Shareware Super Platinum 8/Shareware Super Platinum 8.iso/mac/PROGTOOL/JTXT.ZIP;1/HOOKS.TXT

Text File | 1991-11-26 | 45.7 KB | 1,513 lines

         Hooks and Forks and the Teaching of Elementary Arithmetic

                            Donald B. McIntyre
                          Luachmhor, Church Road
                         Kinfauns, Perth PH2  7LD
                              Scotland - U.K.
                          Telephone:  [phone]

    Introduction:

    After teaching APL to colleagues and students for many years
    [1,2,3], in retirement I am discovering the age-levels at which
    various mathematical concepts can be usefully introduced to
    children with the aid of the executable notation J.   Teachers at
    Edradour School, Pitlochry, are helping me look at the possible use
    of J [4-8] to help children learn arithmetic.   Although lacking
    Zdenek Jizba's experience in the elementary classroom [9], I hope
    the examples given here will encourage teachers to investigate J
    with their pupils.   The underlying notation and syntax of the
    language are explained for the teacher, who must decide how much to
    disclose to particular pupils.

    Algebra passed through three stages:  rhetorical, syncopated, and
    symbolic.   At first words were used, without other symbols.   The
    words were then abbreviated, and eventually the abbreviations
    became so contracted that the origin of the symbols was forgotten.
    New symbols were devised for operations hitherto unknown or not
    formalised [10-12].   J provides a large number of symbols, but the
    teacher (or user) can name these, thus reverting to the rhetorical
    (or perhaps syncopated) stage.   This may at times be a convenience
    for any user;  in particular a teacher may find that children are
    at first more comfortable with words than symbols.

    As we grow we seem destined to recapitulate the intellectual
    history of our race.   Pupils should nevertheless be encouraged to
    use symbols as early as possible.   Dantzig put it well:  "Greek
    thought was essentially non-algebraic, because it was so concrete.
    The abstract operations of algebra, which deal with objects that
    have purposely been stripped of their physical content, could not
    occur to minds which were so intensely interested in the objects
    themselves.   The symbol is not a mere formality;  it is the very
    essence of algebra.   Without the symbol the object is a human





                                    -1-




    perception and reflects all the phases under which the human senses
    grasp it;   replaced by a symbol the object becomes a complete
    abstraction, a mere operand subject to certain indicated
    operations.  ... The symbol has a meaning which transcends the
    object symbolised;  that is why it is not a mere formality.   It is
    the power of transformation that lifts algebra above the level of a
    convenient shorthand." [11, p.80, 87.  Italics in original].

    The J dialect:

    Mathematics is a language [11-14].   Indeed Hogben, in a book of
    great popularity and influence, included a chapter on the grammar
    of mathematical language [13], distinguishing the verbs (functions)
    and nouns (data) of mathematical sentences.   Hui and Iverson et al
    have gone further, recognising and implementing adverbs (monadic
    operators) and conjunctions (dyadic operators) [15];  Bernecky and
    Hui have shown the power of gerunds (function arrays) for parallel
    processing [16];  and Iverson and McDonnell added phrasal forms and
    pronouns [17].   J uses pro-verbs, and similar names assigned to
    adverbs and conjunctions [4].

    J is a powerful dialect of APL available as Shareware implemented
    for a large number of computers [18].   Its roots are in Iverson
    notation [19] and in APL's method of Direct Definition of functions
    [20, 21, 12].   The spelling adopted uses ASCII characters, either
    alone or immediately followed by a period (.) or colon (:).   Thus
    + is plus, +. is  or, and +: is double; % is divide, %. is matrix
    divide, and %: is square root.   As in all dialects of APL, both
    monadic and dyadic meanings are recognised.    Assignment is
    written =. and read "is".
    %%%
    When an expression is executed the result is produced and can be
    assigned to a variable.   If no assignment is made the result is
    displayed.   In the examples below expressions for execution are
    shown after three spaces at the beginning of the line and the
    results are shown on the next line, beginning in the first column
    of the line.   If the expression is a verb or a pro-verb (the name
    of a verb), then the definition of the verb is shown in boxes (see
    examples later in the article).
    %%%









                                    -2-




    Names

    Nouns, verbs, adverbs, conjunctions, and gerunds can all be
    assigned names.   If the name plus is easier to remember than the
    symbol +, simply assign and use the name.   Names used instead of
    nouns are pronouns;  pro-verbs stand for verbs in a similar way;
    and other parts of speech can be named also.   %%%Here I define
    some names for use in the rest of this article.   The rows of the
    table are the defining expressions in J and can be entered like
    this on a computer running J.   In J the comment symbol is NB. and
    anything after that on a line is ignored.   Definitions given here
    are informal;  see the Dictionary for details [4].

    Pro-verbs (or verbs, for short):

          add=. +         NB.   synonyms can be helpful.  See plus
          behead=. }.     NB.   drop first item
          copy=. #        NB.   x # y  is x copies of y.  Compression
          divided_by=. %  NB.   12%3   is   4
          double=. +:     NB.   +: 5   is   10
          floor=. <.      NB.   <. x  drop any fractional part of x
          format=. ":     NB.   width and precision of output
          halve=. -:      NB.   -: 10   is   5
          head=. {.       NB.   take the first item
          increm=. >:     NB.   increm is short for increment.  Add 1
          laminate=. ,:   NB.   join two lists to make a table
          larger_of=. >.  NB.   x >. y   choose the larger value
          left=. [        NB.   return the left argument. Formerly }:
          lesser_of=. <.  NB.   x <. y   choose the smaller value
          less_or_equal=. <:    NB.   1 2 3 <: 2   is  1 1 0
          magnitude=. |   NB.   |y is absolute value of y
          match=. -:      NB.   arguments identical
          minus=. -       NB.   4-6  is  _2
          not=. -.        NB.   converts 0 to 1 and 1 to 0
          off=. 0!:55     NB.   Return to DOS with:   off 0
          one_minus=. -.  NB.   1 - y.   This extends the boolean not
          plus=. +        NB.   synonyms can be helpful.   See add
          power=. ^       NB.   3^2  is  9
          reciprocal=. %  NB.   %2  is  0.5
          residue=. |     NB.   x|y is remainder on dividing x into y
          reverse=. |.    NB.   change a list abcde into edcba
          right=. ]       NB.   return the right argument. Formerly {:
          script=. 0!:2@<:(<@[ 0!:2 <@]) NB.  Read/Write script files
                          NB.   'output.fil' script 'input.fil' OR





