# What does it really mean for something to be “trivial”?

I see this word a lot when I read about mathematics. Is this meant to be another way of saying "obvious" or "easy"? What if it's actually wrong? It's like when I see "the rest is left as an exercise to the reader", it feels like a bit of a cop-out. What does this all really mean in the math communities?

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Very apropos question... –  Ｊ. Ｍ. Apr 28 '12 at 17:39
I'm guessing the downvoter and the one who voted to close as NARQ are one and the same... –  Ｊ. Ｍ. Apr 28 '12 at 18:46
I have the same question, but none of the answers scratched where it itches. In the course of proving "(℩x)(ϕ(x)) is an incomplete symbol," I came across "a=a is trivial." Therefore I'd like to know what trivial means in this context (Principia Mathematica by Whitehead and Russell.) –  George Chen Jan 30 at 22:35

It is true that the meaning of trivial varies as the complexity of the subject increases, or when the area of expertise of the writer is not yours. I find some stuff trivial, which might not be trivial for another person. Even with expert mathematicians, something might be trivial for a number theorist which might not really be trivial for a topologist, for example.

When you find trivial in a book it usually means: "This should be rather easy to see for anyone that has got this far into the theory", or "I think this is easy to see and I don't want to waste my time in proving it", among others. I really suggest you take a look at JM's link, since it has great answers and it is almost the same situation.

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From JM's link: The "What am I, stupid?" comment pretty much sums up my sentiments on this –  AOAOne Apr 28 '12 at 17:50
@AOAOne I don't follow. –  Pedro Tamaroff Apr 28 '12 at 17:51
I feel like it's meant to be a sort of "This should be easy for you to see" comment but more often than not, I am finding myself thinking, "Is it really that easy?" and therefore feeling stupid if it's not immediately intuitive to me as it is for the author. –  AOAOne Apr 28 '12 at 17:52
@AOAOne I see. I don't really like the "It is easy to see..." thing, if it isn't sided by a simple "note that $A$ implies $X$ is $Z$" small explanation, which can make you click. It is as if I said. From $$\frac{{{x^n} - 1}}{{x - 1}} = \sum\limits_{k = 0}^{n - 1} {{x^k}}$$ it is trivial to see that $$\frac{{{b^n} - {a^n}}}{{b - a}} = \sum\limits_{k = 0}^{n - 1} {{b^k}{a^{n - k-1 }}}$$ without hinting let $x=b/a$. –  Pedro Tamaroff Apr 28 '12 at 17:55

It can mean different things. For example:

Obvious after a few moments thought. Clear from a commonly used argument or a short one line proof.

However, it is often also used to mean the most simple example of something. For example, a trivial group is the group of one element. A trivial vector space is the space {0}.

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It either means the canonical obvious thing, for example:

The trivial map usually sends either everything to 0 or to 1 or to itself, depending on the context.

The trivial solutions to $x^n+y^n=z^n$ are when $x=y=z=0$, $x=z=1$ and $y=0$, or $y=z=1$ and $x=0$.

The trivial subgroups of a group are the whole group itself and the one-element subgroup.

Or it means that the author doesn't want to spell out something which is (in the author's view) and easy/obvious fact. I think most commonly, though, it means a student is bluffing on his homework with something he or she can't quite prove.

Of course, if you're reading and you see the that the author calls things trivial that don't appear trivial to you, you should take this as a lesson for your own mathematical writing. I personally make a point of never using the words "clear," "obvious," and "trivial" without first checking to make sure that what I'm saying is indeed obvious. A professor I know insists that you should never use these words: if the fact should be indeed trivial for your intended audience, you can just state the fact itself without mentioning that is obvious.

These seems like an appropriate moment to relate an old joke: While a professor is lecturing, he makes a claim and says "this is obvious." Seeing only blank stares from the graduate students, who have all been lost for at least the past five minutes, he asks, "this is obvious, right?" Getting no response, he starts jotting a few things down on the board, then checks his notes and starts scribbling frantically in the margins. He soon rushes off to his office to do some more verifications. Half an hour later, the class is still sitting there, already a few minutes late for their next class, and the no word from the professor. Finally, right when the students have given up waiting and are about to head out, the professor bursts into the room. "I was right. It is obvious!"

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