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I know that "height" is a uncountable noun when it is a measurement. But in some scenarios I think the form "heights" is better. For example, I want to prove the performance of an algorithm on several binary trees. The trees are of different height. Then I think the following sentence should be correct: "I will prove the performance of the algorithm on the binary trees of various heights".

How you guys think about it? Any suggestion will be appreciated.

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In some cases, height is countable:

[uncountable, countable] the measurement of how tall a person or thing is

In this sense, height means measurement, and measurements are countable. Your sentence with "...various heights" is correct.

Height is uncountable when it refers to the property of tallness:

The height of the mountain did not discourage them.

The mountain has some uncountable amount of height.

Even if we deal with multiple mountains, we can still have a single uncountable height property:

The height of the mountains did not discourage them.

This means the tallness of multiple mountains did not discourage them. It's similar to, "The fact that the mountains were tall did not discourage them." (But this is very awkward phrasing.)

The heights of the mountains did not discourage them.

This means that they measured (or otherwise learned) the individuals heights of multiple mountains (e.g., one was 5000 feet, another was 6000 meters, etc.) and were not discouraged by the sizes of the measurements.

(Note that the plural heights (and rarely in the singular) has yet another possible meaning of "high places," as in, "I have a fear of heights." That is not my intended meaning of "heights" in the above sentence.)


For s similar uncountable property, consider speed:

We need more speed to outrun our opponents.

And, by comparison, consider the countable form of speed:

The runner's speed was 6 kilometers per hour.

The runners all ran at different speeds: 5 kilometers per hour, 6 kilometers per hour, and 7 kilometers per hour.

  • "The speed of the car was amazing. (uncountable property)" - isn't that simply a singular use of (the very much countable word) speed? After all, car isn't uncountable, either, even though it happens to be used in its singular form in that sentence. – O. R. Mapper Dec 1 '14 at 20:15
  • @O.R.Mapper I've added some more mountain examples. I'm currently trying to find a formulate a better example for "speed". – apsillers Dec 1 '14 at 20:30
  • Ok ... I'm still not convinced we're not referring to the (essentially countable) combined height of a set of mountains in "The height of the mountains did not discourage them.", but thanks for the additional examples. – O. R. Mapper Dec 1 '14 at 20:38
  • @O.R.Mapper: the mass noun "speed" shows itself when we say "this car needs more speed". We can instead say, "this car needs a greater speed", treating it as a singular noun, but we don't have to. – Steve Jessop Dec 1 '14 at 21:11
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    @O.R.Mapper: oh, I misunderstood your objection to the examples. I don't think they need to demonstrate that the use in question is unambiguously as a mass noun, merely that there are examples of other uncountable properties can be slotted in the same way as height is. But I agree that they don't demonstrate it. We could just as well substitute "the roof of the car was impressive", certainly a count noun, or "the mathematics of the professor was impressive", certainly a mass noun. – Steve Jessop Dec 1 '14 at 21:23
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It is countable in that sense, so different heights is correct. It is uncountable when we refer to height in general, as in height is important to her.

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Measurement is a countable noun. Units of measurement are thus also countable nouns.

2. [countable, usually plural] the size, length or amount of something
to take somebody's chest/waist measurement
Source: Definition of “measurement” (noun) from the Oxford Advanced Learner's Dictionary

In general, the plural of a unit of measurement indicates the existence of distinct (discrete) values of that measurement in the set being discussed.

For example, an 11-speed bike has 11 speeds. An adjustable ladder might adjust to 3 heights.

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