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Which of the following is most correct?

The distance of the shortest path must not be higher than 10km.
The distance of the shortest path must not be greater than 10km.
The distance of the shortest path must not be bigger than 10km.

Or is there an even better word?

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'Distance' should be replaced by 'length' for idiomatic English, but that doesn't change any of the answers. –  TimLymington Aug 26 '11 at 11:09
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4 Answers

up vote 4 down vote accepted

As height is not being mentioned here, but rather distance on a horizontal scale, "higher" would be inappropriate.

"Bigger" refers to size, not magnitude, and therefore, in this case, is also inappropriate. It's inappropriate because "distance" cannot be measured in size, but in magnitude. "Great length" not "big length".

That leaves, "greater" which is correct.

A better phrasing could be:

The distance of the shortest path must not be more than 10km.

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what is size if not magnitude? –  Matt Эллен Aug 26 '11 at 10:35
    
Size, as in amount. Length is not an amount, but a measure of dimensions –  Thursagen Aug 26 '11 at 10:38
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I disagree with your definitions. However I agree that greater is the better word to use. –  Matt Эллен Aug 26 '11 at 10:46
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According to this NGram, "greater distance" is the most common:

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Next after this is "further", as in "I ran further than he did". You could also say:

The distance of the shortest path must not be further than 10km.

However, I would suggest using "greater" as it seems to fit the context better.

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"greater distance" suggests that something is farther away. I don't think the path is far away, it is long or short. The end of the path might be far away. So I wouldn't say "The distance of X must not be further than....". "Distance of" isn't idomatic, IMO. –  Mr. Shiny and New 安宇 Aug 26 '11 at 12:33
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The distance of the shortest path must not be greater than 10km.

This one is correct.

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While I agree that "greater than" is correct, I'm not sure about the rest of the sentence. Two places have a distance between them, but does a path have a distance? I don't think it does.

The length of the shortest path must not be greater than 10km

or even

The shortest path must not be longer than 10km

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+1 For "longer". –  Gurzo Aug 26 '11 at 11:25
    
+1 For "The shortest path must not be longer than 10km." I feel that this is the most accurate, appropriate way to express the idea in the question. Or alternatively, "The shortest path must be no longer than 10km." –  KitFox Aug 26 '11 at 11:34
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+1 for "paths don't have distance". –  Mr. Shiny and New 安宇 Aug 26 '11 at 12:32
    
"must not be over 10km" even simpler –  z7sg Ѫ Sep 16 '11 at 12:35
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