Please help me to find some natural example of the meaning of a negative $\times$ negative = positive, except velocity, distance etc in physics.
Simplest example to mind is $$a(t) \times v(t)$$ but I am looking for a something else. Thanks in advanced.
If your company loses $\$7$ per year, then $3$ years ago, your company was $\$21$ richer.
A square with side $L$ has area
$$L\cdot L=L^2$$
Then the area of the rectangle with sides $L-a$ and $L-b$ is
$$(L-a)\cdot (L-b)=L^2-La-Lb+ab$$
therefore
$$(-a)(-b)=ab$$
On the plane with a coordinate system, "negative" means a rotation of $180^\circ$ around the origin and possibly a change of the magnitude, given by the multiplication by the absolute value of the negative quantity.
Then "negative $\times$ negative" means a double rotation of $180^\circ$, that is, no rotation at all, and possibly a change of the magnitude, given by the multiplication by the product of the absolute value of the two negative quantities.
I think that the primary reason that negative $\times$ negative $=$ positive is that this yields attractive algebraic behaviour. So, this probably does not qualify as natural as you wished.
That this, and much else of mathematics, often works well in the real world has often been discussed. Eugene Wigner wrote an article on the subject: The Unreasonable Effectiveness of Mathematics in the Natural Sciences (Wikipedia)
Others have given some examples of how the rule fits some real life situations but there are cases where negative results don't make sense.
Consider that you need to travel $100km$ and you expect to be able to travel at $60km/h$ hence $1km$ per minute. So, you expect to complete the journey in $100$ minutes. Suppose that you fail to achieve the $60km/h$ but only achieve $30km/h$. You might calculate the speed you need for the rest of the journey to arrive on time. A little calculation gives this formula: $60 \times (100 - \frac{1}{2}t) / (100 - t)$ where $t$ is the time so far in minutes. As you approach the target time, $t = 100$, the required speed explodes and (informally) is infinity at $t = 100$. What happens after $t = 100$? The formula gives negative values. It suggests that if you drove backwards then you would complete the journal in negative time. Negative speed $\times$ negative time $=$ positive progress - nonsense.
So, you need to consider whether negative numbers make sense in your model. Sometimes they will and sometimes they won't. For example, if you are planning a flight then be careful about negative altitudes.
I think you can simplify the question to show that $(-1)\times(-1)=1$ since positive scalar multiplication in physics is always easy to explain in terms of intensity.
So, we just need to think in the real involution $-:\mathbb{R}\to\mathbb{R}$ and then interpreting in any physical context why under the composition it has order $2$. The minus sign almost always mean "backward" (but you have examples of that in velocity and distance) other times this involution means "lose", for example, if you have a fluid, you can put a positive sign if you are gaining fluid volume and put a negative sign if you are losing (regardless the direction of the fluid). Suppose you are losing a lose of some fluid, then you are gaining that amount of fluid. (It works also with gaining or losing energy)
I hope this viewpoint may help you.
find some natural example ofBefore asking about multiplication, it helped if you gave an example of what you would consider a "natural example" of a negative number, say for example $-1$. – dxiv Aug 13 '18 at 05:33