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In everyday language people often mix up "less than" and "smaller than" and in most situations it doesn't matter but when dealing with negative numbers this can lead to confusion.

I am a mathematics teacher in the UK and there are questions in national GCSE exams phrased like this:

Put these numbers in order from smallest to biggest: 3, -1, 7, -5, 13, 0.75

These questions are in exams designed for low ability students and testing their knowledge of place value and ordering numbers and the correct solution in the exam would be: -5, -1, 0.75, 3, 7, 13.

I think if the question says "smallest to biggest" the correct solution should be 0.75, -1, 3, -5, 7, 13. Even though it doesn't seem to bother most people, I think the precise mathematical language is important and "smallest to biggest" should be avoided but if it is used it should refer to the absolute value of the numbers.

So my question is: Which is bigger, -5 or -1?

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    I don't know if this qualifies as a real question... Nevertheless I think the exam problem is phrased unambiguously as it doesn't mention absolute values in any way.2011-06-01
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    I'd at least add a "soft question" tag to this. Mathematicians will use all sorts of language, informally. When written formally, the distinction between $|x|<|y|$ and $x is a bit clearer. If we were thinking in terms of vectors, we will often say one vector is "bigger" than another if its norm is bigger - a vector is a length and a direction, but you can't really compare directions, just lengths. In the case of the real line as a trivial vector space, this becomes the notion of comparing absolute values.2011-06-01
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    That said, for the problem in question, I'd assume that they want the standard order.2011-06-01
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    I get reminded of this http://menmedia.co.uk/manchestereveningnews/news/s/1022757_cool_cash_card_confusion2011-06-01
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    Obviously the correct answer (from smallest to biggest), at least in this font, is: 7,3,-1,-5,13,0.752011-06-01
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    Let's say I'm tens of thousands of dollars in debt because of student loans and you have exactly one thousand dollars; which of us has "more" money? (This is just one way of looking at it.)2011-09-27
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    @anon: "more" is different from "bigger". $-1$ is clearly "more" than $-5$; the question is whether it is also "bigger". (The OP mentioned this distinction in the first sentence, contrasting "less" and "smaller" -- you're just looking at "more" instead of "bigger" and stepping around the distinction instead of answering it.)2011-09-28
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    @Shree: Eh, I wasn't reading it close enough.2011-09-28
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    @anon: If you are 100 dollars in debt, while I am 1000 dollars in debt, who of us has "more debt"? The meaning of less and more depends on the direction you declare as ascending. - If you have a primitive economy, you may either have no gold nuggets at all, or a positive number of them, so "more" and "less" are meaningful concepts for abstract numbers. But if you introduce financial economy and invent debts, negative numbers become useful, but I think the meaning of numbers changes.2011-09-28

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