                                    -3-




                          NB.   'output.fil' script ''  OR
                          NB.   script 'input.fil'
          show=. ]        NB.   return the right argument
          shape=. $       NB.   length of a list or number of rows
                          NB.     and columns of a table.
          signum=. *      NB.   * _5 0 7    is   _1 0 1
          tally=. #       NB.   number of items.  First item of shape
          times=. *       NB.   3*4  is  12
          times_pi=. o.   NB.   Multiply by pi.   o.1  is  pi
          transpose=. |:  NB.   turn a table (or flat) on its side
          tree=. 5!:4 @ < NB.   display structure of a defined verb;
                          NB.     tree 'mean'
          wholes=. i.     NB.   list integers, starting with 0
                          NB.     i. 10

    Pro-adverbs (or adverbs, for short):

          cross=. ~    NB.  switch or interchange arguments
          fix=. f.     NB.  fix a verb: "compile" it into J symbols
          insert=. /   NB.  insert verb between items
          scan=. \     NB.  apply verb to successively longer subsets

    Pro-conjunctions (or conjunctions, for short):

          atop=. after=. @   NB.  create a verb by bonding two verbs
          rank=. "           NB   explained in context
          with=. &           NB.  bond a noun to a verb
          %%%
    This use of with for bonding a noun to a verb is sometimes called
    Currying, after the mathematician Haskell B. Curry [15].

    Note that (plus insert) is a derived verb, just as run quickly is a
    new verb created from an old one modified by an adverb.   Also
    (plus insert scan) is yet another verb, derived from the first by
    two adverbs, as in run very quickly.

    Conjunctions bond nouns or verbs together to form new verbs as in
    divide by 4, or run and hide.   Thus:
          quarter=. % with 4
    and
           >: @ i.
    is a new verb that generates wholes (integers) starting with 1







                                    -4-




    Plus, Times, Power:

    To add up a list of numbers, insert the verb plus between each
    item:
          1 plus 1 plus 1 plus 1 plus 1
    5
    The operation is written concisely by defining the new verb total,
    and using copy to repeat an item:
          total=. plus insert
          total 5 copy 1
    5
    Obtain partial totals by using scan:
          total scan 10 copy 1
    1 2 3 4 5 6 7 8 9 10

    The sum of six twos is:
          2 plus 2 plus 2 plus 2 plus 2 plus 2
    12
          total 6 copy 2
    12
    Multiplication was, of course, invented to do this conveniently:
          6 times 2
    12

    Similarly:
          2 times 2 times 2 times 2 times 2 times 2
    64
          times insert 6 copy 2
    64

    Exponentiation (raising a number to a power) was invented to make
    this easier:
          2 power 6
    64
    The order of the arguments makes a difference (6 copies of 2 is not
    the same as 2 copies of 6); i.e. power is not commutative:
          times insert 2 copy 6
    36
          6 power 2
    36
    The adverb cross interchanges the arguments:
          2 copy cross 6
    2 2 2 2 2 2






                                    -5-




          2 power~ 6
    36

    Sum of a list:

       show i=. increm wholes 10
    1 2 3 4 5 6 7 8 9 10
       total increm wholes 10
    55
    This total can be obtained (as young Gauss knew) by halving the
    product of the last number and the last number plus 1 [9]:
       i plus reverse i
    11 11 11 11 11 11 11 11 11 11
       halve 10 times 11
    55

       total increm wholes 100
    5050

    Noting that:
       1 2 3 + 100 99 98
    101 101 101

       halve 100 times increm 100
    5050

    And
       total increm wholes 1000
    500500
       halve 1000 times increm 1000
    500500


    Hooks [4, 17]:

    The expressions i plus reverse i and  y times increm y are examples
    of a commonly encountered construction.    Two verbs (call them g
    and h) are applied to the data (call it y) in a special way;
    namely, the result of applying h to y becomes the right argument of
    g, and y  appears again as the left argument of g.   The syntax of
    J enables this to be written concisely as:

           (g h) y     instead of     y g (h y)






                                    -6-




    This construction is called a hook.   Note that g, with arguments
    both to the left and right, is dyadic, whereas h, with an argument
    on the right only, is monadic.   The resulting hook is in this case
    monadic (the argument appears only once).   Thus, instead of
    writing:
          i plus reverse i
    11 11 11 11 11 11 11 11 11 11
    write:
          (plus reverse) i
    11 11 11 11 11 11 11 11 11 11
    Similarly:
          halve 100 times increm 100
    5050
          halve (times increm) 100
    5050
    %%%
    The Sum of the Positive Whole Numbers from 1 to n is given by the
    defined verb spwn:

       spwn=. halve after (times increm)
       spwn 1000
    500500
    %%%
    Parentheses are needed to get the correct hook, because a
    conjunction (after) seizes the item immediately to its right as its
    right argument.

    Dyadic hooks are also common.   They occur when x is modified by
    some function of y.   The syntax is:

               x (g h) y    instead of    x g (h y)

    Here is an example.   How many steps are there if starting at _2 we
    go to 5? [22]   Not counting the _2 we start at, there are 7 steps:
          # _1 0 1 2 3 4 5
    7
    This number is the magnitude of the difference between _2 and 5:
          (magnitude atop (minus insert)) _2 5
    7
    Or using J symbols:
          n=. | @ (-/)
          n _2 5
    7






                                    -7-




    If we proceed in steps other than unit steps, we must divide by the
    step size.   The required verb is a hook combining the two verbs
    into and n.   Name it h:
          into=. %~
          nsteps=. into n
    Then for half steps, we have:
          0.5 nsteps _2 5
    14

    Display nsteps to see that it consists of two adjacent verbs;  i.e.
    a hook.
          nsteps
    ⁄ƒƒƒƒ¬ƒø
    ≥into≥n≥
    ¿ƒƒƒƒ¡ƒŸ
    %%%Fix nsteps, so that subsequent changes to its components (into
    and n), will not affect it.
          nsteps=. nsteps fix

          nsteps
    ⁄ƒƒƒƒƒ¬ƒƒƒƒƒƒƒƒƒƒƒø
    ≥⁄ƒ¬ƒø≥⁄ƒ¬ƒ¬ƒƒƒƒƒø≥
    ≥≥%≥~≥≥≥|≥@≥⁄ƒ¬ƒø≥≥
    ≥¿ƒ¡ƒŸ≥≥ ≥ ≥≥-≥/≥≥≥
    ≥     ≥≥ ≥ ≥¿ƒ¡ƒŸ≥≥
    ≥     ≥¿ƒ¡ƒ¡ƒƒƒƒƒŸ≥
    ¿ƒƒƒƒƒ¡ƒƒƒƒƒƒƒƒƒƒƒŸ

    (( %~ )(| @ ( -/ )))

    This is a display of the definition of nsteps.   It shows how the
    expression will be parsed.   There are two outer boxes, and hence
    this a hook.   Contiguous boxes show that the symbols in them are
    to be grouped together.   We can immediately translate the display
    into the fully parenthesised form written immediately below it.
    There is no harm in using the fully parenthesised form, but
    experimentation will show whether parentheses can be omitted with
    impunity.   In this case the parentheses around -/ are needed to
    prevent the conjunction atop (@) from taking the - instead of (-/)
    as its right argument.    Adverbs are monadic, taking the item to
    the left as the argument.   Thus cross (~) modifies the verb
    divided_by (%) to its left and no parentheses are needed to ensure
    this.   So nsteps can be written with a single set of parentheses.
    No spaces are needed, but I have used one to draw attention to the
    two groups of symbols that make the hook.   As we shall see, three
    groups make a fork.



