Arnold M. Zwicky zwicky at CSLI.STANFORD.EDU
Thu Mar 15 15:20:35 UTC 2007

On Mar 15, 2007, at 5:36 AM, Charlie Doyle wrote:

> Wilson, at a considerably more advanced point in history, the
> 1950s, I was specifically taught to say "lowest common
> denominator," NOT "least common denominator," to avoid the possible
> ambiguity of "least common"!
> --Charlie
> _________________________________________________________
> ---- Original message ----
>> Date: Wed, 14 Mar 2007 18:50:14 -0400
>> From: Wilson Gray <hwgray at GMAIL.COM>
>> Subject: Re: mid-aughts
>> When I was a child during the '40's, I was taught to use "_least_
>> common denominator" and "nought" instead of "_lowest_ common
>> denominator" and "zero." Although I've managed to adjust to
>> "zero," I still say and write "least common denominator." However,
>> I am forced to admit that the number of occasions upon which I
>> need to speak or write of the "least common denominator" is
>> vanishingly small.

to Charlie: if you start worrying about potential ambiguities, you'll
go crazy.

the relevant mathematical concepts are named "least common multiple"
and "greatest common denominator/divisor".  the lcm is also known as
the "lowest common multiple", but "least" is much more frequent here
than "lowest", and was, i think, the original.  yes, "least common"
is potentially ambiguous, but the 'least frequent' reading is stupid
mathematically, and the 'least vulgar' reading is even stupider, so
the potential ambiguity causes no trouble whatsoever for
mathematicians, and shouldn't be problematic for people learning

now, least common denominator (with "lcd" understood compositionally,
in a way parallel to "gcd" and "lcm") is not a particularly useful
concept mathematically.  for two numbers that are prime to one
another (like 4 and 15), the lcd would be 1; otherwise, the lcd would
be the smallest prime that divides them both (for 4 and 18, 2; for 8
and 16, 2; for 10 and 15, 5; etc.).  i don't know of any place where
this concept plays a role in mathematical results.

there *is* a use of "lcd" in arithmetic, but the expression is not
fully compositional there.  it refers to the *lcm* of two or more
denominators, and is used in converting fractions to forms with the
same denominator, so that they can be added or subtracted.

but, of course, "least/lowest common denominator" is an expression of
ordinary language, not mathematics, though it's modeled on technical
terms of arithmetic.  and it's not fully compositional either; the
"least/lowest" is not understood literally, since the lcd isn't the
*smallest* thing shared by everyone, whatever that would be.
instead, "least common denominator" refers to the *most* that's
shared by everyone -- that is, the ordinary-life counterpart to the
*gcd* of arithmetic.  "least/lowest" gets in there to convey that
this shared element is very small/low indeed.

i believe that "lowest" is much more frequent than "least" in the
ordinary-language expression, but that's hard to check by simple

people sometimes rage about "least/lowest common denominator" on the
grounds that it "doesn't make sense" or something of the sort -- that
is, on the grounds that it's not compositional.  hey, idioms are
everywhere; get used to it.


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