Bases By The Numbers - Base 10 to Base B

When discussing the process of base conversion, a logical place to begin is by learning how to first convert our every-day, standard base 10 integers into their corresponding equivalents in other bases.  We will develop a simple PHP function to do this base 10 integer to base B conversion in arbitrary-precision to any number of digits we wish.

The process of integer base conversion is actually very simple and consists of nothing more than collecting up the remainders of a series of reductive integer divisions, in reverse order of their generation (in right-to-left order), and then fetching their corresponding base B digit symbols from within the digits spectrum used for the conversion.

Here, a reductive integer division means that the resulting integer quotient of the division by the base value replaces the original dividend in preparation for the next cycle, if any.  Each division cycle thus reduces the subsequent numerator (X value) in the sequence of divisions.  For example, the integer division of 13/2 results in the integer quotient 6 (Q=6) and a remainder of 1 (R=1).

The process of converting the base 10 integer 13 into binary (base 2):

Let:
X = Base 10 integer to be converted
B = Alternative base into which to convert X
Q = Integer quotient of integer division X/B
R = Integer remainder of integer division X/B

1. Starting with base 10 integer X, we divide it by the base (B), keeping only the integer quotient (Q) and the remainder (R) as demonstrated below.
   
   Starting with ... X=13 and B=2:
   
   $Q = bcdiv($X, $B);
$R = bcmod($X, $B);
$X = $Q;

 X / B = Q, R
13 / 2 = 6, 1
 6 / 2 = 3, 0
 3 / 2 = 1, 1
 1 / 2 = 0, 1


The following random example shows the base 10 integer,  X=78296,  expressed in its corresponding equivalents in all bases from 2 to 36.

Base 10 value X = 78296

B   The same numerical value expressed in base B
2   10011000111011000
3   10222101212
4   103013120
5   10001141
6   1402252
7   444161
8   230730
9   128355
10  78296
11  53909
12  39388
13  2983A
14  20768
15  182EB
16  131D8
17  FFFB
18  D7BE
19  B7GG
20  9FEG
21  89B8
22  77GK
23  6A04
24  5FM8
25  506L
26  4BLA
27  3QAN
28  3FO8
29  362P
30  2QTQ
31  2JEL
32  2CEO
33  25TK
34  1XOS
35  1SW1
36  1OEW

The above example table demonstrates how the same numerical value can be expressed in several base systems.  

The table below was derived by randomly selecting one of the conversions from within the table above, in this case base 20, to demonstrate how its individual digits were derived by converting the base 10 integer, X=78296, into base 20, one digit at a time, using a series of successive integer divisions and reductions and converting the sequential remainders into base 20 digit symbols.  The same process can used with any base to render EXACT arbitrary-precision base 10 integer to base B conversions to any number of digits.

X       /Base  Remainder  Symbol
78296   /20    16         G
3914    /20    14         E
195     /20    15         F
9       /20    9          9


78296 in base 10
=
9FEG in base 20

The Number of Digits in a Base B Integer

It is very interesting to note that we do not have to actually perform any conversion of X to base B to simply predict how many base B digits the result would have if we did!  For example, we may not know what the base 10 integer 78296 looks like when expressed in base 20, but we can still predict that it consists of 4 base 20 digits by using this simple logarithmic formula:


Ydigits = 1 + floor(log(X) / log(B))

In PHP, this formula can be written in very similar form as:

$Ydigits = 1 + floor(log($X) / log($B));

In this case, the log(x) function refers to the natural logarithm.
However, base 10 or any other logarithmic base could be used.



EXAMPLE: For Base 10 Integer X Converted Into Base B Integer Y:

Given base 10 integer, X=78296, to be converted to base B=18:
Base 18 integer equivalent Y = D7BE

Base 18 digits in Y = 1 + floor(log(78296) / log(18)) = 4

The above base conversion table for X = 78296, is reproduced again below, with this version of the table including the counts of the digits (in parentheses).

