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Let $a,b \in \mathbb R^+$ and $b\le a$. By taking $b$ as reference, I have no doubt about using the computation $x = \dfrac{a}{b}$ to calculate how many times $a$ is greater. But when $a,b \in \mathbb R, $ does the $x$ value make sense with

  1. $a$ and $b$ both negative;
  2. $a$ positive and $b$ negative ?

Examples:

  • $ x = \frac{10}{2} = 5$, is 5 five times greater, this is OK
  • $ x = \frac{10}{-2} = -5$, this is not necessarily $|-5|$ times greater because the difference is $ 10-(-2) = 12$, only if I take absolute values for that
  • $ x = \frac{-5}{-8}=0.6$, I'm not sure if this has any meaning.

What about when working with temperatures and wanting to know how many times greater is a temperature is than a reference temperature, for instance: reference $-15^{\circ}$ and the other temperature $-4^{\circ}$ ? In this case do I apply reciprocal and use $x = \frac{b}{a}$ instead?

I read these related questions before posting this question:
"$x$ times as many as" versus "$x$ times more than"?
Finding number $n$ times larger then a given negative number
How many times larger is 5 × 106 than 5 × 102?"How much is it more/small than" calculator..

ryang
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StandardIO
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    I agree with the answer of 311411 and would add two points: [1] Normally, given dimensionless real numbers $a,b$, in my limited experience, the question of how many times greater $a$ is than $b$ refers to a comparison of $|a|$ and $|b|$. [2] As 311411's examples suggest, comparison of dimensioned numbers $a$, $b$ requires that $0$ be delineated. So, when comparing two temperatures, using Kelvin gives a different answer than using Celsius. – user2661923 Mar 14 '21 at 04:30
  • This answer to a question you linked to suggests that you look at the absolute values of both numbers so that you take the ratio of two positive numbers. – David K Mar 14 '21 at 04:31
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    When you mention the temperature $-4^\circ$, it makes me suspect that you are asking the wrong question entirely. $2^\circ$ Celsius is not twice as hot as $1^\circ$ Celsius. There are scientific and engineering applications for a ratio of temperatures, but it is always a ratio of absolute temperatures, so you cannot take ratios of Celsius measurements directly. – David K Mar 14 '21 at 04:34
  • @DavidK yes, but in that answer he defined the operation in that way for convenience, maybe he is considering what user2661023 answered. And the temperature example was only for proposes of introduce the theme in something more related user2661923 introduce: the dimensioned numbers and the importance that $0$ must be delineated. – StandardIO Mar 14 '21 at 06:47
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    Don't blame the temperature example on user2661923. You wrote that into the question before anyone had commented or answered. As for "convenience," virtually every definition in mathematics could be said to be "for convenience," because it is convenient to have definitions that make sense and that you can more easily do mathematics with than definitions that are difficult or impossible to work with. – David K Mar 14 '21 at 15:17
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    "greater" is a troublesome word. It could mean "bigger" or it could mean "take the notable attribute of that number, it's of size $5$ in the negaitve direction... now make it $7$ times more extreme in that attribute... so it is size $35$ in the negative direction". In math we really mean the latter. But "greater" is just a word. DOn't get hung up on it. Words are only words. – fleablood Mar 14 '21 at 19:27
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    Temperature is a weird example also as the "origin" point of $0$ is completely arbitrary. To say, in San Francisco, that $80^\circ F$ is twice as warm as $40^\circ F$ is ... just meaningless. $40^\circ$ isn't warm at all. It's cold. – fleablood Mar 14 '21 at 19:47
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    In germany , quantities are also often wrongly used : If we say : "fünfmal höher als der Eiffelturm" (five times higher than the tour Eiffel) , it should be $6$ times the height of the Eiffelturm , but usually actually $5$ times is meant. No idea why. – Peter Jun 22 '23 at 10:31
  • For a negative number , such a formulation does not make any sense. We should instead describe the absolute value. – Peter Jun 22 '23 at 10:33

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I find your example of temperature interesting. There is a unit of temperature called Kelvin that avoids negative values entirely. Kelvin temperature is also called "absolute temperature". It seems absolute temperature is proportional to energy, and it seems reasonable to say things like "three times the energy".

Conversely, there is nothing wrong with saying "LeBron James is three times the height of my cousin." But I could introduce a ridiculous new unit of length in which Lebron is -102 units while my cousin is -168.

311411
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