# Using differential as a synonym for difference sounds wrong to me. Am I justified or also wrong?

I hear the term "differential" used in many cases where it seems like "difference" should be used instead. It is especially common in sports, some examples are run differential which is the difference between runs scored and runs allowed in a baseball season. Or points differential which is the difference of the average points per game scored by a team and the points scored by an opposing team in the NBA. Based on google's definition of differential, its essentially a synonym of difference. Adjective (seems ok):

of, showing, or depending on a difference; differing or varying according to circumstances or relevant factors.

Noun (seems sketchy):

a difference between amounts of things.

The term seems to be used interchangably with difference and that seems incorrect. The definition of differential as I understood it in school is referring to `an infinitesimal difference between successive values of a variable,` which is commonly associated with the linear operator, the derivative operator, and which also has other related definitions in different contexts in mathematics. But, it all works well with the adjective definition, since all of these measures do indeed depend on a difference, specifically infinitesimal difference.

So is it acceptable to use it as a synonym for difference? The definition from Google seems like, since enough people use it that way, it was just accepted. Or perhaps, historically, differential really did just mean the same thing as difference, and Newton and Leibniz borrowed the term and give it new meaning when they invented Calculus?

• Get back to us when you figure out what to call that gear thingy that allows the drive wheels on a car to rotate at different speeds in a turn. – deadrat Jun 11 '16 at 4:18
• @deadrat rear differential, yet another context, but not a synonym for difference, from the Wikipedia `...an increase in the speed of one wheel is balanced by a decrease in the speed of the other` so it is certainly referring to a gear (er, gear train) with properties related to or dependent on difference, which would still fit. – chiliNUT Jun 11 '16 at 4:25

## 1 Answer

The two are not the same. Conceptually, a difference is the change caused by the set of differentials. Perhaps an example will help. The difference in gearing between two bicycles is a function of the ring gear differential, the rear cassette differential, and the drive wheel radius differential. Once you talk about the difference due to a single variable, it is a differential. When you apply the proper function to calculate the effect of all the differentials, you have the difference. (In calculus, you have the total derivative, a differential is a partial derivative.)

The confusion occurs when referring to inherently single-variable concepts such as time. "What is the time difference/differential between London and New York?" Either can work. There is only one differential, so the difference and the differential are the same. I would use differential. My 90-year-old mom would use difference. If you are tempted to insert "due to" or "caused by" after the word, you should probably be using differential.

Please notice they are both nouns which occasionally get used as adjectives when needing to say "of that noun thingy".

Both terms have idiomatic usage in mathematics which varies branch to branch.

• Nice explanation, but: in line 2 you refer to "[T]he difference in gearing between two bicycles". Did you intend to refer to the two wheels of a bicycle? – TrevorD Jun 11 '16 at 16:21
• I'm using gearing as the distance a bicycle travels with one crank revolution (or speed per crank rpm). Wheel diameter is a factor. Gear ratio is the ring to cassette tooth count ratio. Gearing can be used in other ways as well. – Phil Sweet Jun 11 '16 at 16:41
• That's a bit technical! I assumed it was something to do with one wheel being 'geared' and not the other! – TrevorD Jun 11 '16 at 16:46