How Do You Define Obvious?

Mathematical Obviousness

Mathematicians often say things like “It is obvious that…” or “It can be easily shown that…” when the desired result is relatively easy. But to a normal person, what they are saying is not obvious at all. Other fields often have similar situations. An economist might consider a particular problem trivial, yet even a bright student might not understand it at all.

Consider the following:

$1 + 1 = 2$

$\frac{1}{2} = \frac{3}{6}$

You probably thought that was obvious as well. Even though the two numbers are written differently, they are the same number.

Okay, one more statement. The question is, as always: obvious or not obvious?

$0.999... = 1$

There are three types of people on this issue:

1. Those who think it is true and find it obvious.
2. Those who think it is true and find it not obvious.
3. Those who think it is false.

My opinion, which belongs to that of the first group, is no doubt influenced by the fact that I am studying math. For me, this is the same as saying $1 + 1 = 2$ or saying $1/2 = 3/6$. The two expressions on the left and right sides of the equals sign are written differently, but as the math shows, they exactly equal. Yet many people cannot accept this truth, and are bent on believing 0.999… is not equal to 1.

True Story

Last year I took Math 6120, or graduate Complex Analysis. One day Professor John Hubbard was proving Jensen’s Theorem. I do not remember where in the proof it happened, but at some point, there was a small detail that Hubbard could not immediately prove.

As it turns out, the book he was using left out this particular part of the proof. When Hubbard prepared his notes, he assumed that any small detail the author left out would be “obvious,” and that he would be able to derive it quickly during the lecture.

So he writes the statement on the board and asks, “Why is this obvious?” Nobody in the room has any idea.

It took a while to figure out why that little “obvious” statement was true. And even then, Hubbard had to re-explain, as there were grad students who still did not understand it. Then we found that our justification for why it was “obvious” did not work in general, but luckily did work for this specific problem. When it was all over, this “obvious” result took about 15 minutes of our time.

2 thoughts on “How Do You Define Obvious?”

1. I think that “¿What is mathematically obvious?” is equivalent to “¿What tools I can use?”. For example, your third statement isn’t obvious at all, because 0’999… is an ambiguous notation. If you understand 0’999… like the sequence {0,0’9,0’99,0’999,…} and $\mathbb{R}$ like the construction via Q-Cauchy sequences, yes, it’s obvious. But it’s “false” in some others constructions, e.g., on the hiperreals numbes (and so, I think, in p-adic numbers).
By the way, I repeat, this example is only a notation question.
PS: Wikipedia has a long article on the thread, cf. http://en.wikipedia.org/wiki/0.999

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