                                    -8-




    %%%
          nsteps=. %~ |@(-/)
          0.5 nsteps _2 5
    14

    Two hooks are used in the scaling (or normalising) of a list of
    numbers to make the range from 0 to 1.   Such scaling is often
    needed when preparing graphical display.

    Let list be the name of the list of numbers nine, three, four,
    negative 2, ..., seven:
          list=. 9 3 4 _2 12 1 _4 15 7

    The smallest value will be 0 if we subtract the smallest value.
    This should be quickly recognised as a monadic hook:
          list - <./list
    13 7 8 2 16 5 0 19 11
          (- <./) list
    13 7 8 2 16 5 0 19 11
          q=. minus lesser_of insert
          q=. - <./
          q list
    13 7 8 2 16 5 0 19 11

    Similarly:

    The largest number will be 1 if we divide list by the largest
    value.   This is another monadic hook.
          p=. % >./

    scale is the composite verb in which p is applied to the result of
    q;  i.e.  p is atop q
          scale=. p@q
          0.3 format scale list
    0.684 0.368 0.421 0.105 0.842 0.263 0.000 1.000 0.579

    %%%
    Forks [4, 17]:

    A fork is a succession, or train, of three verbs.   The meaning
    assigned is this:
            (f g h) y    is    (f y) g (h y)           Monadic
          x (f g h) y    is   (x f ) g (x h y)         Dyadic






                                    -9-




    The mean is an example of a monadic fork.   Compute it by dividing
    the total [of the list] by the tally [the number of items in the
    list]:%%%
          y=. 1 2 3 4
          (total y) divided_by (tally y)
    2.5
    Display mean in various ways:

          mean=. total divided_by tally

          mean
    ⁄ƒƒƒƒƒ¬ƒƒƒƒƒƒƒƒƒƒ¬ƒƒƒƒƒø
    ≥total≥divided_by≥tally≥
    ¿ƒƒƒƒƒ¡ƒƒƒƒƒƒƒƒƒƒ¡ƒƒƒƒƒŸ
    %%%
    The string of three boxes shows that mean is a fork.   The same
    information is given by another kind of display, called a parse
    tree.

          tree 'mean'
             ⁄ƒ total
    ƒ mean ƒƒ≈ƒ divided_by
             ¿ƒ tally

    Fix the verb, "compiling" it into primitive J symbols.

          mean=. mean fix%%%
          mean
    ⁄ƒƒƒƒƒ¬ƒ¬ƒø
    ≥⁄ƒ¬ƒø≥%≥#≥
    ≥≥+≥/≥≥ ≥ ≥
    ≥¿ƒ¡ƒŸ≥ ≥ ≥
    ¿ƒƒƒƒƒ¡ƒ¡ƒŸ

          tree 'mean'
             ⁄ƒ / ƒƒƒ +
    ƒ mean ƒƒ≈ƒ %
             ¿ƒ #

          mean 1 2 3 4
    2.5

    mean is a monadic fork because it takes only a right argument.
    The syntax is:





                                   -10-




                (f g h) y   is the same as    (f y) g (h y)

    The arithmetic progression vector (apv) given below is an example
    of a dyadic fork.   This is its syntax:
              x (f g h) y   is the same as    (x f y) g (x h y)

    An example of a fork including a hook as one prong is the verb
    clean, which sets to zero small values resulting from round-off
    errors.   For data take:
          show y=. 1 10 100 1000 10000 into 1.2345
    1.2345 0.12345 0.012345 0.0012345 0.00012345

    Let the right argument y be the data to be cleaned, and let
    tolerance be the left argument x;  then the result is y times the
    truth or falsity (1 or 0) of the proposition that the tolerance is
    less-than-or-equal to the magnitude of y:
           y * (x proposition y)

    The proposition is a dyadic hook:
           0.01 (less_or_equal magnitude) y
    1 1 1 0 0
           0.01 <: (| y)
    1 1 1 0 0

    times is, of course, the central term of the fork:

    The verb clean is therefore:
          clean=. right times (less_or_equal magnitude)

          clean
    ⁄ƒƒƒƒƒ¬ƒƒƒƒƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒø
    ≥right≥times≥⁄ƒƒƒƒƒƒƒƒƒƒƒƒƒ¬ƒƒƒƒƒƒƒƒƒø≥
    ≥     ≥     ≥≥less_or_equal≥magnitude≥≥
    ≥     ≥     ≥¿ƒƒƒƒƒƒƒƒƒƒƒƒƒ¡ƒƒƒƒƒƒƒƒƒŸ≥
    ¿ƒƒƒƒƒ¡ƒƒƒƒƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ

          tree 'clean'

              ⁄ƒ right
              √ƒ times
    ƒ clean ƒƒ¥        ⁄ƒ less_or_equal
              ¿ƒƒƒƒƒƒƒƒ¡ƒ magnitude

    %%%





                                   -11-




          x=. 0.01
          y=. 1 10 100 1000 10000 into 1.2345

          x clean y
    1.2345 0.12345 0.012345 0 0
          x clean -y
    _1.2345 _0.12345 _0.012345 0 0

    If we left off the parentheses then less_or_equal would be the
    central term of a fork, which along with right would make a hook.
          f=. right times less_or_equal magnitude
          f
    ⁄ƒƒƒƒƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒø
    ≥right≥⁄ƒƒƒƒƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒ¬ƒƒƒƒƒƒƒƒƒƒø≥
    ≥     ≥≥times≥less_or_equal≥magnitude ≥≥
    ≥     ≥¿ƒƒƒƒƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒ¡ƒƒƒƒƒƒƒƒƒƒŸ≥
    ¿ƒƒƒƒƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ
          f=. f fix
          f
    ⁄ƒ¬ƒƒƒƒƒƒƒƒø
    ≥]≥⁄ƒ¬ƒƒ¬ƒø≥
    ≥ ≥≥*≥<:≥|≥≥
    ≥ ≥¿ƒ¡ƒƒ¡ƒŸ≥
    ¿ƒ¡ƒƒƒƒƒƒƒƒŸ

