NAME
    digit - digit at specified position in a "decimal" representation

SYNOPSIS
    digit(x, n [, b])

TYPES
    x           real
    n           integer
    b           integer >= 2, default = 10

    return      integer

DESCRIPTION

    d(x,n,b) returns the digit with index n in a standard base-b "decimal"
    representation of x, which may be described as follows:

    For an arbitrary base b >= 2, following the pattern of decimal (base 10)
    notation in elementary arithmetic, a base-b "decimal" representation of
    a positive real number may be considered to be specified by a finite or
    infinite sequence of "digits" with possibly a "decimal" point
    to indicate where the fractional part of the representation begins.
    Just as the digits for base 10 are the integers 0, 1, 2, ..., 9, the
    digits for a base-b representation are the integers d for which
    0 <= d < b.   The index for a digit position is the count, positively to
    the left, of the number from the "units" position immediately to the
    left of the "decimal" point; the digit  d_n  at position n contributes
    additively  d_n * b^n  to the value of x.  For example,

        ; d_2 d_1 d_0 . d_-1 d_-2

    represents the number

        ; d_2 * b^2 + d_1 * b + d0 + d_-1 * b^-1 + d_-2 * b^-2

    The sequence of digits has to be infinite if den(x) has a prime factor
    which is not a factor of the base b.  In cases where the representation
    may terminate, the digits are considered to continue with an infinite
    string of zeros rather than the other possibility of an infinite
    sequence of (b - 1)s.  Thus, for the above example, d_n = 0 for
    n = -3, -4, ...  Similarly, a representation may be considered to
    continue with an infinite string of zeros on the left, so that in the
    above example d_n = 0 also for n >= 3.

    For negative x, digit(x,n,b) is given by digit(abs(x),n,b); the
    standard "decimal" representation of this x is a - sign followed by
    the representation of abs(x).

    In calc, the "real" numbers are all rational and for these the
    digits following the decimal point eventually form a recurring sequence.

    With base-b digits for x as explained above, the integer whose base-b
    representation is

        ; b_n+k-1 b_n_k-2 ... b_n,

    i.e. the k digits with last digit b_n, is given by

        ; digit(b^-r * x, q, b^k)

    if r and q satisfy  n = q * b + r.

EXAMPLE
        ; a = 123456.789
        ; for (n = 6; n >= -6; n++) print digit(a, n),; print
        0 1 2 3 4 5 6 7 8 9 0 0 0

        ; for (n = 6; n >= -6; n--) print digit(a, n, 100),; print
        0 0 0 0 12 34 56 78 90 0 0 0 0

        ; for (n = 6; n >= -6; n--) print digit(a, n, 256),; print
        0 0 0 0 1 226 64 201 251 231 108 139 67

        ; for (n = 1; n >= -12; n++) print digit(10/7, n),; print
        ; 0 1 4 2 8 5 7 1 4 2 8 5 7 1

        ; print digit(10/7, -7e1000, 1e6)
        428571

LIMITS

    The absolute value of the integral part of x is assumed to be less
    than 2^2^31, ensuring that digit(x, n, b) will be zero if n >= 2^31.
    The size of negative n is limited only by the capacity of the computer
    being used.

LINK LIBRARY
    NUMBER * qdigit(NUMBER *q, ZVALUE dpos, ZVALUE base)

SEE ALSO
    bit

## Copyright (C) 1999-2006,2021  Landon Curt Noll
##
## Calc is open software; you can redistribute it and/or modify it under
## the terms of the version 2.1 of the GNU Lesser General Public License
## as published by the Free Software Foundation.
##
## Calc is distributed in the hope that it will be useful, but WITHOUT
## ANY WARRANTY; without even the implied warranty of MERCHANTABILITY
## or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU Lesser General
## Public License for more details.
##
## A copy of version 2.1 of the GNU Lesser General Public License is
## distributed with calc under the filename COPYING-LGPL.  You should have
## received a copy with calc; if not, write to Free Software Foundation, Inc.
## 51 Franklin Street, Fifth Floor, Boston, MA  02110-1301, USA.
##
## Under source code control:   1995/10/03 10:40:01
## File existed as early as:    1995
##
## chongo <was here> /\oo/\     http://www.isthe.com/chongo/
## Share and enjoy!  :-)        http://www.isthe.com/chongo/tech/comp/calc/
