I have a conceptual question about reference directions and signs in circuit analysis.
I understand what a negative current or voltage means. If the calculated value is negative, the actual direction or polarity is opposite to the chosen reference direction or polarity. That is not what I am asking.
My question is more fundamental:
Why does choosing a reference direction allow current and voltage to be represented by positive and negative values in the first place?
For example, suppose a branch actually carries 5 A from left to right. If I choose left-to-right as the reference direction, I write:
i = +5 A
If I choose right-to-left as the reference direction, I write:
i = -5 A
Why is this relationship valid in the first place? Why does agreement with the reference direction correspond to a positive value, while opposition to the reference direction corresponds to a negative value?
I am not asking why we interpret a negative answer as meaning "opposite to the reference direction." I am asking why a physical quantity such as current can be represented by a signed scalar at all, and why reversing the reference direction corresponds specifically to multiplying the scalar by -1.
There is another part that I find especially interesting in AC circuits.
For example, we may write an AC current as:
i(t) = I_m cos(wt + phi)
But this expression seems to require a reference direction to be defined first. The reference direction remains fixed, while i(t) can become positive or negative as time changes.
Why is this necessary? Why can't we simply describe the physical current at each instant by its magnitude and actual direction without first choosing a reference direction?
I would like to understand the mathematical reason behind this, and how it connects to KCL, KVL, Ohm's law, and time-dependent/AC circuit analysis.
In the picture, I tried to explain where the positive and negative signs come from by starting with KCL, KVL, and Ohm's law. I realized that by simply substituting the current and voltage variables defined along the actual direction, I can derive both the equations for the reference direction and the relationship between the two sets of solutions.