    This would be quite wrong!   Consider why this is so.   A dyadic
    hook like this can be expanded:

          x (g h) y     expands to    x g (h y)
    In our case:
          g=. ]
          h=. *<:|

          x (g h) y
    1 0 0 0 0
          x g (h y)
    1 0 0 0 0

    Because g is the dyadic verb right it returns its right argument;
    so x can make no contribution to the result, which must depend
    solely on the monadic function h applied to y.

          h y
    1 0 0 0 0





                                   -12-




    Defining the following pro-verbs:
          p=.*
          q=. <:
          r=. |
    Expand the fork h:
          (p q r) y
    1 0 0 0 0
          (p y) q (r y)
    1 0 0 0 0
    But
          r y
    1.2345 0.12345 0.012345 0.0012345 0.00012345
    And
          p y
    1 1 1 1 1
          (* y) <: (|y)
    1 0 0 0 0
    The monadic times (*) is called the signum.   The result is 1, 0,
    or _1 depending upon whether the argument is positive, zero, or
    negative.   Hence with this definition of f

          f=. right times less_or_equal magnitude

    the left argument is ignored and the result is 1 if the signum of y
    is less than or equal to the absolute value of y.   This is true
    unless y is a positive number lying between 0 and 1.   This might
    be a useful function in another situation, but it is not what we
    want for clean!

    Iverson's utility verbs by and over provide a convenient way to
    illustrate the result.   See the Appendix for an explanation.

       x=. 9 0.5 0 _0.5 _9
       y=. _9 _1.5 _1 _0.5 0 1e_3 0.3 0.4 0.999 1 2 10 100

       over=. ({.,.@;}.)@":@,
       by=. ' '&;@,.@[,.]












                                   -13-





       x by y over x f"0 1 y
    ⁄ƒƒƒƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒø
    ≥    ≥_9 _1.5 _1 _0.5 0 0.001 0.3 0.4 0.999 1 2 10 100≥
    √ƒƒƒƒ≈ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ¥
    ≥   9≥ 1    1  1    1 1     0   0   0     0 1 1  1   1≥
    ≥ 0.5≥ 1    1  1    1 1     0   0   0     0 1 1  1   1≥
    ≥   0≥ 1    1  1    1 1     0   0   0     0 1 1  1   1≥
    ≥_0.5≥ 1    1  1    1 1     0   0   0     0 1 1  1   1≥
    ≥  _9≥ 1    1  1    1 1     0   0   0     0 1 1  1   1≥
    ¿ƒƒƒƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ

    The rank conjunction (") here instructs f to use rank-0 cells
    (atoms, units, or scalars) from the left argument with rank-1 cells
    (lists, longs, or vectors) of the right argument.

    We can return to the definition of clean with added understanding.
    It is helpful to use spaces in order to draw attention to the three
    verbs of the fork, but spaces are not required:%%%
         clean=. ] * (<:|)

         clean
    ⁄ƒ¬ƒ¬ƒƒƒƒƒƒø
    ≥]≥*≥⁄ƒƒ¬ƒø≥
    ≥ ≥ ≥≥<:≥|≥≥
    ≥ ≥ ≥¿ƒƒ¡ƒŸ≥
    ¿ƒ¡ƒ¡ƒƒƒƒƒƒŸ


    The Number Line:

    Negative numbers once seemed absurd or fictitious [23, p.252], but
    a number line makes it easy for a child to grasp a concept that
    once taxed the greatest mathematicians.   In a few years of
    instruction we are each led through the stages that took our
    ancestors generations to achieve.   Experiments should help:

          show i=. wholes 10
    0 1 2 3 4 5 6 7 8 9
          10 plus i
    10 11 12 13 14 15 16 17 18 19

    Number lines are produced by the following fork:
          nl=. wholes minus floor atop halve





                                   -14-





          nl
    ⁄ƒƒƒƒƒƒ¬ƒƒƒƒƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒø
    ≥wholes≥minus≥⁄ƒƒƒƒƒ¬ƒ¬ƒƒƒƒƒø≥
    ≥      ≥     ≥≥floor≥@≥halve≥≥
    ≥      ≥     ≥¿ƒƒƒƒƒ¡ƒ¡ƒƒƒƒƒŸ≥
    ¿ƒƒƒƒƒƒ¡ƒƒƒƒƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ

    Or, more concisely in J symbols:
          nl=. i. - <.@-:
    ⁄ƒƒ¬ƒ¬ƒƒƒƒƒƒƒƒƒø
    ≥i.≥-≥⁄ƒƒ¬ƒ¬ƒƒø≥
    ≥  ≥ ≥≥<.≥@≥-:≥≥
    ≥  ≥ ≥¿ƒƒ¡ƒ¡ƒƒŸ≥
    ¿ƒƒ¡ƒ¡ƒƒƒƒƒƒƒƒƒŸ
          nl 11
    _5 _4 _3 _2 _1 0 1 2 3 4 5

    Various number lines can be produced simultaneously by using the
    rank conjunction (") to specify that rank-0 cells (atoms, units, or
    scalars) on the left are to be combined with rank-1 cells (lists,
    longs, or vectors) on the right:

          1 10 100 1000 times"0 1 nl 11
       _5    _4    _3    _2    _1 0    1    2    3    4    5
      _50   _40   _30   _20   _10 0   10   20   30   40   50
     _500  _400  _300  _200  _100 0  100  200  300  400  500
    _5000 _4000 _3000 _2000 _1000 0 1000 2000 3000 4000 5000

    The argument for nl should be odd.  To ensure that an even argument
    is made odd, define odd as a monadic hook:
    %%%
       odd=. plus one_minus@(2&residue)
       odd
    ⁄ƒƒƒƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒø
    ≥plus≥⁄ƒƒƒƒƒƒƒƒƒ¬ƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒø≥
    ≥    ≥≥one_minus≥@≥⁄ƒ¬ƒ¬ƒƒƒƒƒƒƒø≥≥
    ≥    ≥≥         ≥ ≥≥2≥&≥residue≥≥≥
    ≥    ≥≥         ≥ ≥¿ƒ¡ƒ¡ƒƒƒƒƒƒƒŸ≥≥
    ≥    ≥¿ƒƒƒƒƒƒƒƒƒ¡ƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒŸ≥
    ¿ƒƒƒƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ
    %%%
       odd=. + -.@(2&|)
       odd





