Trivial

The adjective “trivial” is a bit puzzling when used in math. In general usage it refers to something lacking importance (example: that fact is trivial). Sometimes, the math usage seems to agree with the common usage.

However, more often, it seems something being trivial means it is the one example that doesn’t behave the same as all the others. Let’s make one up.

All integers are either positive or negative except for the trivial case of the integer being zero.

Let’s make up another.

The addition of a second number to a first number changes the value, except for the trivial case where the second number is zero.

The adjective trivial may first surface in Linear Algebra.

We may be trying to solve something like the following:

ax + by + cz = 0

where a,b,c are constants with values given to us and we want a solution for the values of x,y,z. Setting them all to zero, x=y=z=0 make the equation true regardless of what the values are for a,b,c. You might joke that this is cheating. Well, if we are doing the calculation for a practical reason, the solution of all zeroes most likely will not help the customer. It’s a waste of time. It’s distraction. In other examples where there is a trivial solution, it seems the trivial solution is the one solution that doesn’t fit in with an idea that the teacher wants to teach. We can’t accuse that trivial solution of being a lie–it is not, but we can scream in its face that it is meaningless, worthless and completed hated by everyone.

Appendix A

Velocity changes your position over time except for the trivial case where velocity equals 0.

A symmetry operation moves or changes an object in some way except for the trivial case with a symmetry operation is the identity operation.