FYI: Ken Iverson credited Linda with elucidating the mysterious concept of
outer product. http://www.jsoftware.com/papers/autobio.htm , Section 5.
Linda Alvord said to introduce the matrix as an outer product — an idea
that the rest of us thought outrageous, until Linda pointed out that the
kids already knew the idea from familiar addition and multiplication tables.
Function tables, a medium in which a child can wade but an elephant can
drown:
% 1 + +/~ i.7
1 0.5 0.333333 0.25 0.2 0.166667 0.142857
0.5 0.333333 0.25 0.2 0.166667 0.142857 0.125
0.333333 0.25 0.2 0.166667 0.142857 0.125 0.111111
0.25 0.2 0.166667 0.142857 0.125 0.111111 0.1
0.2 0.166667 0.142857 0.125 0.111111 0.1 0.0909091
0.166667 0.142857 0.125 0.111111 0.1 0.0909091 0.0833333
0.142857 0.125 0.111111 0.1 0.0909091 0.0833333 0.0769231
% 1 + +/~ i.7x
1 1r2 1r3 1r4 1r5 1r6 1r7
1r2 1r3 1r4 1r5 1r6 1r7 1r8
1r3 1r4 1r5 1r6 1r7 1r8 1r9
1r4 1r5 1r6 1r7 1r8 1r9 1r10
1r5 1r6 1r7 1r8 1r9 1r10 1r11
1r6 1r7 1r8 1r9 1r10 1r11 1r12
1r7 1r8 1r9 1r10 1r11 1r12 1r13
-/ .* % 1 + +/~ i.7x
1r2067909047925770649600000
% -/ .* % 1 + +/~ i.7x
2067909047925770649600000
q: % -/ .* % 1 + +/~ i.7x
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 5 5 5 5 5 7 7 7 7
7 7 7 11 11 11 13
~. q: % -/ .* % 1 + +/~ i.7x
2 3 5 7 11 13
~. q: % -/ .* % 1 + +/~ i.8x
2 3 5 7 11 13
~. q: % -/ .* % 1 + +/~ i.9x
2 3 5 7 11 13 17
~. q: % -/ .* % 1 + +/~ i.10x
2 3 5 7 11 13 17 19
p:^:_1 +: 10x
8
p: i. p:^:_1 +: 10x
2 3 5 7 11 13 17 19
p: i. p:^:_1 +: 20x
2 3 5 7 11 13 17 19 23 29 31 37
~. q: % -/ .* % 1 + +/~ i.20x
2 3 5 7 11 13 17 19 23 29 31 37
On Sun, Nov 15, 2015 at 1:05 AM, Linda A Alvord <[email protected]>
wrote:
> How would you explain to a 4th grade student why subtraction and division
> of fraction is so difficult for J. Here is a dialog I wrote to review all
> the big ideas students should remember from third grade. Tables show how
> some verbs like + and times are commutative and * and % are not. If the
> verb
> is commutative you need only "master" a little more than half of the table.
>
>
>
>
>
>
>
> +/~i.5
>
>
>
> 0 1 2 3 4
>
> 1 2 3 4 5
>
> 2 3 4 5 6
>
> 3 4 5 6 7
>
> 4 5 6 7 8
>
>
>
>
>
>
>
> -/~i.5
>
>
>
> 0 _1 _2 _3 _4
>
> 1 0 _1 _2 _3
>
> 2 1 0 _1 _2
>
> 3 2 1 0 _1
>
> 4 3 2 1 0
>
>
>
>
>
>
>
> */~i.5
>
>
>
> 0 0 0 0 0
>
> 0 1 2 3 4
>
> 0 2 4 6 8
>
> 0 3 6 9 12
>
> 0 4 8 12 16
>
>
>
>
>
>
>
> %/~i.5
>
>
>
> 0 0 0 0 0
>
> _ 1 0.5 0.333333 0.25
>
> _ 2 1 0.666667 0.5
>
> _ 3 1.5 1 0.75
>
> _ 4 2 1.33333 1
>
>
>
>
>
>
>
> 1r3%i.5
>
>
>
> _ 1r3 1r6 1r9 1r12
>
>
>
>
>
>
>
> +/~ 1r3%i.5
>
>
>
> _ _ _ _ _
>
> _ 2r3 1r2 4r9 5r12
>
> _ 1r2 1r3 5r18 1r4
>
> _ 4r9 5r18 2r9 7r36
>
> _ 5r12 1r4 7r36 1r6
>
>
>
>
>
> -/~ 1r3%i.5
>
>
>
> |NaN error
>
> | -/~1r3%i.5
>
>
>
>
>
> */~ 1r3%i.5
>
>
>
> _ _ _ _ _
>
> _ 1r9 1r18 1r27 1r36
>
> _ 1r18 1r36 1r54 1r72
>
> _ 1r27 1r54 1r81 1r108
>
> _ 1r36 1r72 1r108 1r144
>
>
>
>
>
>
>
> %/~ 1r3%i.5
>
>
>
> |NaN error
>
> | %/~1r3%i.5
>
>
>
>
>
>
>
> t=: 13 :'>:/~i.y'
>
>
>
> t 5
>
>
>
> 1 0 0 0 0
>
> 1 1 0 0 0
>
> 1 1 1 0 0
>
> 1 1 1 1 0
>
> 1 1 1 1 1
>
>
>
>
>
>
>
> (t 5)*+/~i.5
>
>
>
> 0 0 0 0 0
>
> 1 2 0 0 0
>
> 2 3 4 0 0
>
> 3 4 5 6 0
>
> 4 5 6 7 8
>
>
>
>
>
>
>
> (t 5)**/~i.5
>
>
>
> 0 0 0 0 0
>
> 0 1 0 0 0
>
> 0 2 4 0 0
>
> 0 3 6 9 0
>
> 0 4 8 12 16
>
>
>
>
>
>
>
> (t 5)*+/~ 1r3%i.5
>
>
>
> _ 0 0 0 0
>
> _ 2r3 0 0 0
>
> _ 1r2 1r3 0 0
>
> _ 4r9 5r18 2r9 0
>
> _ 5r12 1r4 7r36 1r6
>
>
>
>
>
>
>
> (t 5)**/~ 1r3%i.5
>
>
>
> _ 0 0 0 0
>
> _ 1r9 0 0 0
>
> _ 1r18 1r36 0 0
>
> _ 1r27 1r54 1r81 0
>
> _ 1r36 1r72 1r108 1r144
>
>
>
>
>
>
>
> Linda I
>
>
>
>
>
> ----------------------------------------------------------------------
> For information about J forums see http://www.jsoftware.com/forums.htm
>
----------------------------------------------------------------------
For information about J forums see http://www.jsoftware.com/forums.htm
