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For example, we could say HTML is the lowest common denominator on the web", because one can be sure all web browsers are able to render HTML (but not Flash or Java). If I want my web page to show properly everywhere, I know I can use maximum set of features shared between all web browsers, i.e. HTML. Yet, we don't say HTML is the greatest common divisor, which would be a correct analogy.

Why is the mathematical term lowest common denominator misused in everyday usage?

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I am not sure that in the everyday language greatest common divisor would have any meaning. –  kiamlaluno Jan 29 '11 at 15:40
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You could just say the literal "largest common subset [of features]", and be both right and understood. –  ShreevatsaR Jan 31 '11 at 17:14
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@kiamlaluno "Greatest common factor" makes much more sense. –  tchrist Jan 29 '12 at 3:36

8 Answers 8

up vote 11 down vote accepted

Because people think of something as "low" and "common", both of which it is, and then get themselves mixed up and say "lowest common denominator" which is not what they mean. (This vaguely similar to the way people say "steep learning curve" based on informal ideas of steep surfaces being difficult to climb, without thinking carefully of what a learning curve is and what steepness actually means for one.)

In short, the typical usage of "lowest common denominator" is a logical mistake. To speculate why it happens, my guess is that it's brought about by wanting (rightly) to use the terms "low" and "common", and the temptation of using (incorrectly) the technical-sounding term "lowest common denominator". That, and the fact that one has seen others use the phrase in that sense!

Using something that's actually logically right — like "highest common factor" — sounds positive rather than negative, so this is less used. (Although, in your HTML example, you could use something like "largest common subset of features", and be both right and understood.)


That's the answer, but explanation of the literal meaning…

To take the example of TV shows — because the citation given on Wiktionary is "Reality TV really is appealing to the lowest common denominator in audiences" — it is often said of low-quality populist shows that they are dumbed down, and at the "lowest common denominator", in order to cater to a large population. In this case, it is true that the quality may be low, and that the intention is to make something of a quality whose acceptability is common to the large population. But the resulting quality of the show is actually the greatest common factor! It is at the greatest level of quality that is still common (that's why it's low, because it needs to be common). If they were actually picking the lowest common quality, it would be even lower, or zero.

[Mathematical background: In mathematics, there are two related concepts: the greatest/highest common divisor/factor of two integers, say 15 and 24, is the greatest factor common to both, in this case 3. (Note that 3 is lower than 15 and 24.) The lowest common multiple of the integers is the smallest multiple common to both; in this case 120. (Note that 120 is greater than 15 and 24.) So the lowest common multiple is greater than the numbers, and the greatest common factor is lower than the numbers. This itself should be enough to suggest that if what you want to talk of is something low, you must use "greatest common…", not "lowest common…". But most people don't think so much.

Lowest common denominator is another term which is used when adding fractions: if you're adding the fractions 1/15 and 1/24, you convert them both to the lowest common denominator, which is the lowest common multiple of the denominators. In this case, the lowest common denominator is 120, and you write 1/15 as 8/120 and 1/24 as 5/120, so that you can add them: 1/15 + 1/24 = 8/120 + 5/120 = 13/120.
Even in this case, the lowest common denominator (120) is larger than the original denominators (15 and 24).]

The actual "lowest common factor" of any set of integers is 1, irrespective of what the numbers are, so it's not a very useful term. Similarly, the literal meaning of "lowest common denominator", when used in its usual context, refers to a quality that is always zero (or the minimum possible) irrespective of the population: the lowest common denominator among high-school graduates, the lowest common denominator among the whole population, and the lowest common denominator among people with PhDs would all be the same: abysmal.

Edit: Someone on Wikipedia has already explained this:

In common non-mathematical usage, the term "least common denominator" is often misused for the concept of the greatest common divisor. For example, a graphic toolkit which rendered features like lines and polygons into either Microsoft VML or standard SVG might choose to implement only the maximum set of graphic attributes common to both destination formats, which is an easy analogy to the concept of the greatest common divisor (The greatest common divisor of 12 and 18 is 6, which is the largest factor evenly dividing both numbers). If the systems being compared are very similar, then the common functionality can be a powerful subset (as the greatest common divisor of 375 and 250 is 125), while if the systems are very dissimilar the common capabilities might be very minimal (as the greatest common divisor of 270 and 98 is only 2). With additional systems (or numbers), the set of features common to all cannot grow and often shrinks (likewise for finding the greatest common divisor for a series of numbers).