                                   -15-




    ⁄ƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒø
    ≥+≥⁄ƒƒ¬ƒ¬ƒƒƒƒƒƒƒø≥
    ≥ ≥≥-.≥@≥⁄ƒ¬ƒ¬ƒø≥≥
    ≥ ≥≥  ≥ ≥≥2≥&≥|≥≥≥
    ≥ ≥≥  ≥ ≥¿ƒ¡ƒ¡ƒŸ≥≥
    ≥ ≥¿ƒƒ¡ƒ¡ƒƒƒƒƒƒƒŸ≥
    ¿ƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ

          odd 2 4 6 8 laminate 1 3 5 7
    3 5 7 9
    1 3 5 7
    %%%
    This works by dividing the argument by 2 and subtracting the
    remainder (residue) from 1.   The result is then added to the
    argument.   Because the remainder on division by 2 is either 0 or 1
    (depending on whether the number is even or odd, subtracting it
    from 1 changes a 0 to a 1 and a 1 to a zero.   A list (vector) of
    0s and 1s is called logical (taking 1 as true and 0 as false) or
    boolean (after George Boole, who taught us how much can be done
    with an algebra restricted to these two numbers).   In logic the
    verb not converts true to false and false to true.   Boolean
    algebra is, however, part of algebra, and its 0 and 1 can be
    combined with ordinary numbers by algebraic operations.   Because
    there is no difference between the verbs one_minus and not (other
    than their domain), J uses the same symbol (-.) for both.
    %%%
    Define the verb nline, which applies nl after using odd:
       nline=. nl @ odd
       nline"0 right 12 13
    _6 _5 _4 _3 _2 _1 0 1 2 3 4 5 6
    _6 _5 _4 _3 _2 _1 0 1 2 3 4 5 6
    %%%
    Using the verb nsteps (defined above to determine the number of
    steps) produce the required number of wholes (integers):%%%
          g=. i.@>.@>:@nsteps
          0.5 g _2 5
    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
          0.5 g _2 5.2
    0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
    %%%
    The arithmetic progression vector is a fork containing a fork that
    in turn contains a hook.   The outside fork is:  the first item of
    the right argument ({.@]) plus the fork [*g, which in turn is the
    left argument times the dyadic application of g.%%%





                                   -16-




          apv=. {.@] + [*g

          apv
    ⁄ƒƒƒƒƒƒƒƒ¬ƒ¬ƒƒƒƒƒƒƒø
    ≥⁄ƒƒ¬ƒ¬ƒø≥+≥⁄ƒ¬ƒ¬ƒø≥
    ≥≥{.≥@≥]≥≥ ≥≥[≥*≥g≥≥
    ≥¿ƒƒ¡ƒ¡ƒŸ≥ ≥¿ƒ¡ƒ¡ƒŸ≥
    ¿ƒƒƒƒƒƒƒƒ¡ƒ¡ƒƒƒƒƒƒƒŸ
          0.5 apv _2 5
    _2 _1.5 _1 _0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
    %%%
    Using J primitives (or fixing apv with fix):
    %%%
       apv=. {.@] + [ * i.@>.@>:@(%~ |@-/)


    Place value:

    In learning arithmetic one of the first tasks is to understand
    place notation;  i.e. the values assigned to digits at each
    successive place in a number.   With this in mind we explore as
    follows:
       times insert scan 10 10 10 10 10 10
    10 100 1000 10000 100000 1000000

    Because successive multiplications are produced by power:
       10 power 1 2 3 4 5 6
    10 100 1000 10000 100000 1000000

    Looking at this result a pupil might wonder what the result of 10
    power 0 would be:
       10 power 0 1 2 3 4 5 6
    1 10 100 1000 10000 100000 1000000

    which leads in turn to the exploration of negative values:
       10 power _4 _3 _2 _1 0 1 2 3 4 5 6
    0.0001 0.001 0.01 0.1 1 10 100 1000 10000 100000 1000000

    The reciprocals are place values:
          reciprocal 10 power _4 _3 _2 _1 0 1 2 3 4 5 6
    10000 1000 100 10 1 0.1 0.01 0.001 0.0001 1e_5 1e_6

    The value assigned to a place is given by the verb place:
       place=.10 with power





                                   -17-




       place 3
    1000

    Because we write numbers so that the larger place values are to the
    left (ancient Egyptians usually did the opposite), the values of
    successive places are:

       reverse place wholes 7
    1000000 100000 10000 1000 100 10 1

    Continuing the series to the right:
          |. 10^ 1 apv _4 4
    10000 1000 100 10 1 0.1 0.01 0.001 0.0001


    Rounding:

    Having grasped place notation, pupils are ready for rounding.
    This must be done mentally, but at an appropriate stage the formal
    process should be introduced.   Because J is an executable
    mathematical notation, its use has the advantage that computing
    skills are taught at the same time.

       100 *<.0.5+438%100       NB.   Round 438 to the nearest 100
    400

       10 *<.0.5+26%10          NB.   Round 26 to the nearest 10
    30

       0.1 *<.0.5+12.345%0.1    NB.   Round 12.345 to the nearest 0.1
    12.3

       0.2 *<.0.5+12.345%0.2    NB.   Round 12.345 to the nearest 0.2
    12.4

    Rounding will, in general, be to an integer multiple of a given
    number.   This is defined formally as:
          round=. '' : 'x. * <. 0.5+ y.%x.'

    where x. and y. are place-holders for the left and right arguments
    respectively.   The empty string preceding the conjunction (:)
    means that we are defining the dyadic case only.   Because the
    arguments are referred to explicitly, this is an example of
    explicit definition, as opposed to the tacit definitions used
    exclusively above.   It can be translated to tacit form by the
    adverb :11  [24]:



                                   -18-





          'x. * <. 0.5+ y.%x.' : 11
    ⁄ƒ¬ƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒø
    ≥[≥*≥⁄ƒƒ¬ƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒø≥
    ≥ ≥ ≥≥<.≥@≥⁄ƒƒƒƒƒƒƒƒƒ¬ƒ¬ƒƒƒƒƒƒƒø≥≥
    ≥ ≥ ≥≥  ≥ ≥≥⁄ƒƒƒ¬ƒ¬ƒø≥@≥⁄ƒ¬ƒ¬ƒø≥≥≥
    ≥ ≥ ≥≥  ≥ ≥≥≥0.5≥&≥+≥≥ ≥≥]≥%≥[≥≥≥≥
    ≥ ≥ ≥≥  ≥ ≥≥¿ƒƒƒ¡ƒ¡ƒŸ≥ ≥¿ƒ¡ƒ¡ƒŸ≥≥≥
    ≥ ≥ ≥≥  ≥ ≥¿ƒƒƒƒƒƒƒƒƒ¡ƒ¡ƒƒƒƒƒƒƒŸ≥≥
    ≥ ≥ ≥¿ƒƒ¡ƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ≥
    ¿ƒ¡ƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ

