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I often have a hard time figuring out exactly what is meant when exponential/ly is used outside of mathematics. This is especially true for exponential decrease.

Take the sentence:

The number of fish in the lake has decreased exponentially over the past couple of years

Mathematically this would mean a while ago the decrease was rapid, but rate of decrease has decreased (e−x, left figure), which is a good thing (assuming we want fish in the lake). However I believe that the intent of the sentence is the number of fish in the lake is decreasing at an increasing rate (−ex, right figure).

e^-x -e^x

Is it true that decreased exponentially often means negative exponential increase when used outside of mathematics? And if so what are good ways of clearly specifying which kind of decrease you are talking about.

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    In common use, "asymptotic" would be used for the first case, with "exponential" reserved for the second case. – Hot Licks Jul 17 '15 at 12:58
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    Firstly, at What is a better word for exponentially? there is an answer showing that the word is often used in a sense less precise than the mathematical one (but echoing it). Then, by analogy, 'decreased exponentially' would be a paraphrase of 'decreased at a/n colossal / incredible / alarming ... rate'. // Is any increase / decrease in real life truly exponential in the strict sense (though radioactive decay gets close)? Isn't an exponential model just that? – Edwin Ashworth Jul 17 '15 at 13:01
  • Colloquially, it just means the number of fish decreased drastically. It does allude to the fact that the decrease is trending in an almost exponential fashion. However, It doesn't mean we are trying to quantify exactly how many fish have disappeared. – Alex W Jul 17 '15 at 13:16
  • When you say "Mathematically this would mean a while ago the decrease was rapid, but rate of decrease has decreased" that is not true in a certain sense: Mathematically, the ratio rate of decrease does not decrease. It is decreasing at X% per unit of time. – Avon Jul 17 '15 at 13:17
  • As @edwinAshworth points out exponentially is actually rather rarely used correctly. For one if you say that the money you have put into your savings account is increasing exponentially people would be surprised though it's technically correct. So unless a mathematician of physician is using the word you should probbably assume it's used as an intensifier meaning large or fast. (btw. lot's of things in life are truly exponential locally, bacteria growth, credit card debts, lots of probabilty and certainly radioactive decay). It's just most of them are finitary in the end. – DRF Jul 17 '15 at 13:21
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It means the right figure. I'm pretty sure most people who use the term exponentially aren't referencing any equation in their head.

Exponential has a very precise definition, but people just throw the word around to mean change very quickly, because this is how they have heard it used in the past and it sounded good.

In college I studied applied mathematics. In my experience, many people who use these words have very little knowledge of the real definition. Instead of focusing on the semantics of the sentence, you should use context of the conversation to determine what they mean.

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    Exactly. There are technical usages of words, and their non-technical usages. In a technical conversation, use the technical meaning. In a non-technical discussion, cut 'em some slack. – Mitch Jul 17 '15 at 14:25
  • I agree that most people probably don't have a function in mind, but I am just wondering how many of the functions features people include. Would you say that exponential simply means 'change very quickly', or that it also encompasses 'at an increasing rate' – Toke Faurby Jul 17 '15 at 17:09
  • Without more context its hard to tell. Certainly what you suggest is definitely true sometimes. Given the provided sentence, your suggestion and mine are both possible. I still think that colloquially its imprecise and that you need to hear the context to understand their definition. – Carlos Bribiescas Jul 18 '15 at 2:43
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Take the sentence:

The number of fish in the lake has decreased exponentially over the past couple of years

Mathematically this would mean a while ago the decrease was rapid, but rate of decrease has decreased

I disagree with your premise.

Reason

The basic structure of the sentence is: "The number of fish has decreased exponentially."

This does not mean that the rate of decrease has decreased as you state. It means that the rate of decrease has steadily increased over the period of time. There has been an acceleration.

You are confusing "decrease" with "rate of decrease".

Non-mathematical use

I agree with Alex W who says it could be used to mean "has decreased drastically."

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    I have to disagee with your disagreement. The OP's interpretation of exponential decrease is pretty established in maths. It is exponetial in that it decreases at a rate proportional to its current value – Tushar Raj Jul 17 '15 at 13:26
  • What Tushar Raj said. While I agree about the non-mathematical use. The (correct)(mathematical) meaning of the sentence is what the OP points out. – DRF Jul 17 '15 at 13:27
  • @TusharRaj - I disagree with your disagreement of my disagreement. In your comment you say "*it " decreases at a rate proportional ..." but you haven't stated what "it" is. This is crucial. Perhaps you could give your own answer - I'd genuinely be interested. – chasly from UK Jul 17 '15 at 13:32
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    An exponential increase in the number of fish being lost from the lake is a very different thing to an exponential decrease in the number of fish in the lake. That people may mean the former when they say the latter might be true but they are wrong to. – Avon Jul 17 '15 at 13:41
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    But chasly that is exactly what you are saying people mean by exponential decrease: "the rate of decrease has steadily increased over the period of time." That is an exponential increase in the numbers being lost. The right hand graph in the OP's question. – Avon Jul 17 '15 at 13:48

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