Intro 2. Ambivalence

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2. Ambivalence
Cover the comments on the right and provide your own.

       7-5                        The function in the sentence 7-5 applies to two
    2                             arguments to perform subtraction, but in the
       -5                         sentence -5 it applies to a single argument to
    _5                            perform negation.

                                  Adopting from chemistry the term valence, we
                                  say that the symbol - is ambivalent, its effective
       7%5                        binding power being determined by context.
    1.4

       %5                         The ambivalence of - is familiar in arithmetic;
    0.2                           it is here extended to other functions.

       3^2
    9

       ^2                         Exponential (that is, 2.71828^2)
    7.38906

       a=: i. 5                   The function integer or integer list
       a
    0 1 2 3 4                     List or vector

       a i. 3 1                   The function index or index of
    3 1

       b=: 'Canada'               Enclosing quotes denote literal characters
       b i. 'da'
    4 1

       $ a                        Shape function
    5

       3 4 $ a                    Reshape function
    0 1 2 3                       Table or matrix
    4 0 1 2
    3 4 0 1

       3 4 $ b
    Cana
    daCa
    nada

       %a                         Functions apply to lists
    _ 1 0.5 0.333333 0.25         The symbol _ alone denotes infinity

Exercises

2.1 | Enter the following sentences (and perhaps related sentences using different arguments), observe the results, and state what the two cases (monadic and dyadic) of each function do:
       a=: 3 1 4 1 5 9
       b=: 'Canada'
       #a
       1 0 1 0 1 3 # a
       1 0 1 0 1 3 # b
       /: a
       /: b
       a /: a
       a /: b
       b /: a
       b /: b
       c=: 'can''t'
       c
       #c
       c /: c
2.2 | Make a summary table of the functions used thus far. Then compare it with the following table (in which a bullet separates the monadic case from the dyadic, as in Negate • Subtract):
|  | . | :
`+` | • Add | • Or
`-` | Negate • Subtract
`*` | • Times | • And
`%` | Reciprocal • Divide
`^` | Exponential • Power | • Log
`<` | • Less Than | • Lessor Of
`>` | • Greater Than | • Greater Of
`=` | • Equals |  | Is (Copula)
`i` |  | Integers • Index Of
`$` | Shape • Reshape
`/` |  |  | Grade • Sort
`#` | Number Of Items • Replicate
2.3 | Try to fill some of the gaps in the table of Exercise 2.2 by experimenting on the computer with appropriate expressions. For example, enter`^.10` and`^. 2.71828` to determine the missing (monadic) case of`^.` and enter`%: 4` and`%: -4` and`+%: -4` to determine the case of`%` followed by a colon. However, do not waste time on matters (such as, perhaps, complex numbers or the boxed results produced by the monad`<`) that are still beyond your grasp; it may be better to return to them after working through later sections. Note that the effects of certain functions become evident only when applied to arguments other than positive integers: try`<.1 2 3 4` and`<.3.4 5.2 3.6` to determine the effect of the monad`<.` .
2.4 | If`b=: 3.4 5.2 3.6` ,`` then`<.b` yields the argument`b` rounded down to the nearest integer. Write and test a sentence that rounds the argument`b` to the nearest integer. Answer:`<.(b+0.5)` or`<.b+0.5` or`<.b+1r2`
2.5 | Enter`2 4 3 $ i. 5` to see an example of a rank`3` array or report (for two years of four quarters of three months each).
2.6 | Enter`?9` repeatedly and state what the function`?` does. Then enter`t=: ?3 5 $ 9` to make a table for use in further experiments. Answer:`?` is a (pseudo-) random number generator;`?n` produces an element from the population`i.n`

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