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Sometimes when we discuss limits verbally, we may say that a variable $x$ being "very small" (assuming that $x$ is a real number). But this could mean any one of the following:

  1. The number $x$ is a positive number with very small absolute value (for example, $x=10^{-10}$ may be considered as "very small").
  2. The number $x$ is a negative number with very large absolute value (for example, $x=-10^{10}$ may be considered as "very small").

Of course in writing the meaning is usually clear from the context. However, how should one avoid verbal ambiguity when teaching a class? I have heard people using the following terminologies to describe a negative number with large absolute value:

  1. very small.
  2. very negative.
  3. very negative large.
  4. negative with very large absolute value.

The fourth one is probably most clear but is also more or less verbose. Now the question is:

What is the most recommended terminology for such a number?

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    $\begingroup$ "Very large negative" sounds good. $\endgroup$ – Joel Reyes Noche Feb 27 '18 at 3:58
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    $\begingroup$ Personally I always use "very small" to mean very small in magnitude and "very negative" to mean large in absolute value but negative. It just matches my intuitions about what "small" means better. I agree with Joel Reyes Noche though, you should change the title, as it is right now it looks like a student question. $\endgroup$ – Nate Bade Feb 27 '18 at 4:58
  • $\begingroup$ @JoelReyesNoche, thank you for your suggestion! It has been edited. $\endgroup$ – Zuriel Feb 27 '18 at 10:57
  • $\begingroup$ Not an answer, just an opinion: I call the number very negative or very positive number if it has large absolute value. And I call number a tiny positive or tiny negative, if the absolute value is very small. $\endgroup$ – Thinkeye Feb 28 '18 at 14:43
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    $\begingroup$ @Thinkeye, I like the way that you call them! It is very clear. $\endgroup$ – Zuriel Feb 28 '18 at 18:02
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It depends a little on context and how careful I am trying to be:

  • If I were being very careful, I might call such a number "a negative number with a large absolute value" or "a negative number with large magnitude."
  • If I am being a little less careful (which I often am when discussing asymptotic behaviour of, for example, rational functions), I'll generally say "a very large negative number" or "a large number to the left of zero" (like I said, we're probably graphing, so left and right make sense).
  • Even more sloppily, it is just "large number" (maybe the sign doesn't matter at all!) or "a number that is far from zero" or "far to the left of zero."

That being said, I think that the more important distinction is between "small" numbers and "large" numbers. I try to be careful to always use "large" to refer to numbers of great magnitude (i.e. numbers far from zero, in any direction—this generalizes nicely to vector spaces), and "small" to refer only to numbers that are near zero (again, this continues to work well in vector spaces).

The other place this comes up is when dealing with inequalities. One might reasonably say that $-10$ is smaller than $-9$, but I would try to avoid that language. Instead, I would describe $-10$ as being less than $-9$ (and refer to neither as being "smaller than" the other). Since the inequality $-10 < -9$ is read "$-10$ is less than $-9$," there should be no ambiguity.

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    $\begingroup$ Thank you for your detailed and helpful answer! I have never thought about the difference between "smaller than" and "less than" before. It is good to see the difference. $\endgroup$ – Zuriel Feb 27 '18 at 15:05
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As far as the OP liked my comment, I put this as an answer even if it is only opinion based:

option 2: very negative

I call the number very negative or very positive (number) if it has large absolute value. And I call number a tiny positive or tiny negative, if the absolute value is very small.

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I tend to say "extreme negative number" for numbers like -10^10

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I use number 2, very negative. Concise and clear.

Just small related FYI. Realize that in some contexts, the wording takes care of the issue (loss versus profit, export versus import).

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