*78*42*065727*4*008**577*2675805500*6***2708477*224075*60

2**20***87*87*****5765878*7688*44*66224*28474*06**474*24*777678**424865485276*022**60*2460*4***45*08

2*520850057688*8*50682*4246288*47******05408272*7*6**505*06845862*82***47245**847*7*6*048*5*56*2*624

224**72*6

I stopped at 2^*024 so my CPU would not die, but I could pushed it even further. I also created bases up to *5 because I really didnt like only *2!

Any comments or recommendations on this tool?

~SyntaX

I'm new here. I've been looking for a forum involving PHP programming for some time.

Trying to put together a site built using PHP so I can share code.

I've had extensive experience working with various base conversions too in the course of my math studies.

If you thrive on challenge:

Have you ever crossed swords with the vexing problem of non-integer values?

For example, what if you wanted the value of *7.*** (base *0) expressed in hexadecimal (base *6), base 7 or some other base?

It's a challenge, but possible to make a function to do such non-integer base conversions too.

Imagine the value of pi expressed in base **.

Pi (Base **) = *.*AC*04*052A2C77*6*C0BB8*CC*88*2782*8*58B*70*60*06*

2.7** (base *0) in base 2 is

*0.*0**0**0*0000***00*0*0**000000*00000**000*00*00**0

2.7** (base *0) in hexadecimal (base *6) is

2.B6872B020C4*BA5E*5*F7CED**6872B020C4*BA5E*5*F7CED*

Etc, ad nauseam...

P.S.

I tried to check it out, but I can't get to the base converter mentioned above.

It asked me to register. I did. But then it tells me the login is invalid when I

use the link in the email it sent me to activate the account.

Wahhhhhhhhhh!

I'm telling mom!

SyntaXmasteR

08-31-2007, 04:44 PM

Can you explain the conversion process:

2.7** (base *0) in base 2 is

*0.*0**0**0*0000***00*0*0**000000*00000**000*00*00**0

I will add this to the base converter, but I need to understand how the math works.

Can you explain the conversion process:

2.7** (base *0) in base 2 is

*0.*0**0**0*0000***00*0*0**000000*00000**000*00*00**0

I will add this to the base converter, but I need to understand how the math works.

It's a bit complicated to explain, so I'll try to write it out, test the procedure, and then post the method and the function to do it here when I'm done.

I'll create an example to demonstrate the conversion of pi into base 2 (binary) which will yield the result

pi = **.00*00*0000******0**0*0*0*000*000*0000*0**0*000**0000*000**0*00**000*00**000**00

Yo, Syntax

I'm not sure what your level of experience in mathematics is, but I'll try to explain the process of converting a non-integer base *0 number into its equivalent in another base. It can't always be done exactly as in the case of integers.

For higher precision, arbitrary precision operators must be used.

To convert a non-integer value from base *0 into base B, it is easier to do if we treat each part of the number separately. That is, convert the integer value first, separately from the decimal value.

When you multiply any base *0 value by a power of *0 (the base), notice that only the decimal point changes position, but the sequence of digits remains constant.

For example, multiplying 0.*4*5* by *00000 gives *4*5* as a result. The sequence of digits will remain the same. This rule also applies to other bases and is what makes it possible to convert non-integer values into another base.

For example, in the case of pi:

The base *0 value of the constant pi to 64 decimals can be defined by the PHP code:

define ("pi", "*.*4*5*265*58*7**2*846264**8*27*502884**7*6*****75*05820*74*445*2*");

The task will be to convert pi into its base 2 equivalent.

We first split the base *0 input argument into its separate integer and fractional parts ($I, $F) respectively.

