This page is concerned with the math meaning of the word. See appendix Q below for a different meaning for these two words.
…the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually.
The above definition is a Swiss Army knife.
It will be valuable to dissect it at some point, but perhaps it is better to first see the principal in an example.
Assume we have a black box we’ve named Doubler. We know 2+3 is 5. We put 5 into the box and out comes 10.
Let’s do this again, only this time, lets put Doubler on the numbers before we do the addition:
- Doubler(2)=4
- Doubler(3)=6
Now add: 4+6=10
Doubler(2+3)=Doubler(2) + Doubler(3)
This works with all numbers, and you might guess that we wouldn’t be talking about it unless it showed up in a bunch of places so we’re going to go more general:
F(a+b)=F(a) + F(b)
Here’s how to play the game: define how you’re going to get your ‘a’ and ‘b’ and then F can be anything for which the above is true.
The function is called linear and the whole idea is called the Superposition Principle.
Quick note of importance: linearity carries with it two ideas, additivity and homogeneity:
- f(a+b) = f(a) + f(b) is additivity
- f(ca) = c f(a) is homogeneity and c is a scalar
Another reference mentions that additivity and homogeneity together give the superposition principle.
Appendix A
In differential equations “superposition principle” means that if f(x) is a solution to a linear homogenous differential differential equation and g(x) is also a solution to that equation, then f(x) + g(x) is also a solution.
Appendix Q
A second meaning for the two words “superposition principle” is found in Quantum Mechanics.
The superposition principle is that a system is in all possible states prior to the measurement and that the act of the measurement cause a collapse at which point it falls into one of the possible states.
“Choice is the death of all other possibilities.”
Vectoreverything 