This approach of making use of only the greatest subset of function common to all supported systems is often disparaged when the common feature set is sparse or weak (by analogy, having a small "greatest common divisor"). In this context colloquial usage has conflated the concept of "greatest common divisor" with the familiar sounding jargon of "least common denominator", which seems to emphasize smallness of overlap through the word "least", but actually refers to a different and inappropriate mathematical concept.

Edit 2: In case it helps, below is a vague picture illustrating what I mean. (Anyone please feel free to make a better image and replace this.)

Highest common factor, etc.

When most people say "lowest common denominator", they mean the second horizontal line, near the bottom: it's very low, because it must be common to so many, but it's also not zero. If you take "lowest common denominator" in its mathematical sense of being an lcm (of some set of existing denominators), then it's actually higher than the elements of the set, and is the highest line. If you use "denominator" to mean just "trait", then the "lowest common denominator" is literally the lowest common trait, which is the very bottom-most line: it's always zero (or 1 in mathematics, if you're talking of factors), independent of your original population, and even lower than it needs to be.

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The problem with this answer is that “lowest common denominator” is something that applies to fractions whereas “greatest common divisor” applies to integers. In the usage which is objected to, the analogy is that people (or whatever) are fractions. “Greatest common divisors” or “factors” apply to integers not fractions, and thus are not the same analogy. –  nohat Jan 29 '11 at 20:09
    
@nohat: The lowest common denominator of a set of fractions is just an integer, which is just the lowest common multiple of the denominators of the fractions (which are all integers). If the people are all fractions, and the l.c.d. is an integer… isn't that a type error or something? :-) [I'm not sure how the fractions analogy can be made to work, actually.] –  ShreevatsaR Jan 29 '11 at 20:29
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@ShreevatsaR, it's not a type error at all. The lowest common denominator is not one of the items in the set, it is a trait that they all share. In the analogy, fractions are complex objects, denominators are traits of those objects, and the l.c.d. is the simplest trait that all objects in the set share. –  nohat Jan 29 '11 at 20:37
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In most instances of the usage, including in the one from the OP, l.c.d. means the simplest (relevant) trait they all share. In the context of the OP's example, HTML is the simplest trait shared by web browsers. I think the idea that l.c.d. means the most complex set of shared traits is a red herring brought up by that dubious bit of writing from Wikipedia (which I have removed). –  nohat Jan 29 '11 at 20:45
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@nohat: Well, I think (independent of Wikipedia) that "lowest common denominator" is actually used to mean the most complex set of shared traits, not the simplest. The actual simplest is always the empty set independent of the population, and thus a useless concept. :-) I've added a rough figure to illustrate what I mean; I'm curious to know if more people think the lowest (solid) line is meant. –  ShreevatsaR Jan 29 '11 at 21:29

It's an analogy.

When comparing disparate things, you can't use simple, direct comparisons. Whether that comparison is between individuals, between distinct populations or, in the case of your example, between different technologies, there is no apples-to-apples comparison. So how can you compare apples to oranges and get away with it?

Well, we all learned in school that you can compare things that are different not just in magnitude but in substance. Rational numbers with different denominators can be rewritten as numbers with a common denominator, and can then be treated as things which differ only in magnitude (specifically, in the magnitude of the numerator). The lowest (or least) common denominator is the number (or term) that expresses the greatest commonality between the elements to be compared; the once-disparate elements have been distilled down to the simplest expression of their sameness.

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Any common denominator would work, so why "lowest"? –  ShreevatsaR Feb 16 '11 at 19:43
    
@ShreevatsaR Because the speaker wants the negative "low culture" connotation of "Lowest?" –  user867 Nov 15 '12 at 6:56

I don't think it's being misused. To add several fractions, you convert them to a common denominator. To add 1/2 and 1/3, you could convert them to 6/12 and 4/12 respectively; their sum is obviously 10/12, or 5/6.

The lowest common denominator is the lowest to which all of your addends can be converted.

