Digit/Bit Representation of Integers in any Base

Integer number representations in other Bases.

Formally, for every element \(N =\)x[i], compute the (vector of) “digits” \(A\) of the base \(b\) representation of the number \(N\), \(N = \sum_{k=0}^M A_{M-k} b ^ k\).
Revert such a representation to integers.

Usage

digitsBase(x, base = 2, ndigits = 1 + floor(1e-9 + log(max(x,1), base)))
# S3 method for class 'basedInt'
as.integer(x, ...)
# S3 method for class 'basedInt'
print(x, ...)

as.intBase(x, base = 2)
bi2int(xlist, base)

Arguments

x

For digitsBase(): non-negative integer (vector) whose base base digits are wanted.

For as.intBase():
a list of numeric vectors, a character vector, or an integer matrix as returned by digitsBase(), representing digits in base base.

base

integer, at least 2 specifying the base for representation.

ndigits

number of bits/digits to use.

...

potential further arguments passed to methods, notably print.

xlist

a list of integer vectors with entries typically in 0:(base-1), such as resulting from digitsBase().

Value

For digitsBase(), an object, say m, of class "basedInt" which is basically a (ndigits x n) matrix where m[,i] corresponds to x[i], n <- length(x) and attr(m,"base") is the input base.

as.intBase() and the as.integer method for basedInt objects return an integer vector.
bi2int() is the low-level workhorse of as.intBase().

Note

Some of these functions existed under names digits and digits.v in previous versions of the sfsmisc package.

Author

Martin Maechler, Dec 4, 1991 (for S-plus; then called digits.v).

Examples

digitsBase(0:12, 8) #-- octal representation
#> Class 'basedInt'(base = 8) [1:13]
#>      [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10] [,11] [,12] [,13]
#> [1,]    0    0    0    0    0    0    0    0    1     1     1     1     1
#> [2,]    0    1    2    3    4    5    6    7    0     1     2     3     4
empty.dimnames(digitsBase(0:33, 2)) # binary
#> Class 'basedInt'(base = 2) [1:34]
#>                                                                     
#>  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1
#>  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0 0
#>  0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 0 0
#>  0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 1 0 0
#>  0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0 1 1 0 0
#>  0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1

## This may be handy for just one number (and default decimal):
digits <- function(n, base = 10) as.vector(digitsBase(n, base = base))
digits(128982734)     # 1 2 8 9 8 2 7 3 4
#> [1] 1 2 8 9 8 2 7 3 4
digits(128, base = 8) # 2 0 0
#> [1] 2 0 0

## one way of pretty printing (base <= 10!)
b2ch <- function(db)
        noquote(gsub("^0+(.{1,})$"," \\1", 
                apply(db, 2, paste, collapse = "")))
b2ch(digitsBase(0:33, 2))  #->  0 1 10 11 100 101 ... 100001
#>  [1]  0      1      10     11     100    101    110    111    1000   1001 
#> [11]  1010   1011   1100   1101   1110   1111   10000  10001  10010  10011
#> [21]  10100  10101  10110  10111  11000  11001  11010  11011  11100  11101
#> [31]  11110  11111 100000 100001
b2ch(digitsBase(0:33, 4))  #->  0 1 2 3 10 11 12 13 20 ... 200 201
#>  [1]  0   1   2   3   10  11  12  13  20  21  22  23  30  31  32  33 100 101 102
#> [20] 103 110 111 112 113 120 121 122 123 130 131 132 133 200 201

## Hexadecimal:
i <- c(1:20, 100:106)
M <- digitsBase(i, 16)
hexdig <- c(0:9, LETTERS[1:6])
cM <- hexdig[1 + M]; dim(cM) <- dim(M)
b2ch(cM) #->  1  2  3  4  5  6  7  8  9  A  B  C  D  E  F 10 11 ... 6A
#>  [1]  1  2  3  4  5  6  7  8  9  A  B  C  D  E  F 10 11 12 13 14 64 65 66 67 68
#> [26] 69 6A

## IP (Internet Protocol) numbers coding:  <n>.<n>.<n>.<n>  <-->  longinteger
ip_ntoa <- function(n)
        apply(digitsBase(n, base = 256), 2, paste, collapse=".")
ip_ntoa(2130706430 + (0:9))# "126.255.255.254" ... "127.0.0.7"
#>  [1] "126.255.255.254" "126.255.255.255" "127.0.0.0"       "127.0.0.1"      
#>  [5] "127.0.0.2"       "127.0.0.3"       "127.0.0.4"       "127.0.0.5"      
#>  [9] "127.0.0.6"       "127.0.0.7"      
## and the inverse:
ip_aton <- function(a)
        bi2int(lapply(strsplit(a, ".", fixed=TRUE), as.integer), 256)

n <- 2130706430 + (0:9)
head(ip <- ip_ntoa(n))
#> [1] "126.255.255.254" "126.255.255.255" "127.0.0.0"       "127.0.0.1"      
#> [5] "127.0.0.2"       "127.0.0.3"      
head(ip_aton(ip))
#> [1] 2130706430 2130706431 2130706432 2130706433 2130706434 2130706435
stopifnot( n == ip_aton(ip_ntoa(n )),
          ip == ip_ntoa(ip_aton(ip)))


## Inverse of digitsBase() : as.integer method for the "basedInt" class
as.integer(M)
#>   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20 
#>   1   2   3   4   5   6   7   8   9  10  11  12  13  14  15  16  17  18  19  20 
#>  21  22  23  24  25  26  27 
#> 100 101 102 103 104 105 106 
## or also as.intBase() working from strings:
(cb <- apply(digitsBase(0:33, 4), 2, paste, collapse = ""))
#>  [1] "000" "001" "002" "003" "010" "011" "012" "013" "020" "021" "022" "023"
#> [13] "030" "031" "032" "033" "100" "101" "102" "103" "110" "111" "112" "113"
#> [25] "120" "121" "122" "123" "130" "131" "132" "133" "200" "201"
##-> "000" "001" ..... "200" "201"
all(0:33 == as.intBase(cb, base = 4))
#> [1] TRUE