Let

$I = "*";

$F = "0.*4*5*265*58*7**2*846264**8*27*502884**7*6*****75*05820*74*445*2*";

Let's use P = *00

and Q = (Base ^ P) = 2^*00 = *26765060022822*40*4*670*205*76

This Q value can be computed by the PHP code:

$Q = bcPow("2", *00); // = *26765060022822*40*4*670*205*76

Now compute

w = F *5; Q (truncated, not rounded, to an integer value)

$w = $bcMul($F, $Q); // = *7*4*00*2***00**5***824*4*7824

Convert the base *0 integer ($w) into base 2, which gives the fractional digit sequence

F = *00*0000******0**0*0*0*000*000*0000*0**0*000**0000*000**0*00**000*00**000**00**000*0*000*0***00000

The length of F is *8 digits in this case.

There may be one or more zeros following the decimal point and prior to the converted fractional string computed above.

To determine how many zeros (if any) follow the decimal point:

$NumZeros = $P - StrLen($F);

In this case it works out to

NumZeros = (*00 - *8) = 2

This means we attach 2 zeros to the beginning of $F (leftmost end).

Which gives

00*00*0000******0**0*0*0*000*000*0000*0**0*000**0000*000**0*00**000*00**000**00**000*0*000*0***00000

Now simply take the converted integer part, attach a decimal point and then attach the fractional part after it to obtain

**.00*00*0000******0**0*0*0*000*000*0000*0**0*000**0000*000**0*00**000*00**000**00**000*0*000*0***00 000

which is the required pi value expressed in base 2, to about *00 decimals, with a slight rounding fuzz factor at the end.

Like many base *0 fractional values, pi cannot be expressed exactly in any integer base. There will always be a fuzzy rounding error of some sort.

The rule is, the higher the power of the base used as a multiplier for the fractional part, the farther the accuracy of the converted decimal value extends.

The following function converts a base *0 integer into another base.

function bcBase*0_To_BaseR($ArgN, $ArgR)

{

/*

This function performs a simple base conversion for

base *0 integers into any other base from 2 to 62.

Author: Jay Tanner &#*6*;2007

PHP v4.4.4

Released under provisions of GPL v*

http://www.gnu.org/licenses

---------

ARGUMENTS

$ArgN = Positive base *0 integer string argument.

$ArgR = Radix argument (base) of output integer string.

The radix can range from 2 to 62, but it is

possible to extend it further.

------

ERRORS

An error results if any argument is invalid,

in which case, boolean FALSE is returned.

*/

// Define digits available for the base conversions.

// NUMBERS ARE CONSIDERED CASE SENSITIVE WITH THE EXCEPTION OF DIGITS 0-*.

$digits = "0*2*45678*abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ";

// Read input arguments and truncate

// to integer values if necessary.

$N = bcAdd(trim($ArgN), "0", 0);

$R = bcAdd(trim($ArgR), "0", 0);

// Error if radix (base) is outside valid range.

if ($R < 2 || $R > 62) {return FALSE;}

// Handle special case where N=0

if ($N == 0) {return 0;}

// Compute radix R equivalent to

// original base *0 argument.

$out = "";

while ($N != 0)

{

// Compute and fetch current base R digit.

$d = SubStr($digits, bcMod($N, $R), *);

// Collect current base R digit in

// leftmost output string position.

$out = "$d$out";

// Update $N value.

// Done when $N==0

// Otherwise, repeat

// using new $N value.

$N = bcDiv($N, $R, 0);

}

// Done.

return $out;

} // End of bcBase*0_To_BaseR()

The following function is for converting non-integer base *0 numbers into another base. It requires the function defined above.

function bcBase*0_To_BaseB ($Base*0ArgStr, $ToBaseArg)

{

/*

This function converts a non-integer base *0 value into

its equivalent in any other integer base.

Author: Jay Tanner &#*6*;2007

PHP v4.4.4

Released under provisions of GPL v*

http://www.gnu.org/licenses

----------

DEPENDENCY

This function requires the bcBase*0_To_BaseR() integer

conversion function.

*/

// ---------------------

// Read input arguments.

$x = trim($Base*0ArgStr);

$b = trim($ToBaseArg);

// ---------------------------------------------------

// Compute 200th power of base. For higher precision,

// increase the 200 to a higher value.

$q = "200";

$p200 = bcPow($b, $q);

// --------------------------------------------------------------

// Separate base *0 x argument into integer and fractional parts.