Base 10 value X = 78296

B   Same numerical value expressed in base B (with digits count)
2   10011000111011000 (17)
3   10222101212 (11)
4   103013120 (9)
5   10001141 (8)
6   1402252 (7)
7   444161 (6)
8   230730 (6)
9   128355 (6)
10  78296 (5)
11  53909 (5)
12  39388 (5)
13  2983A (5)
14  20768 (5)
15  182EB (5)
16  131D8 (5)
17  FFFB (4)
18  D7BE (4)
19  B7GG (4)
20  9FEG (4)
21  89B8 (4)
22  77GK (4)
23  6A04 (4)
24  5FM8 (4)
25  506L (4)
26  4BLA (4)
27  3QAN (4)
28  3FO8 (4)
29  362P (4)
30  2QTQ (4)
31  2JEL (4)
32  2CEO (4)
33  25TK (4)
34  1XOS (4)
35  1SW1 (4)
36  1OEW (4)

For Higher Bases

For the higher bases, (greater than 10), we traditionally use the sequential (not case sensitive) letters of the English alphabet as extended digit symbols, such as the letters "A" to "F", as used in hexadecimal numbers.  Normally, there is no single digit with a value of 15, so we use the letter "F" to represent it.  When dividing by 16 and the remainder is 15, we use the symbol "F" to represent that digit value in hexadecimal and higher bases.  For numbers expressed in any base from 2 to 10, the digit symbols are the same as the digits themselves.  We only apply the sequential letters to the higher bases when we run out of normal digits to represent their values, that is, base 11 and above.


Below is a simple PHP function designed to perform the complete, exact arbitrary-precision base 10 integer to base B conversion process.  Note that since we are doing arbitrary-precision computations, the function can handle integer base conversions extending out to hundreds or even thousands of digits with EXACT accuracy.

 function bcBase_10_Int_To_Base_B ($X10, $B)
{
 $X = $X10;
 if (bccomp($X, 0) == 0) {return "0";}

 $DigitsSpectrum = substr("0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ",0,$B);

 $XB = "";

  while (bccomp($X, 0) > 0)
 {
  $XBDigitSymbol = substr($DigitsSpectrum, bcmod($X,$B),1);
  $XB = substr($DigitsSpectrum, $XBDigitSymbol . $XB;
  $X = bcdiv($X,$B);
 }
 return $XB;
}

In the above function, to generate the specific digits spectrum string for any given base from B=2 to B=36, we use the following code line:

$DigitsSpectrum = substr("0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ",0,$B);

Since our standard base 10 digits spectrum consists of all the numerical digits from 0 to 9, the base 10 digits spectrum string takes the form, "0123456789".

If we were converting into binary (B=2), then the binary digits spectrum string would only consist of the 2 digits, "01".

If we were converting into octal (B=8), the digits spectrum would take the form, "01234567".

If we were converting into hexadecimal (B=16), the hexadecimal digits spectrum string would take the form, "0123456789ABCDEF".

In the case of B=24, the digits spectrum would take the form, "0123456789ABCDEFGHIJKLMN", and so on.

The 0-based position index of a digit in the digits spectrum corresponds to the value of the remainder of the integer division of the current base 10 integer, X, by the current base (B).  This division is computed and the corresponding base B digit symbol extracted from the digits spectrum by the line:

$XBDigitSymbol = substr($DigitsSpectrum, bcmod($X, $B), 1);

An integer division here means that both dividend (X) and divisor (B) are integers and that any fractional part is dropped without rounding.  The division of 127/16 would result in exactly 7, and not 7.9375, as in floating point arithmetic nor rounded up to 8.  Any decimal fraction that results from the division is simply ignored and dropped without exception.

Within the function, the value of the remainder of the integer division, X/B, is computed by the statement:

bcmod($X,$B)

This value is the index of the base B digit symbol within the digits spectrum.

$X = "78296";
$B = 31;

print bcBase_10_Int_To_Base_B($X, $B);

// The printed result should be:
   2JEL