    This is a fork with * as the central verb.   Because the fork ]%[
    can be written %~  we have:
       round=. [ * <.@(0.5&+@(%~))

       round
    ⁄ƒ¬ƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒø
    ≥[≥*≥⁄ƒƒ¬ƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒø≥
    ≥ ≥ ≥≥<.≥@≥⁄ƒƒƒƒƒƒƒƒƒ¬ƒ¬ƒƒƒƒƒø≥≥
    ≥ ≥ ≥≥  ≥ ≥≥⁄ƒƒƒ¬ƒ¬ƒø≥@≥⁄ƒ¬ƒø≥≥≥
    ≥ ≥ ≥≥  ≥ ≥≥≥0.5≥&≥+≥≥ ≥≥%≥~≥≥≥≥
    ≥ ≥ ≥≥  ≥ ≥≥¿ƒƒƒ¡ƒ¡ƒŸ≥ ≥¿ƒ¡ƒŸ≥≥≥
    ≥ ≥ ≥≥  ≥ ≥¿ƒƒƒƒƒƒƒƒƒ¡ƒ¡ƒƒƒƒƒŸ≥≥
    ≥ ≥ ≥¿ƒƒ¡ƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ≥
    ¿ƒ¡ƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ

       10 100 1000 round 146464
    146460 146500 146000

       y=. 2 4 6 8
       show x=. y round 146463
    146464 146464 146466 146464

    Verify that these results are indeed (integer) multiples of y
       x%y
    73232 36616 24411 18308

       y=.3 7 11 15 25 33 125
       show x=. y round 146464
    146463 146461 146465 146460 146475 146454 146500
       x%y
    48821 20923 13315 9764 5859 4438 1172






                                   -19-




    Rounding 438 to the nearest 100
       (10^2) * <.0.5+438%10^2
    400

    Round to given number of decimal places
       rdp=. '' : '(10^x.) %~ <. 0.5 + y. * 10^x.'
       0 1 2 3 4 5 rdp o.1
    3 3.1 3.14 3.142 3.1416 3.14159

    Rewriting in tacit form:
       rdp=. 10&^@[ %~ <.@(0.5&+@(] * 10&^@[))
    This contains two forks, but the last one (] * 10&^@[) can be
    written as a hook ((* 10&^)~), though this necessitates additional
    parentheses.
       rdp=. 10&^@[ %~ <.@(0.5&+@((* 10&^)~))
       0 1 2 3 4 5 rdp o.1
    3 3.1 3.14 3.142 3.1416 3.14159

       rdp
    ⁄ƒƒƒƒƒƒƒƒƒƒƒƒƒƒ¬ƒƒƒƒƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒø
    ≥⁄ƒƒƒƒƒƒƒƒ¬ƒ¬ƒø≥⁄ƒ¬ƒø≥⁄ƒƒ¬ƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒø≥
    ≥≥⁄ƒƒ¬ƒ¬ƒø≥@≥[≥≥≥%≥~≥≥≥<.≥@≥⁄ƒƒƒƒƒƒƒƒƒ¬ƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒø≥≥
    ≥≥≥10≥&≥^≥≥ ≥ ≥≥¿ƒ¡ƒŸ≥≥  ≥ ≥≥⁄ƒƒƒ¬ƒ¬ƒø≥@≥⁄ƒƒƒƒƒƒƒƒƒƒƒƒ¬ƒø≥≥≥
    ≥≥¿ƒƒ¡ƒ¡ƒŸ≥ ≥ ≥≥     ≥≥  ≥ ≥≥≥0.5≥&≥+≥≥ ≥≥⁄ƒ¬ƒƒƒƒƒƒƒƒø≥~≥≥≥≥
    ≥¿ƒƒƒƒƒƒƒƒ¡ƒ¡ƒŸ≥     ≥≥  ≥ ≥≥¿ƒƒƒ¡ƒ¡ƒŸ≥ ≥≥≥*≥⁄ƒƒ¬ƒ¬ƒø≥≥ ≥≥≥≥
    ≥              ≥     ≥≥  ≥ ≥≥         ≥ ≥≥≥ ≥≥10≥&≥^≥≥≥ ≥≥≥≥
    ≥              ≥     ≥≥  ≥ ≥≥         ≥ ≥≥≥ ≥¿ƒƒ¡ƒ¡ƒŸ≥≥ ≥≥≥≥
    ≥              ≥     ≥≥  ≥ ≥≥         ≥ ≥≥¿ƒ¡ƒƒƒƒƒƒƒƒŸ≥ ≥≥≥≥
    ≥              ≥     ≥≥  ≥ ≥≥         ≥ ≥¿ƒƒƒƒƒƒƒƒƒƒƒƒ¡ƒŸ≥≥≥
    ≥              ≥     ≥≥  ≥ ≥¿ƒƒƒƒƒƒƒƒƒ¡ƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ≥≥
    ≥              ≥     ≥¿ƒƒ¡ƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ≥
    ¿ƒƒƒƒƒƒƒƒƒƒƒƒƒƒ¡ƒƒƒƒƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ

    Rounding to a given number of decimal places is, of course, only a
    special case of round:
          (10^-i.6) round o.1
    3 3.1 3.14 3.142 3.1416 3.14159

    The hook is obvious in the following:
       h=. (]*<.@(0.5&+@%)) 10&^
       rnd=. h~








                                   -20-




       0 _1 _2 _3 _4 _5 rnd o.1
    3 3.1 3.14 3.142 3.1416 3.14159

       x=. i.6
       y=. 146464 14646 1464 146
       table=. (10^x) round"0 1 y
    %%%
    Notice that we simultaneously round many numbers to several
    different places and produce a table of rounded values.
    (Iverson's utility verbs by and over are explained in the
    Appendix)%%%

       y by x over transpose table
    ⁄ƒƒƒƒƒƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒø
    ≥      ≥     0      1      2      3      4      5≥
    √ƒƒƒƒƒƒ≈ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ¥
    ≥146464≥146464 146460 146500 146000 150000 100000≥
    ≥ 14646≥ 14646  14650  14600  15000  10000      0≥
    ≥  1464≥  1464   1460   1500   1000      0      0≥
    ≥   146≥   146    150    100      0      0      0≥
    ¿ƒƒƒƒƒƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ

       table match x rnd "0 1 y
    1

    Letting f be the fork and h the hook:
       f=. ] * <.@(0.5&+@%)
       h=. (f 10&^)~
       table -: x h"0 1 y
    1


    Examples of rounding exercises from a school workbook [25, p.14,
    15, 22]:

       n,: 10 round n=.12 26 165 14 38 43 56 65 97 145 235
    12 26 165 14 38 43 56 65  97 145 235
    10 30 170 10 40 40 60 70 100 150 240

    n,: 100 round n=.170 438 650 160 250 463 729 607 896 717 91 332 548
    170 438 650 160 250 463 729 607 896 717  91 332 548
    200 400 700 200 300 500 700 600 900 700 100 300 500

