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 everyday, 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 arbitraryprecision 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 righttoleft 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

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=7831480735744 , expressed in its corresponding equivalents in all bases from 2 to 36 .
Base 10 value X = 7831480735744
B The same numerical value expressed in base B
2 1110001111101101000101000001001100000000000
3 1000201200102100122210022121
4 1301331220220021200000
5 2011302333032020434
6 24353422123214024
7 1435543251623011
8 161755050114000
9 30650370583277
10 7831480735744
11 254A348788816
12 A65963077314
13 44A675675A93
14 1D108D533008
15 D8AAC60B7B4
16 71F68A09800
17 3F0B6FC75AC
18 238BGFGF0CG
19 15525E421B2
20 F5I6HEBJ74
21 9I14132K51
22 6AFEK3LFH6
23 4802D8IAGF
24 2N3CA3B9IG
25 2182IFH24J
26 1BD1D58M4G
27 10JIB9HL8G
28 KKBG9N3E8
29 FJ01N4I57
30 BS2N4M1S4
31 95K55GF5E
32 73TKA1600
33 5IP0FWU7S
34 4D3J2UMMC
35 3GP8WMSJT
36 2RXQDFDOG
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 4 , to demonstrate how its individual digits were derived by converting the base 10 integer, X=7831480735744 , into base 4 , one digit at a time, using a series of successive integer divisions and reductions and converting the sequential remainders into base 4 digit symbols. The same process can used with any base to render EXACT arbitraryprecision base 10 integer to base B conversions to any number of digits.
X /Base Remainder Symbol
7831480735744 /4 0 0
1957870183936 /4 0 0
489467545984 /4 0 0
122366886496 /4 0 0
30591721624 /4 0 0
7647930406 /4 2 2
1911982601 /4 1 1
477995650 /4 2 2
119498912 /4 0 0
29874728 /4 0 0
7468682 /4 2 2
1867170 /4 2 2
466792 /4 0 0
116698 /4 2 2
29174 /4 2 2
7293 /4 1 1
1823 /4 3 3
455 /4 3 3
113 /4 1 1
28 /4 0 0
7 /4 3 3
1 /4 1 1
7831480735744 in base 10
=
1301331220220021200000 in base 4
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 7831480735744 looks like when expressed in base 4 , but we can still predict that it consists of 22 base 4 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=7831480735744 , to be converted to base B=2 :
Base 2 integer equivalent Y = 1110001111101101000101000001001100000000000
Base 2 digits in Y = 1 + floor(log(7831480735744) / log(2)) = 43
The above base conversion table for X = 7831480735744 , is reproduced again below, with this version of the table including the counts of the digits (in parentheses).
Base 10 value X = 7831480735744
B Same numerical value expressed in base B (with digits count)
2 1110001111101101000101000001001100000000000 (43)
3 1000201200102100122210022121 (28)
4 1301331220220021200000 (22)
5 2011302333032020434 (19)
6 24353422123214024 (17)
7 1435543251623011 (16)
8 161755050114000 (15)
9 30650370583277 (14)
10 7831480735744 (13)
11 254A348788816 (13)
12 A65963077314 (12)
13 44A675675A93 (12)
14 1D108D533008 (12)
15 D8AAC60B7B4 (11)
16 71F68A09800 (11)
17 3F0B6FC75AC (11)
18 238BGFGF0CG (11)
19 15525E421B2 (11)
20 F5I6HEBJ74 (10)
21 9I14132K51 (10)
22 6AFEK3LFH6 (10)
23 4802D8IAGF (10)
24 2N3CA3B9IG (10)
25 2182IFH24J (10)
26 1BD1D58M4G (10)
27 10JIB9HL8G (10)
28 KKBG9N3E8 (9)
29 FJ01N4I57 (9)
30 BS2N4M1S4 (9)
31 95K55GF5E (9)
32 73TKA1600 (9)
33 5IP0FWU7S (9)
34 4D3J2UMMC (9)
35 3GP8WMSJT (9)
36 2RXQDFDOG (9)
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 arbitraryprecision base 10 integer to base B conversion process. Note that since we are doing arbitraryprecision 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.
A simple test call to the above function might resemble the following:
$X = "7831480735744";
$B = 27;
print bcBase_10_Int_To_Base_B($X, $B);
// The printed result should be:
10JIB9HL8G
Even Higher Bases
For Base 2 to Base 36
As previously defined above, the case insensitive digits spectrum (up to base 36) is:
"0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ "
=
"0123456789abcdefghijklmnopqrstuvwxyz "
For Bases > 36
If we allow case sensitivity, where lowercase "a" is NOT identical to uppercase "A", and both symbols represent entirely different values, then we can use the uppercase letters to extend the digits spectrum by yet another 26 letters, giving us a CASE SENSITIVE digits spectrum consisting of 62 unique digits.
The case sensitive base 62 digits spectrum is:
"0123456789abcdefghijklmnopqrstuvwxyzABCDEFGHIJKLMNOPQRSTUVWXYZ "
Note that the lowercase letters from "a" to "f" must be used to represent hexadecimal values under this convention, since the uppercase letters, "A" to "F", are NOT equivalent symbols here.
Allowing case sensitivity this way, lets us handle bases from 2 to 62. Of course, we could extend the digit symbols indefinitely to represent even higher bases.
SPECIAL NOTE:
 When displaying numbers in higher bases, we need to be careful which font we choose to display or print the digit symbols. For example, care must be taken so that digits like "
0 " (zero) and "1" (one) are not mistaken for letters, like "O" or the lower case "L" or uppercase "I".
Since in some font styles, certain numerical digits can closely resemble certain letters, we need to read such higherbase numbers carefully, and a wellchosen alphanumeric font can help reduce this kind of confusion.
To avoid this problem, either the 'Consolas' or 'Inconsolata' font is recomended because they show a slash through the digit zero (0 ) which easily distinguishes the digit from the letter 'O ' when higher bases are used. For those who do not have either font or something similar, they are freely available on the Internet.
