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Trivial — Definition, Meaning & Examples

Trivial

A solution or example that is ridiculously simple and of little interest. Often, solutions or examples involving the number 0 are considered trivial. Nonzero solutions or examples are considered nontrivial.

For example, the equation x + 5y = 0 has the trivial solution x = 0, y = 0. Nontrivial solutions include x = 5, y = –1 and x = –2, y = 0.4.

Worked Example

Problem: Consider the homogeneous equation 3x + 7y − 2z = 0. Identify the trivial solution and find one nontrivial solution.
Step 1: A homogeneous equation (one where the right side is 0) always has the solution where every variable equals 0. Substitute x = 0, y = 0, z = 0.
3(0)+7(0)−2(0)=0✓3(0) + 7(0) - 2(0) = 0 \quad \checkmark
Step 2: This all-zero solution is the trivial solution. It works, but it tells us nothing interesting about the relationship among x, y, and z.
x=0,  y=0,  z=0(trivial)x = 0,\; y = 0,\; z = 0 \quad \text{(trivial)}
Step 3: To find a nontrivial solution, choose values for two variables and solve for the third. Let x = 1 and y = 1.
3(1)+7(1)−2z=0  ⟹  10=2z  ⟹  z=53(1) + 7(1) - 2z = 0 \implies 10 = 2z \implies z = 5
Step 4: Verify: 3(1) + 7(1) − 2(5) = 3 + 7 − 10 = 0. This solution is nontrivial because at least one variable is nonzero.
x=1,  y=1,  z=5(nontrivial)x = 1,\; y = 1,\; z = 5 \quad \text{(nontrivial)}
Answer: The trivial solution is x = 0, y = 0, z = 0. One nontrivial solution is x = 1, y = 1, z = 5.

Another Example

Problem: Consider the equation x(x − 4) = 0. Which solution is trivial and which is nontrivial?
Step 1: Apply the zero-product property: either x = 0 or x − 4 = 0.
x=0orx=4x = 0 \quad \text{or} \quad x = 4
Step 2: The solution x = 0 is the trivial solution because it involves zero and is immediately obvious from the factored form. The solution x = 4 is nontrivial — it conveys actual information about the equation.
Answer: x = 0 is the trivial solution; x = 4 is the nontrivial solution.

Frequently Asked Questions

What does 'trivial' mean in math?
In math, 'trivial' describes a result, solution, or case that is immediately obvious and carries little useful information. The most common example is the all-zero solution to a homogeneous equation. Calling something trivial does not mean it is wrong — it is a valid result, just not an interesting or informative one.
Why do mathematicians care whether a solution is trivial or nontrivial?
Because the existence of a nontrivial solution often reveals important structural information. For instance, knowing that a system of equations has only the trivial solution tells you the equations are independent, while the existence of nontrivial solutions tells you there is redundancy. In many proofs and applications, the key question is whether anything beyond the obvious zero solution exists.

Trivial vs. Nontrivial

A trivial solution or case is the obvious, uninteresting one — typically involving all zeros or doing nothing. A nontrivial solution is any solution that goes beyond this obvious case, meaning at least one variable is nonzero or the result is not immediately self-evident. Both are valid; the distinction is about how much useful information the solution provides.

Why It Matters

Distinguishing trivial from nontrivial results is central to many areas of mathematics. In linear algebra, whether a homogeneous system has only the trivial solution determines if the coefficient matrix is invertible. In number theory and differential equations, the interesting work almost always lies in finding nontrivial solutions, since the trivial one is guaranteed.

Common Mistakes

Mistake: Thinking 'trivial' means 'wrong' or 'invalid.'
Correction: A trivial solution is perfectly valid — it satisfies the equation. The word 'trivial' just means the solution is obvious and uninteresting, not that it should be discarded.
Mistake: Assuming any solution involving a zero is automatically trivial.
Correction: A solution is trivial only when all variables are zero (in the standard usage for homogeneous systems). A solution like x = 5, y = 0 is nontrivial because not every variable is zero.

Related Terms