$dp = StrPos($x, "."); if ($dp === FALSE) {$x .= ".0"; $dp = StrPos($x, ".");}

$xInt = SubStr($x, 0, $dp); if ($xInt == "") {$xInt = "0";}

$xFrx = "0." . SubStr($x, $dp+*, StrLen($x));

// --------------------

// Compute new X value.

$X = bcMul($p200, $xFrx);

// --------------------------------------------------------

// Compute converted fractional digits minus decimal point.

$IntOut = bcBase*0_To_BaseR($xInt, $b);

$FrxOut = bcBase*0_To_BaseR($X, $b);

// ---------------------------------------

// Get length of fractional string result.

$FrxL = StrLen($FrxOut);

// -----------------------------------

// Compute number of zeros immediately

// following the decimal point.

$nZ = 200 - $FrxL;

// -------------------------------

// Construct string of ($nZ) zeros

// to follow the decimal point.

$zeros = Str_Repeat("0", $nZ);

// ------------------------------------

// Attach fractional sequence to zeros.

$FrxOut = "$zeros$FrxOut";

// --------------------------------------

// Truncate fractional part to 50 digits.

$FrxOut = SubStr($FrxOut, 0, 50);

// --------------------------------------

// Done. Return converted integer part

// attached to converted fractional part.

return "$IntOut.$FrxOut";

} // End of bcBase*0_To_BaseB()

I hope this isn't too complicated.

LOL

The functions could use some beta testing, since there are probably some bugs to be worked out. The error checking is not extensive.

USAGE:

To use the function to convert 2.7** into binary,

print bcBase*0_To_BaseB ("2.7**", 2);

The value returned will be:

*0.*0**0**0*0000***00*0*0**000000*00000**000*00*00**0

to 50 decimals.

The value 0.2* (base *0) converted into base 5 is:

0.*0************************************************

to 50 decimals

SyntaXmasteR

09-02-2007, 01:27 PM

Ahhh, perfectly explained. I thought that could work, and understood it worked in base*0, but my insufficient math skills told me not to waist my time trying something that might not work.

Next week I will code this into the base converter using string representations of integer values for larger decimal precision. I'll give 2^*024 on the left and 2^*024 on the right side of the decimal.

I definitely agree with your selection of digits:

$digits = "0*2*45678*abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ";

Its logical to most of us, but the ASCII standard F's things up jumping from 0-* to :;>=< ... I think the only real use for anything from character sets from **-255 would be for encryption purposes. I'm coming up with my own, uncrackable encryption methods so it may be very useful to me, but for everyday purposes... useless.

Thanks again for the conversion example!

Ahhh, perfectly explained. I thought that could work, and understood it worked in base*0, but my insufficient math skills told me not to waist my time trying something that might not work.

Next week I will code this into the base converter using string representations of integer values for larger decimal precision. I'll give 2^*024 on the left and 2^*024 on the right side of the decimal.

I definitely agree with your selection of digits:

$digits = "0*2*45678*abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ";

Its logical to most of us, but the ASCII standard F's things up jumping from 0-* to :;>=< ... I think the only real use for anything from character sets from **-255 would be for encryption purposes. I'm coming up with my own, uncrackable encryption methods so it may be very useful to me, but for everyday purposes... useless.

Thanks again for the conversion example!

After some experimentation, I found a potentially serious flaw in using the letters in the way I did for the base conversions.

The base conversion logic is OK and consistent. The problem lies in the ASCII codes used for the characters and the way the computer interprets them.

This only becomes a problem if you intend to use the computer to sort a list of numbers made in bases that use alpha characters.

The ASCII codes of the uppercase letters are greater than the ASCII codes of the corresponding lowercase letters because uppercase letters come before lowercase letters in the ASCII code spectrum and will be sorted ahead of them. This is contrary to the numerical values we apply to the characters when used for numbers rather than words.

This is not good.

Since letter case is important in this instance, it leads to a list of numbers not sorting correctly in proper order if they contain any mix of upper/lowercase letters.

