Real Life Examples

Example 1: Height and Depth

Suppose you are climbing a hill and gained 5 metres of height. When you undo your operation, you will again reach to the ground from where you started. This is normal addition and subtraction which involves accepting quantities and rejecting quantities.

$0+5m-5m=0$

But what if we want to maintain that position and reset our measurement?

In this case we must acquire a negative height. How? By rejecting our current measurement.

To reject a measurement, one must do subtraction. Rejecting a measurement from a system is similar to subtract the same quantity from the system which in result resets the system to its initial point but without a change in its reference. But this rejection can also be achieved by the gain of a negative height, changing its reference. In other words, the reference line will now shift upwards.

$0+5m+\dot{5}m=0$

If we assign a quantity with a behavior to go out of the system, then holding that quantity for a while behaves like accepting that quantity along with that tendency.

Either the representation is positive, or it is negative, they represent the same physical quantity. Positive and negative numbers do not mean that the quantity is of different types, it's just about opposite behaviours.

Example 2: Force (Linear situation)

Suppose a force of 5N is acting on a body in certain direction. We can represent this as:

$Net \ \ Force=0N+5N=5N$

Now, if we remove this force from the system, then it can be represented as:

$Net \ \ Force=0N+5N-5N=0N$

But, instead of removing this force, if you add another force in opposite direction, it can be represented as:

$Net \ \ Force=0N+5N+\dot{5}N=0N$

If you remove $5N$ force from a stable system, the wise should understand this that there must exist a force in opposite direction to $5N$. Also, if someone removes $\dot{5}N$ force from the stable system, the wise should understand this situation that there must exist a force in opposite direction to $\dot{5}N$.

Hence, a wise person considers $-5N$ as same as $\dot{5}N$ and $-\dot{5}N$ as same as $5N$.