Counting in different bases
Added 2016-11-15 03:02:21 +0000 UTCHey there $4+ers, I had a bit of fun today trying to illustrate the concept of counting in different bases, relevant for an upcoming video.
Comments
Given the way you animated the ternary and binary bases, you could also delve into non-integer bases, starting with fractional bases. The simplest would be Base 1.5, and explaining it which connects back to the "traditional" way of translating between bases (exponents of the base then adding them up). The first time I was introduced to fractional bases, it felt magical that the exponents would be non-integers, but all the numbers you were making would be integers. Looking forward to the next official video!
Henri Taarisen
2016-11-15 20:02:01 +0000 UTCI left this comment elsewhere recently: I thought folks interested in math education might find this interesting. Steve Stephenson, a retired engineer (and later retired schoolteacher) spent several years researching historical abacuses / counting boards, and thinking about how they might be used for complicated computations, taking the design of the Salamis Tablet (<a href="https://en.wikipedia.org/wiki/Salamis_Tablet)" rel="nofollow noopener" target="_blank">https://en.wikipedia.org/wiki/Salamis_Tablet)</a> as inspiration. This website <a href="http://ethw.org/Ancient_Computers" rel="nofollow noopener" target="_blank">http://ethw.org/Ancient_Computers</a> and this Youtube playlist <a href="https://www.youtube.com/view_play_list?p=545ABCC6BA8D6F44" rel="nofollow noopener" target="_blank">https://www.youtube.com/view_play_list?p=545ABCC6BA8D6F44</a> present his findings. His ideas are somewhat speculative about the details, but what I like about it is that (a) he takes ancient technologies seriously instead of brushing them aside as obsolete relics [nearly every modern description of Roman numerals aimed at a popular audience talks about how terribly cumbersome they are, without bothering to explain their context or purpose, and mostly use them as a straw-man foil for Hindu–Arabic numerals which are presented as clearly superior], (b) he tries to reconstruct/invent working solutions to problems that would come up when attempting to do real concrete computations, approaching it as an engineer rather than a historian. In the context of a school, I think it would be great to teach this kind of arithmetic for a few reasons: 0) It requires almost nothing in the way of supplies. Just a pile of pebbles and some lines drawn on paper (or lines drawn in the dirt with a stick). 1) There is practically no rote memorization involved, but lots of pattern matching, discovery, and thinking about the concrete meaning of numbers. 2) Unlike a fixed-frame abacus, the counting board allows for unreduced numbers to sit on the board, and shows very clearly and explicitly the relation between reduced and unreduced numbers. The method of reducing a number to a standard form can be handled using tiny obvious steps in many possible orders. Other arithmetic operations can also be done in a variety of possible paths, as long as each step is valid. This is the core of algebraic thinking. 2a) The counting board can make use of “balanced” positional numeration, i.e. the use of negative numbers within a particular place value instead of only complete integers being positive/negative, and there is obvious symmetry between positive/negative versions – just flip everything across the divider line; I think the loss of this idea in Hindu–Arabic arithmetic as usually practiced and on fixed-frame abacuses is a real step back for conceptual understanding of more sophisticated later mathematics. 3) The counting board is a fabulous piece of our cultural history as humans, and it’s a shame that a mere 4–5 centuries of disuse (after millennia of use) have been enough to almost completely wipe it out of mainstream awareness (except as backgammon boards, and in the etymology of various words/phrases). Moreover, understanding the counting board makes it easy to understand what Roman numerals are for and how they function. Namely, they are the written record of finished computations, not a tool for actively performing arithmetic. 4) In Steve’s version, there is an explicit treatment of an exponent as an integer, instead of as just a movable position of a decimal point. This is a good stepping stone to scientific notation, the floating point arithmetic used by computers, and logarithms. 5) The counting board can be fluidly/easily adopted to many different number bases for different purposes, unlike Hindu–Arabic numerals. Students figuring out how to properly change the “rules” of the board to deal with new number bases will have to think deeply about the relationships involved and the fundamental nature of positional numeration. 6) In a modern era when electronic calculators can handle the rote performance of computation, it’s more important than ever to focus thinking on the *meaning* of numerical relationships, and the *design* of algorithms. The counting board does this very well. Pen-and-paper arithmetic might turn out to be more efficient if you need to multiply 1000 pairs of 4-digit numbers, but these days nobody needs to do that.
Jacob Rus
2016-11-15 03:06:42 +0000 UTC
