Wow.

I love Feynman also; hadn’t seen this,

I DO remember being terrorized by the bases, however.  A waking nightmare.  As I recall, it caused me to lose any working knowledge of base TEN; which was not made to clear in any case,

As for SPUTNIK, that is another story entirely…

Feynman’s statement about bases, taken literally, is wrong, but his intended meaning is correct.  It is quite right to say that there is no reason for a general student to learn alternate bases, except to reinforce (by giving a broader context) what base 10 means. 

But translating from one base to another is NOT a useless thing.  Your computer regularly and automatically translates the numbers you enter in base 10 into base 2, its native language.  And in case the point escapes you, certain people have to UNDERSTAND this process very well, and to do it all the time in their work.  But certainly not everyone.  Yet everyone in our society should know “in general terms” what’s going on, and so should have a basic understanding of both base 2 and base 10, even if they don’t do the translation themselves.  It’s basic literacy for the world you live in; like knowing in general terms that gasoline is mixed with oxygen, injected into steel cylinders, flashed by electric sparks, driving pistons before the byproducts of the combustion are exhausted, and that’s what makes cars go.

You don’t think that gasoline is a magical substance, on the level of medieval superstition—you have a general knowledge of what’s going on, even though only a small number of people really need to master the details.  Similarly, you are illiterate about the world around you if you think computers are magic boxes that somehow turn numbers into electricity, which “knows” the answer to everything because electricity has a mind of its own, or something of that sort.

Many important problems pertaining to physics (as Feynman well knew), math, engineering, statistics, economics and computer science are only properly understood in terms of the binary representation of certain numbers intrinsic to the problem.  That’s a pretty good reason to know how to translate into base 2.

Computer scientists, in order to “speak” machine code, must be able to quickly convert into, and from, hexadecimal, which is base 16.  Of course, that’s a small leap from base 2 because 16 = 2^4.  Still, it’s another important base.

Basic cultural and technical literacy in the 21st Century is impossible without awareness of kilobytes, megabytes, gigabytes, terabytes, and bits.  What is a bit?  A binary digit.  A byte?  eight bits.  A kilobyte?  1000 bytes?  NO!  it is 1024 = 2^10 bytes.  Why?  Well, it’s a natural part of how computers work.  the “kilo” part, which literally says “one thousand” is only an approximation.  Similarly, a megabyte is not one million bytes, but 1,048,576 = 2^20.

An odd group in Britain years ago tried to get the government to officially change to base 12, the “duodecimal system”.  Good thing they didn’t, but it was an interesting proposition, and not all of their reasons were pure nonsense.

I am a lover of different bases, my favourites being base 3 in “symmetric form”, in which the possible digits are 0, 1 and -1 (there are computational reasons why this is a lovely way to “do” numbers) and 2+i (where i^2 = -1; if you don’t do complex numbers ... maybe forget about this one) with digits 0, 1, i, and j = -i.

But I do not advocate the teaching of these things to EVERYONE.  In that I agree wholeheartedly with Feynman.  He was quite right that excessive formality was the mortal failing of the old “new math”.  But as far as I know this is no longer the issue.  Today’s “new math” has almost the opposite problem; established mathematical standards are tossed out the window and the math students learn have about the same foundation as a piece of wood bobbing along in the open ocean.  We need a return, not to the old formality of old “new math” but to the well-defined structure and skills base of standard, time-tested math that anyone can learn and that provides both the basic mathematical literacy for modern life but the practical skills needed for daily activity.