# Finding number $n$ times larger then a given negative number

This is an interesting question, and I can't really answer it.

The number which is $5$ times larger than $2$ is $10$ ($2 \times 5 = 10$)

But, which number is $5$ times larger then $-2$?

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It depends on the definition of "$n$ times larger" (for some real number $n$).

If we define $a$ to be $n$ times larger than $b$ to be $|a| = n|b|$, then we see that there are many numbers (arguably infinitely many) which are "$5$ times larger" than $-2$ :

• $-10$ (because it is 5 times longer than $-2$ on the real number line)
• $10$ (because again, it is 5 times longer than $-2$ on the real number line)
• $-10i$ (because it is $5$ times longer than $-2$ on the Argand diagram)
• $10i$ (because it is also $5$ times longer than $-2$ on the Argand diagram)
• In general, $10e^{i\theta}$ $\forall \theta \in \mathbb{R}$ (because they are all 5 times longer than $-2$ on the Argand diagram)

I'm sure there can be many more answers if you were to go into hypercomplex numbers, e.g. quaternions, octonions. But then again, it all depends on how you define a number to "n times larger" than another.

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I think the expression '$5$ times larger' isn't well defined in this case. It is even ambiguous in the positive case: Are you asking for $5$ times that number or $2 + 5\times 2$.

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Good point! I think that this depends on the language you are speaking. In english, there is an ambiguity that you speak of. In slavic languages (at least the ones I know of), saying "$10$ je petkrat večje od $2$" (which translates into "$10$ is five times larger than $2$") is not ambiguous at all. – 5xum Mar 3 '14 at 15:00
When I think about it, it is indeed ambiguous when translated from my language to English (in my language it says "Koji broj je 5 puta veći od broja -2", Srpski, Serbian). I guess what I am actually asking can be translated to "find the number 5 times the given number, but larger than the given number". – dizarter Mar 3 '14 at 15:30

I think this question is a clear proof that some question, although they seem logical, are not really valid questions.

I think that when we say "object $x$ is five times larger than object $y$", what we imply in this sentence is that the size of both $x$ and $y$ is positive, and that the factor between the sizes is $5$. Therefore, for negative numbers, I don't think we should be talking about something being $n$ times larger than something else.

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Here you go: $$5\cdot(-2)=-10$$

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But $-10$ is smaller than $-2$! Indeed, OP's question can be really confusing and ambiguous. – Yiyuan Lee Mar 3 '14 at 15:04