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Let $a,b\in R$ and $a<b.$ Can we give a precise definition to say "$b$ is $m$- times greater than $a$?" in general. For the positive $0<a<b$ it is clear. "We say that $b$ is $m$-times greater than $a$ if $\frac{b}{a}=m.$" How about if both are negative or one negative the other one is positive?

MIYY
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    This means $b = ma$. – Toby Mak Apr 15 '21 at 11:02
  • @Toby Mak. I don't think so – MIYY Apr 15 '21 at 11:02
  • I would agree with Toby. What makes you disagree, MIYY? Do you have a particular example for this? – Matti P. Apr 15 '21 at 11:08
  • @Matti P. Let's take $a=-8$ and $b=-2$ try to write $b$ is 4 time bigger than $a$ – MIYY Apr 15 '21 at 11:19
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    Good point, actually. It's so simple to accidentally assume that variables are positive ... – Matti P. Apr 15 '21 at 11:22
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    @MIYY: Please include specific concerns (eg, negatives) as part of the question itself, so that people don't waste time giving responses you're prepared to rebut. (I'd thought your concern might relate to how "$b$ is $100%$ bigger than $a$" means $b=a+a=2a=a\cdot 200%$, yet some people (wrongly) say "$a$ increased by $300%$" to mean $a\to a\cdot 300%$.) ... In any case, common language is ambiguous, so formal translation can depend upon context. It might be good to ask how best to informally translate $b=ma$, since "$m$-times bigger" may not always sound right (eg, with negatives). – Blue Apr 15 '21 at 12:10
  • @Blue My concern is the following. Two different negative numbers are given $a<b<0$ how to determine that $b$ is $m$-times bigger than $a$? We are not going to use percentages. How about if $a<0<b$? – MIYY Apr 15 '21 at 14:02
  • Generally, for any $a<b$ can we give a precise definition to say "$b$ is $m$ times bigger than $a$"? – MIYY Apr 15 '21 at 14:09
  • @MIYY: Put your concerns and clarifications into the body of the question. Comments are easily overlooked (and can be hidden). – Blue Apr 15 '21 at 14:17
  • It's in natural language, so the possibility of misinterpretation is open. It sounds like you could draw a distinction between "bigger" talking about magnitude and "greater" talking about number line ordering. But it would be something more likely to be apparent from context rather than a "rule" of what the words mean. – Joffan Apr 15 '21 at 14:30
  • You seem to be hanging-up on the semantics of treating "bigger than" as a synonym for "greater than"; the resolution might be: Don't do that. ... Personally, I try to use bigger/smaller than in an "absolute (value)" sense, as they connote comparison in size; whereas greater/less than are agreed-upon conventions for order. Thus, I'd say $-8$ is "less than", yet "bigger than", $-2$. Not everyone would agree with my terminological choices here, but at least they avoid the semantic issue. ... Even so, I'm not sure I'd say "$-8$ is $4$-times bigger than $-2$" out loud; it sounds weird. :) – Blue Apr 15 '21 at 14:31
  • Is "$b$ is $m$ times $a$" not satisfactory? – user Apr 15 '21 at 14:48
  • @MIYY If $b$ is $m$ times greater than $a$, then $b > a$. If both $a$ and $b$ are less than zero, then that will not be true as $-b < -a$. As Matti P. and Blue pointed out, you can think in terms of absolute values. – soupless Apr 15 '21 at 14:49

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I would say that "b is m times as large as a" means that b= ma but that "b is m times larger than a" means that b= a+ ma= (m+1)a.

user247327
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