       10 100 1000 round 8478





                                   -21-




    8480 8500 8000

    Round the numbers n to the place values given by p:
       n=. 8217 4096 7358 6105 8654 5583 7950 6008
       p=. 10 100 1000
       table=.|: p round"0 1 n

       n by p over table
    ⁄ƒƒƒƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒø
    ≥    ≥  10  100 1000≥
    √ƒƒƒƒ≈ƒƒƒƒƒƒƒƒƒƒƒƒƒƒ¥
    ≥8217≥8220 8200 8000≥
    ≥4096≥4100 4100 4000≥
    ≥7358≥7360 7400 7000≥
    ≥6105≥6110 6100 6000≥
    ≥8654≥8650 8700 9000≥
    ≥5583≥5580 5600 6000≥
    ≥7950≥7950 8000 8000≥
    ≥6008≥6010 6000 6000≥
    ¿ƒƒƒƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ


    Concluding Remarks:

    This paper illustrates some aspects of the J dialect of APL using
    examples related to the teaching of elementary arithmetic.    It
    demonstrates that hooks and forks are ubiquitous, and shows how to
    use them in reading and writing J.   Examples are given of
    displays that show how expressions are parsed.   These are
    invaluable aids in understanding the language.

    Definitions are given in tacit form.   They are functional as
    advocated by Backus [26].

    Because J is evolving, it is essential to note which version is
    used.  Examples are included of changes that were made between
    versions. The files Status.Doc and Xenos.Doc, provided with the
    system, document changes from earlier versions.   All examples
    included in this paper have been executed with Version 3.5x4.

    As from Version 3.5x1, 1991 8 26, the two remaining "system
    commands" have been replaced by verbs created by the external
    conjunction (!:).
         )script       replaced by 0!:2





                                   -22-




         )sscript      replaced by 0!:3
         )off          replaced by 0!:55
    It is convenient to include the following verbs in the file
    profile.js, which is loaded automatically at the start of a
    session.
         script=. 0!:2@< : (<@[ 0!:2 <@])
         off=. 0!:55

    script is an ambi-valent verb that permits specification of output
    and input script files;  e.g.   'output.fil' script 'input.fil'

    Because off is a verb, it must have an argument.   To terminate a
    session and return to DOS, enter off 0

    Remembering Klein's Elementary Mathematics from an Advanced
    Standpoint [27], in this paper I have treated topics in elementary
    arithmetic from a relatively advanced viewpoint.   I hope teachers
    will find stimulation for themselves and be able to select examples
    suitable for pupils at various levels.   Even very young children
    can use some of the examples, if only to check their answers.   The
    exercise could provide an excellent introduction to both computing
    and mathematics.   The literature of J is still limited.   This
    paper may be useful as an introduction for anyone wishing to try J.


    Acknowledgements:
    %%%
    Kenneth Iverson, Roger Hui, and Eugene McDonnell (Iverson Software
    Inc.) generously provided invaluable support on numerous occasions.
    Mrs. Rosemary Roache and Julian Romanes (Edradour School) gave
    insight into the teaching of elementary mathematics.   Anthony
    Camacho and Graham Woyka encouraged me to write about learning J.
    Anthony Camacho's suggestions improved the paper.
    %%%















                                   -23-




    Appendix: Displaying a table with over and by:

    Iverson has provided the utility verbs over and by [5, p.7; 28,
    p.272].   Users new to J will find them well worthy of study in
    their own right.   The following discussion may be helpful.

       over=. ({.,.@;}.)@":@,
       by=. ' '&;@,.@[,.]

    Notice that over contains the fork  {. ,.@; }. and by is built upon
    the fork  ' '&;@,.@[) ,. ]

    While reference [8] was in press, the symbols for Ravel Items and
    Raze were interchanged (Version 3.2, June 1991).   Because the
    monadic form Ravel Items changed from ";" in J3 to the form ",."
    (as in J3.5x4, 1991 10 22), the versions of over and by given in
    [8] must now be written as follows:
          over=. ,. @ ({. ; }.) @ ": @ ,
          by=. (,~"_1 ' '&; @ ,.)~

    In these versions, over contains the fork  {. ; }. and by is built
    on the hook  (,~"_1) (' '&;@,.)
    J's displays are invaluable aids in learning to read and write
    verbs like these.   Compare two versions of by each in two types of
    display:

          by=. ' '&;@,.@[,.]
          by
    ⁄ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ¬ƒƒ¬ƒø
    ≥⁄ƒƒƒƒƒƒƒƒƒƒƒƒƒƒ¬ƒ¬ƒø≥,.≥]≥
    ≥≥⁄ƒƒƒƒƒƒƒ¬ƒ¬ƒƒø≥@≥[≥≥  ≥ ≥
    ≥≥≥⁄ƒ¬ƒ¬ƒø≥@≥,.≥≥ ≥ ≥≥  ≥ ≥
    ≥≥≥≥ ≥&≥;≥≥ ≥  ≥≥ ≥ ≥≥  ≥ ≥
    ≥≥≥¿ƒ¡ƒ¡ƒŸ≥ ≥  ≥≥ ≥ ≥≥  ≥ ≥
    ≥≥¿ƒƒƒƒƒƒƒ¡ƒ¡ƒƒŸ≥ ≥ ≥≥  ≥ ≥
    ≥¿ƒƒƒƒƒƒƒƒƒƒƒƒƒƒ¡ƒ¡ƒŸ≥  ≥ ≥
    ¿ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ¡ƒƒ¡ƒŸ

          tree 'by'
                             ⁄ƒ
                       ⁄ƒ & ƒ¡ƒ ;
                 ⁄ƒ @ ƒ¡ƒ ,.
           ⁄ƒ @ ƒ¡ƒ [
    ƒ by ƒƒ≈ƒ ,.
           ¿ƒ ]




                                   -24-






          by=. (,~"_1 ' '&; @ ,.)~
          by
    ⁄ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ¬ƒø
    ≥⁄ƒƒƒƒƒƒƒƒƒƒƒƒ¬ƒƒƒƒƒƒƒƒƒƒƒƒƒƒø≥~≥
    ≥≥⁄ƒƒƒƒƒ¬ƒ¬ƒƒø≥⁄ƒƒƒƒƒƒƒ¬ƒ¬ƒƒø≥≥ ≥
    ≥≥≥⁄ƒ¬ƒø≥"≥_1≥≥≥⁄ƒ¬ƒ¬ƒø≥@≥,.≥≥≥ ≥
    ≥≥≥≥,≥~≥≥ ≥  ≥≥≥≥ ≥&≥;≥≥ ≥  ≥≥≥ ≥
    ≥≥≥¿ƒ¡ƒŸ≥ ≥  ≥≥≥¿ƒ¡ƒ¡ƒŸ≥ ≥  ≥≥≥ ≥
    ≥≥¿ƒƒƒƒƒ¡ƒ¡ƒƒŸ≥¿ƒƒƒƒƒƒƒ¡ƒ¡ƒƒŸ≥≥ ≥
    ≥¿ƒƒƒƒƒƒƒƒƒƒƒƒ¡ƒƒƒƒƒƒƒƒƒƒƒƒƒƒŸ≥ ≥
    ¿ƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒƒ¡ƒŸ

          tree 'by'
                       ⁄ƒ ~ ƒƒƒ ,
                 ⁄ƒ " ƒ¡ƒ _1
                 ≥           ⁄ƒ
    ƒ by ƒƒ ~ ƒƒƒ¥     ⁄ƒ & ƒ¡ƒ ;
                 ¿ƒ @ ƒ¡ƒ ,.





























