Least common denominator

For another comment about something a recent speaker said, we look at the guy yesterday who made a reference to least common denominator, and included a graphic that showed the fraction 9 / 12, then displayed it as 3*3 / 4*3, and concluded with 3 / 4. There are two problems with the graphic.

One is that it’s gratuitous. It has nothing to do with the colloquial meaning of least common denominator, which doesn’t relate to fractions or mathematics at all. In English rhetoric, it refers to a common kernel that can serve or satisfy everyone involved. Alternatively, it can be used disparagingly to refer to someone or something from which every distinguishing and distinguished characteristic has been removed, leaving only a common bit that’s dull and useless.

Some presenters seem to like sticking graphics on every Powerpoint slide they show — sometimes several per slide — whether or not the graphics add anything to the understanding of the slides. Presenters who do that think the graphics make their presentations snazzier.

They don’t.

But the other problem with the graphic is from a mathematical point of view: it’s not illustrating the concept of least common denominator at all. It’s an illustration of greatest common factor. When we reduce a fraction, as in the graphic, we find the greatest common factor of the numerator (the top of the fraction) and the denominator (the bottom) — the largest number we can find that goes evenly into both numbers, that divides both numbers with a remainder of zero. When we cancel that greatest common factor out, what’s left is the fully reduced fraction.

We use the least common denominator to compare (or add or subtract) two or more fractions.

Which is greater?: 5 / 12 ... or ... 9 / 20 ?

To answer that using fractions, we need to convert them into fractions with a common denominator, and we customarily use the least common denominator — the smallest number that is a multiple of both denominators. In this case, 12 = 4 * 3, and 20 = 4 * 5, so the least common denominator would be 4 * 3 * 5 = 60. Multiply both the numerator and denominator by the same amount, and we get 5 / 12 = 25 / 60, and 9 / 20 = 27 / 60. And, so, because 25 is less than 27, 5 / 12 is less than 9 / 20. And the difference between the two is (27 - 25) / 60 = 2 / 60 = 1 / 30 (which we reduced by finding the greatest common factor of 2 and 60).

I have no quibble with the colloquial use of least common denominator as a language idiom, with a meaning that doesn’t relate to the mathematical one (though I do think the usage is trite). But when you bring mathematics into it, please get the maths right.