At some point one moves on and uses powerful enough computer system like APL, J, or Smalltalk which support rational numbers. :-) -djl
On Jun 15, 2012, at 3:14 PM, Andre van Delft wrote: > Fascinating. > How did Iverson do division? > > Op 15 jun. 2012, om 23:08 heeft David Leibs het volgende geschreven: > >> Speaking of multiplication. Ken Iverson teaches us to do multiplication by >> using a * outer product to build a times table for the digits involved. >> +-+--------+ >> | | 3 6 6| >> +-+--------+ >> |3| 9 18 18| >> |6|18 36 36| >> |5|15 30 30| >> +-+--------+ >> >> Now you sum each diagonal: >> (9) (18+18) (18+36+15) (36+30) (30) >> 9 36 69 66 30 >> And just normalize as usual: >> >> 9 36 69 66 30 >> 9 36 69 69 0 >> 9 36 75 9 0 >> 9 43 5 9 0 >> 13 3 5 9 0 >> 1 3 3 5 9 0 >> >> The multiplication table is easy and just continued practice for your >> multiplication facts. >> >> You don't need much more machinery before you have the kids doing Cannon's >> order n systolic array algorithm for matrix multiply, on the gym floor, with >> their bodies. This assumes that the dance teacher is coordinating with the >> algorithms teacher. Of course if there isn't something relevant going on >> that warrants matrix multiply then all is lost. I guess that's a job for the >> motivation teacher. :-) >> >> -David Leibs >> >> On Jun 15, 2012, at 12:57 PM, Pascal J. Bourguignon wrote: >> >>> 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 {}. >>> _______________________________________________ >>> fonc mailing list >>> fonc@vpri.org >>> http://vpri.org/mailman/listinfo/fonc >> >> _______________________________________________ >> fonc mailing list >> fonc@vpri.org >> http://vpri.org/mailman/listinfo/fonc > > _______________________________________________ > fonc mailing list > fonc@vpri.org > http://vpri.org/mailman/listinfo/fonc
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