David Leibs <david.le...@oracle.com> writes:
> I have kinda lost track of this thread so forgive me if I wander off
> in a perpendicular direction.
>
> I believe that things do not have to continually get more and more
> complex. The way out for me is to go back to the beginning and start
> over (which is what this mailing list is all about). I constantly go
> back to the beginnings in math and/or physics and try to re-understand
> from first principles. Of course every time I do this I get less and
> less further along the material continuum because the beginnings are
> so darn interesting.
>
> Let me give an example from arithmetic which I learned from Ken
> Iverson's writings years ago.
>
> As children we spend a lot of time practicing adding up
> numbers. Humans are very bad at this if you measure making a silly
> error as bad. Take for example:
>
> 365
> + 366
> ------
>
> this requires you to add 5 & 6, write down 1 and carry 1 to the next
> column then add 6, 6, and that carried 1 and write down 2 and carry a
> 1 to the next column finally add 3, 3 and the carried 1 and write down
> 7 this gives you 721, oops, the wrong answer. In step 2 I made a
> totally dyslexic mistake and should have written down a 3.
>
> Ken proposed learning to see things a bit differently and remember the
> digits are a vector times another vector of powers. Ken would have
> you see this as a two step problem with the digits spread out.
>
> 3 6 5
> + 3 6 6
> ------------
>
> Then you just add the digits. Don't think about the carries.
>
> 3 6 5
> + 3 6 6
> ------------
> 6 12 11
>
> Now we normalize the by dealing with the carry part moving from right
> to left in fine APL style. You can almost see the implied loop using
> residue and n-residue.
> 6 12 11
> 6 13 0
> 7 3 0
>
> Ken believed that this two stage technique was much easier for people
> to get right. I adopted it for when I do addition by had and it works
> very well for me. What would it be like if we changed the education
> establishment and used this technique? One could argue that this sort
> of hand adding of columns of numbers is also dated. Let's don't go
> there I am just using this as an example of going back and looking at
> a beginning that is hard to see because it is "just too darn
> fundamental".
It's a nice way to do additions indeed.
When doing additions mentally, I tend to do them from right to left,
predicting whether we need a carry or not by looking ahead the next
column. Usually carries don't "carry over" more than one column, but
even if it does, you only have to remember a single digit at a time.
There are several ways to do additions :-)
Your way works as well for substractions:
3 6 5
- 3 7 1
-----------
0 -1 4
0 -10 + 4 = -6
3 7 1
- 3 6 5
-----------
0 1 -4
10 -4 = 6
and of course, it's already how we do multiplications too.
> We need to reduce complexity at all levels and that includes the
> culture we swim in.
Otherwise, you can always apply the KISS principle
(Keep It Simple Stupid).
--
__Pascal Bourguignon__ http://www.informatimago.com/
A bad day in () is better than a good day in {}.
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