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I often have trouble speaking good English when negative numbers are involved. Would it be correct to say that negative five is less than positive two? If not, what expression is most appropriate?

The issue, of course, is that the absolute value of negative five is greater than the absolute value of positive two.

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closed as off topic by Bradd Szonye, Kris, MετάEd, Kristina Lopez, Hellion May 21 '13 at 14:57

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This question is probably more appropriate for English Language Learners. –  Bradd Szonye May 21 '13 at 5:56
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@BraddSzonye, I assure you I'm a native speaker, and that many native speakers have this same issue. –  goblin May 21 '13 at 5:57
    
Hm, in that case it would help to know why you doubt it's the right wording. Does “one is less than two” sound natural to you? Is your concern because the absolute value of +2 is less than –5? –  Bradd Szonye May 21 '13 at 6:00
    
@BraddSzonye, yes that's precisely my concern. I edited the question to make this explicit. –  goblin May 21 '13 at 6:01
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You really shouldn't use "less" with negative numbers unless you're doing mathematics, in which case -5 is indeed less than -2. But in plain English, say that -19°C is "colder than" 5°C, somebody owing $1,000 is "deeper in debt" than somebody owing $50, 50 feet below sea level is "lower" than 25 feet above sea level, and so forth. Otherwise you are liable to confuse people. –  Peter Shor May 21 '13 at 13:01

4 Answers 4

I'm not entirely sure this is an English question. It may not be on-topic on any SE site.

The value −5 is less than +2. However, as Bradd has commented, it's an abstract quantity as far as English is concerned.

There should be no difficulty with this number-line:

...  3    4    5    6    7  ...
  ___|____|____|____|____|___
   <-less            more->

So just extend it to the left:

... -2   -1    0    1    2  ...
  ___|____|____|____|____|___
   <-less            more->

You can easily see that if you have a number x which is to the left of a number y then x < y. It doesn't matter how far apart they are.

We're not talking absolute values here. Absolute values are always positive, so in their case there is nothing on the number-line to the left of zero.

     0    1    2    3    4  ...
     |____|____|____|____|___
        <-less        more->

Because negative numbers are an abstract concept, if you attempt to visualise them it's easy to get confused and end up comparing absolute values. For example, a pile of balls is easily visualised as reducing to zero as you remove balls. How do you go negative? A negative heap is a hole, so −1 is a hole the size of a ball, −2 a larger hole, and so on. But by saying "larger hole" you are comparing absolute values. Use the number line: don't attempt to realise the abstract.

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Number lines. Surely the ultimate metaphor. Did you work for SMP in a previous existence? –  Edwin Ashworth May 21 '13 at 8:21
    
Were they the green and orange Alpha and Beta books? Or the ones with computer punched-tape on the cover? Happy memories! (If a little distant now) –  Andrew Leach May 21 '13 at 8:31
    
@Andrew Negative numbers are not necessarily abstract. They are used in quantifying things where 0 does not mean "the absence of." For example, consider temperature as measured with Celsius or Fahrenheit scales. 0 degrees doesn't mean an absence of heat. It is a particular reference point on the scale. Negative values indicate values below the reference point. –  HTG May 21 '13 at 14:11

I can't help weighing in on this one since I teach algebra and calculus at a university. The math symbol '<' is read 'is less than' so a mathematical sentence of the form

a < b

would be read "a is less than b" and would indicate that the number 'a' is farther to the left on a standard number line than is the number 'b'. So the answer to the original question is

In the language of Mathematics, it is correct to say '-5 is less than 2'.

I hope you will forgive me, but since I'm a teacher, I can't stop there. The original post indicated a trouble with English when dealing with negative numbers. If you were able to stick to using only the precise definitions, then there would be little ambiguity.

But nobody does that (and if they did, it would be hard for listeners to follow). The careful precision of the mathematical language makes it difficult to listen to, so we translate from Mathematics into English as we speak, and in doing so we naturally use other phrasings, synonyms. And that is where confusion starts to creep in.

In the context of this question, the most common confusion arises when people substitute the phrase "smaller than" for "less than."

Is -5 smaller than 2?

Well, if by 'smaller than' you mean a number's size (also called magnitude or absolute value), then no. The number -5 has size 5 which is larger than a number with size 2. The negative sign is simply telling you where to find it in relation to the number 0. If you mean 'farther to the left on a number line,' then yes, -5 is smaller than 2.

Summary: When comparing numbers, do you want to compare their relative locations (left-right on a number line) or their relative sizes? If the former, then use "less than". If the latter, use the whichever you like, but include the term "size" or "magnitude" or "absolute value".

And thank you for letting me expound on one of my favorite topics: The intersection between the languages of Math and English.

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.........maths. –  Edwin Ashworth May 22 '13 at 16:18

Yes, it is a correct statement. To avoid confusion, it may be clearer to say

"-5" is more negative than "-2".

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That's an amusing comparison, like saying “B is more pregnant than A” when A and B have been pregnant 5 weeks and 5 months respectively. –  jwpat7 May 21 '13 at 14:57
    
@jwpat7 as an engineering student who has taken a lot of math courses, I can tell you that this expression is used quite often. –  Ataraxia May 21 '13 at 14:59

You can always say X is less than Y - this is fine in English. Whether it is mathematically correct or not is a mathematical question and nothing to do with the English language. The question would be the same in other languages, therefore it is not an English language question.

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I disagree, since the specific confusion stated here is how to express the correct idea in English. The trouble arises because we have so many ways (in English) of conveying the same basic idea, but in Mathematics a slight change in wording can have dramatic changes in meaning. Perhaps this is true in other spoken languages, but as you alluded to, this site is about English. –  HTG May 21 '13 at 13:51
    
Exactly - I am not sure why you say that you disagree as you have essentially agreed with your reasoning! The issue does not arise from the English language, but language in general and how it should be expressed. In the end, as it is a mathematical situation, using mathematical language is probably the answer. –  Sam May 21 '13 at 13:58
    
Sorry, I thought your answer implied, "the question isn't an appropriate question for this site." That's what I disagreed with as the question concerned the correct use of English in this context. –  HTG May 21 '13 at 14:06

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