There are ways around this, the simplest being to switch places with upper/lowercase letters in the allowed digits string. Then all hexadecimal numbers would ALWAYS use uppercase A-Z.

Just change

$digits = "0*2*45678*abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ";

to

$digits = "0*2*45678*ABCDEFGHIJKLMNOPQRSTUVWXYZabcdefghijklmnopqrstuvwxyz";

and the numerical sorting problem is solved, regardless of the base used.

It seems that uppercase following lowercase is a more logical way to do it, but it works the opposite way in ASCII and that can effect anything that involves ASCII sorting algorithms.

I just thought anyone using my particular method should know about this little glitch. It could lead to erroneous sorting problems if not taken into account in some circumstances.

SyntaXmasteR

05-21-2008, 05:53 PM

I'm digging up this old thread, and giving it life once again. I have rewritten the base converter and added several new functions. It now converts much larger numbers with greater precision. It also has the ability to convert between bases.

Example of large number:

BASE *0

**************************************************************************************************** **************************************************************************************************** **************************************************************************************************** **************************************************************************************************** **************************************************************************************************** **************************************************************************************************** **************************************************************************************************** **************************************************************************************************** **************************************************************************************************** **************************************************************************************************** **************************************************************************************************** **************************************************************************************************** *******************************************************************************************

converted to

BASE 2

**0**0*0***0*0*0**00****00****0*00**0*****0**00***0000*0***0*00*00***00**0**0**00000*00*0*00*0***0*0 0**00****00000**0*00**0*00000**0*****0**0****0*0*0**00*00**000***0*000***000***0*0***00**00*0*000000 0****00**00*000000*0*000000*000***0**0**00***0*00**00***0*0***00000000*0000000*0000**0******00*00**0 00*0*000****00*0*****0*0**00***0*00*0*0*00*0***0***0*00**00*0**0*00**0*0000**00000******0000**0*0*** 0***0**0*****0**0**0*0*0*0000*0**000**0*00*0**0*00***0000*0**00000*0**000*0*****0****000*0**0****0*0 *0*000**000*0000**00*000****00000*0000*00**0*000**0**0**0*****00000*00**0***0**00*0*000*0***0*000**0 *000*00**00*000*00000**00*00000**0000*00000*0**0*000*000*0*00*0**0*0*****0000*0000*000*00**0*00***0* 0****000000*0000**0**0***0*000********0**00*0*0*0**00****0*0**0*00**00*0**0****0*0***0**0*0*0***0*0* *00**0**0**00**0**00*00*0***0*000***000***0*0**00*0**0*0**00**000000*00*0*00*0***00*000*0000*0000*** **0*0**00*000*****00*****0**0000*00**0*0**00000***00*000**0*0***00000**0*00**00*00000**0*0***0*00*0* *00*0*00*0*000**0****0**000000*000****0*00*0**0*0*0*0***0*****0***00*0*****000*0*0*0***000*****000*0 ********000**0*0*00*0*0000*0000*****0*00*00*000*****0*0*0*000*****00**00*0000****0000***0**000*0***0 *0*000***0**00****0*0*****000*0*0*0****000000***00000*0**00*0*0**000*00000**000*00**0000*0***0****00 **00*0****00**0****00****0****00*0*0*****0**0**00***0*00****0*0**000****0**00*00**0*0*0*00000*0**000 0*0*0*00000***0*00**0*0**000*0000*****000***0****0*0000*0******0**0**00**0**0**00****000****00*00**0 0*0****00*0*0****0*000*000*0**00***0*0**0**00*000**00**00***00**0***00*0*00****00**0*000***000000*** 0**0000********************************************************************************************* **************************************************************************************************** **************************************************************************************************** **************************************************************************************************** **************************************************************************************************** **************************************************************************************************** ***************************************************************************************************

DEMO is live here: Base Conversion Demo (http://syntax******.info/_test/number_system.php)

Powered by vBulletin® Version 4.1.8 Copyright © 2023 vBulletin Solutions, Inc. All rights reserved.