                                   -25-




    References

        [1]  Donald B. McIntyre, Introduction to the Study of Data
             Matrices, In: Models of Geologic Processes:  an
             Introduction to Mathematical Geology.  Short Course
             Lecture Notes, Philadelphia, November 1969. Edited by
             Peter Fenner.   American Geological Institute, Washington
             D.C. (1969) p.A1-A44, B1-B17, C1-C4.

        [2]    Donald B. McIntyre, The Architectural Elegance of Crystals
             made clear by APL, In:  Conference Proceedings, APL Users
             Meeting, Toronto, September 1978.   Sponsored by I.P.
             Sharp Associates, Toronto (1978) p.233-250.

        [3]    Donald B. McIntyre, APL in a Liberal Arts College, In:
             Conference Proceedings, APL Users Meeting, Toronto,
             October 1980.  Sponsored by I.P. Sharp Associates, Toronto
             (1980), p.544-574.

        [4]    Kenneth E. Iverson, ISI Dictionary of J, Version 3.3 with
             Tutorials, Iverson Software inc., Toronto (1991) 32pp.

        [5]    Kenneth E. Iverson, Programming in J, Iverson Software
             inc., Toronto (1991) 71pp.

        [6]    Kenneth E. Iverson, Tangible Math, Iverson Software inc.,
             Toronto (1991) 33pp.

        [7]    Kenneth E. Iverson, A Personal View of APL, IBM Systems
             Journal, Vol. 30, Number 4 (1991) In Press.

        [8]    Donald B. McIntyre, Mastering J, APL91 Conference
             Proceedings, Stanford, California, August 1991.   APL
             Quote Quad Vol. 21 Number 4 (August 1991), p.264-273.

        [9]    Zdenek V. Jizba, Science Education in California, Vector
             vol.8 number 2 (1991) p.22-24.

        [10] Florian Cajori, A History of Mathematical Notations,  The
             Open Court Publishing Company, La Salle, Illinois. Volume
             1 (1951) 451pp. First published 1928;  Volume 2 (1952)
             367pp.  First published 1929.







                                   -26-




        [11]    Tobias Dantzig, Number: the Language of Science, London,
             George Allen and Unwin, Ltd.   2nd edition (1942) 320pp.
             First Published 1936, 4th edition 1962.

        [12]    Donald B. McIntyre, Language as an Intellectual Tool:
             From hieroglyphics to APL, IBM Systems Journal, Vol. 30,
             Number 4 (1991) In Press.

        [13]    Lancelot Hogben, Mathematics for the Million, London,
             George Allen and Unwin, Ltd (1936) 678pp.

        [14]    Kenneth E. Iverson, Notation as a Tool of Thought, 1979
             Turing Award Paper, Communications of the A.C.M., Vol.23,
             Number 8 (August, 1980) p.444-465

        [15]    Roger K.W. Hui, Kenneth E. Iverson, E.E. McDonnell, and
             Arthur T. Whitney, APL\?, APL90 Conference Proceedings,
             Copenhagen, Denmark, August 1990.  APL Quote Quad Vol. 20,
             Number 4 (July 1990) p.192-200.

        [16]    Robert Bernecky and Roger K.W. Hui, Gerunds and
             Representations, APL91 Conference Proceedings, Stanford,
             California, August 1991.  APL Quote Quad Vol. 21, Number 4
             (August 1991) p.39-46.

        [17]    Kenneth E. Iverson, and E.E. McDonnell, Phrasal Forms,
             APL89 Conference Proceedings, New York City, August 1989.
             QuoteQuad, Volume 19, Number 4, (1989) p.197-199.

        [18]    J is available from Iverson Software Inc., 33 Major
             Street, Toronto, Ontario, Canada M5S  2K9.   Phone (416)
             925-6096; Fax [phone].

        [19]    Kenneth E. Iverson, A Programming Language, John Wiley and
             Sons, Inc., New York (1962) 286pp.

        [20]    Kenneth E. Iverson, Elementary Analysis, APL Press,
             Swarthmore, Pennsylvania (1976) 219pp.

        [21]    Donald B. McIntyre, Experience with Direct Definition One-
             liners in Writing APL Applications, Conference
             Proceedings, APL Users Meeting, Toronto, September 1978.
             Sponsored by I.P. Sharp Associates, Toronto (1978) p.281-
             297.





                                   -27-





        [22] William K. Clifford, The Common Sense of the Exact
             Sciences, Edited with a preface by KArl Pearson.  Newly
             edited and with an introduction by James R. Newman.
             Preface by Bertrand Russell.  With bibliography of W.K.
             Clifford.   Alfred A. Knopf, New York (1946), Sigma Books,
             Ltd., London (1947) 249pp..   Reprinted by Dover
             Publications, Inc., New York (1955).   Chapter 1, Number,
             Section 11, Steps.

        [23] Morris Kline, Mathematical Thought from Ancient to Modern
             Times, Oxford University Press, New York (1972) 1238pp.

        [24]    Roger K.W. Hui, Kenneth E. Iverson, Eugene E. McDonnell,
             Tacit Definition, APL91 Conference Proceedings, Stanford,
             California, August 1991.   APL Quote Quad Vol. 21 Number 4
             (August 1991), p.264-273.

        [25]    Roy Hollands, Ginn Mathematics Level 4, Teachers' Resource
             Book, Ginn and Company Ltd, Aylesbury, Bucks. (1983),
             Third impression 1985, 160pp.   Intended for 8 to 9 year
             old children.

        [26] John Backus, Can programming be liberated from the Von
             Neumann style?   A functional style and its algebra of
             programs.   1977 Turing Award Lecture.  Communications of
             the ACM, Vol. 21, number 8 (1978) p.613-641.

        [27]    Felix Klein, Elementary Mathematics from an Advanced
             Standpoint, Third Edition first published 1924-1925.
             English translation published by Dover Publications Inc.,
             New York. 2 Volumes, (1939) 274pp. and 214pp.

















                                   -28-