Similar-looking formula

The equivalent resistance of a parallel circuit

can be determined by

.

A similar-looking formula found in a basic mathematics question involving parallel lines as shown below:

.

The length of the middle line segment, , depends on the lengths of outer lines segments and is independent of the distance between them.

In a circuit, we can consider more than two resistors:

And the equivalent resistance can be determined by

and it helps me to make an analogy between the ‘middle length’ and equivalent resistance like

.

Another formula in Physics pops up in my mind:

it is the lens formula, a similar-looking formula.

For convex lens, the object and image distances are 1 and 2 respectively (say), but if the image is virtual, we need to plug in the formula to get the focal length,

What is the so-called insight of the adding-minus-sign move in the basic mathematics question?

Just think of the following case: the given line segments (of lengths ) on the opposite sides of the ‘base line’. That is

We can still apply

.

e.g. Taking negative length for the line segment under the base line, thus

Yield , meaning the length is 2 and its position is under the base line.

(Refer to the figure above, we can just regard as the middle length, and we’ll have to get rid of negative thing.)

The adding-minus-sign move can also be found when applying section formula, but I seldom use it now.

Nothing special.

Exercise
Prove all the formulae seen in this post.