A concept (or argument or piece of entertainment) has to be expressed in a way that it is accessible to each member of the intended audience. This is labeled, by way of analogy, the "common denominator" -- and whatever expression is easiest, or most denigrated, is the "lowest" common denominator.

The only real error comes in when someone says reducing to the lowest common denominator. No reduction is involved: when you convert the fraction to the l.c.d., you aren't changing its value, only the way it's written.

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Your mathematical explanation is fine, and the meaning of "common denominator" is fine, but, again, no one is picking "whatever expression is easiest, or most denigrated", but only the highest among the "common denominators". I don't think those who use the phrase mean to say that someone is picking an argument/expression even lower than the highest common one, since that's already low enough to be common… the actual lowest common denominator is independent of the intended audience, which is not what those who use the phrase mean. –  ShreevatsaR Jan 29 '11 at 14:58
    
This would be valid if the expression meant "making the same concept accessible to all people". For instance in a group of people from different countries, English might be the only language they all know; thus translating everything to English first is the "least" they must do to understand each other. However the common usage of this phrase falls under your "error" case; it refers to the selection of a set of concepts which are acceptable to all. –  Modern Hacker Jan 29 '11 at 14:58
    
To those upvoting this: can someone explain to me why the intended meaning is the "easiest, or most denigrated" among the "common denominators"? The original phrase "lowest common denominator" I can understand (even if I can't interpret it literally), but this claim I truly fail to comprehend. –  ShreevatsaR Jan 29 '11 at 19:36
    
When someone uses "lowest common denominator", they are referring (typically) to a work that they regard as base, crude, or exploitative. When I said "most denigrated", I just meant that products or genres that other people say "appeal to the lowest common denominator" are in my opinion often just socially designated whipping boys. A movie about incest and murder would be considered l.c.d. trash by people who think operas or plays on those subjects would be Constitutionally entitled to Federal funding. –  Malvolio Jan 29 '11 at 21:55
    
Yes, I agree about being "base", "common", "denigrated", etc. — just not about "most denigrated". –  ShreevatsaR Jan 29 '11 at 22:02

this is how i was able to make some sense of this to myself:

Greatest Common Den. (GCD) of 8 & 12 = 4 (limited set of possibilities that are a part of 8 and 12 only)

Lowest Common Den. (LCD) of 8 & 12 = 24 (finite possibilities referencing all numbers which are the larger set both 8 & 12 are a part of)

GCD of cap, gap, & cope = p (followed by letter 'a')(limited set of possibilities restricted to capgapcope only)

LCD of cap, gap, & cope = capoge (finite set, 6 of a possible 26 letters. :. alphabet is the set)

so far it can be inferred that GCD is for a limited set that can only be made up of the members or elements in question or a well defined set. we can also say that the GCD cannot be defined unless we know all the members of the set. Because the GCD is a part of each of the members of the set

while LCD is for a more expansive set that the members or elements could be or are a part of, with the possibility of having more members yet unknown ~ or an undefined set. alternately the LCD can be arrived at without knowledge of all members of that set. and as new members come into existence, we can ask if they could possibly belong to that particular set.

keep in mind also that GCD and LCD are not exclusive properties. Based on my understanding, it also helps to make a qualifier where

GCD becomes G(of members that make up a set)CD and is therefore always a member of the set

and LCD as L(of set made up of members)CD and as such does not have to come from the members in question but as a property of a set

Anyway, to apply this analogy to its colloquial use i played around with an abstraction like Ignorance and, using the parameters above, tried to come up with a scenario where ignorance is the GCD and LCD of some hypothetical members. it is at this point that my brain sprung a leak.

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You mean Greatest Common Divisor (GCD), right? Also, I think Lowest Common Denominator (LCD), applies only to fractions, so 24 is LCD of 1/8 and 1/12. You may be thinking of Lowest Common Multiple (LCM) - LCM(8,12) = 24. –  Danko Durbić Feb 2 '11 at 17:18

A “common denominator”, in the extended usage, is a trait or theme that all elements of a group have in common. The “lowest common denominator” is the denominator that it is most practical or easiest to use when summing or comparing fractions. In the analogy, the “lowest common denominator” is the trait or theme that is most practical or easiest to take advantage of or use.

