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

Solution 1:

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 $2 + 5\times 2$.

Solution 2:

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.