The suggestion to use “greatest common factor” instead kind of misses the point that the analogy is that people or web browsers (or whatever) are fractions, not integers. Sure, if we wanted to make the analogy that web browsers are integers, then speaking of the “greatest common factor” would make sense. But that analogy has no currency, so the suggestion is dead on arrival.

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Independent of integers or divisors or fractions, isn't it strange to use lowest-common-something for " maximum set of features shared between all web browsers" (OP's words, emphasis mine)? –  ShreevatsaR Jan 29 '11 at 21:36
    
OP simply misunderstood the quote. HTML is a single feature which is the lowest common denominator (of features) on the web. –  nohat Jan 29 '11 at 22:32
    
After two days' break, I think I understand and appreciate this explanation better: what you're saying is that the thing being called the "lowest common denominator" is not actually the lowest common anything, but is instead merely an analogy: just as, for a set of fractions, representing them in terms of the lcd is an expression that works for all the fractions, so also does (figuratively) l.c.d mean something that works for everyone. Is this right? But if so, since any common denominator works for the fractions, I'm not sure what the function of "lowest" is: what, exactly, is lowest? –  ShreevatsaR Jan 31 '11 at 9:03

Because, even in rich sophisticated western societies, huge numbers of people are almost innumerate and therefore unfamiliar with simple arithmetical expressions of this sort.

I just asked a university-educated friend "what is the lowest common denominator of a half and a third?" - Some conversation ensued before I got an answer. Try it.

Journalists and broadcasters who use the phrase ought to agree that by doing so they are are pandering to the lowest common denominator of their audience. ;-)

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The lowest common denominator of halves and thirds is sixths, which is smaller than both of them. –  nohat Jan 29 '11 at 20:32
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No, the lowest common denominator of 1/2 and 1/3 is 6 (see en.wikipedia.org/wiki/Lowest_common_denominator). –  Danko Durbić Feb 2 '11 at 17:10

It is just the deliberate use of a set phrase to suggest negative features. Compare

  • Broadcast television is aimed at the lowest common denominator of its audience's tastes
  • Broadcast television is aimed at the highest common factor of its audience's tastes

Highest common factor is an alternative way of saying greatest common divisor (in fact the one I was taught at school) but sounds so much more positive

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Right, this is the reason why the phrase arises: there's a strong temptation to say "lowest" greater than "highest", because we want to indicate that the object in question is low, not high. It just happens to mean the opposite. :-) –  ShreevatsaR Jan 29 '11 at 17:43

Because people mix up the concepts, and use the one that sounds more like what they want to express. The lowest common denominator sounds like it's smaller than the greatest common divisor, when in fact it's the opposite.

It's not a very good expression to use outside mathematics, as it sounds better when it's used incorrectly.

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Sorry, but do you know what either of those terms means? You are saying for a set of numbers S, GCD(S) < LCM(S). In fact GCD only applies to integers while LCM applies to fractions, so you are comparing apples with oranges. –  Modern Hacker Jan 29 '11 at 14:38
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@Modern: No, both gcd and lcm are for integers. And it is true that for a set of numbers S, gcd(S) < lcm(S) (with equality only when all integers in the set are equal). In fact, gcd(S) ≤ min(S) ≤ max(S) ≤ lcm(S), i.e. the gcd is (obviously) less the smallest of the numbers, and the lcm is larger than the largest of the numbers. [It's true that lowest common denominator is used for fractions, but lowest common denominator is just lcm of the denominators, so it's still an lcm.] –  ShreevatsaR Jan 29 '11 at 15:04
    
@Shree: Sorry I meant to type LCD (least common denominator), don't know where least common multiple came from. Yes LCM is for integers; the LCD of any set of integers is trivially 1. –  Modern Hacker Jan 29 '11 at 15:14
    
@Modern Hacker: Yes, I know what those terms mean, than you very much. The LCD doesn't apply to fractions, it applies to the divisors of fractions, which are integers. –  Guffa Jan 29 '11 at 22:26
    
So you mean to say the greatest common divisor of the denominators of a set of fractions is smaller than the least common multiple of the denominators. –  Modern Hacker Jan 30 '11 at 0:07

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