# Succinct description of situations where naively obvious is correct, but for far more complicated reasons?

What is the name for a situation where the obvious thing turns out to be true, but the reasoning is more complicated?

In abstract algebra we can say the rational numbers - the fractions, $$\mathbb{Q}$$ form a field. Then we can adjoin the number $$x = \sqrt{-1}$$ which solves $$x^2 + 1 = 0$$, and we can have the field $$\mathbb{Q}(\sqrt{-1}) = \mathbb{Q}[x]/(x^2 + 1)$$.

Doesn't it make sense that the ring of integers should be $$\mathbb{Z}[i]$$. We are going to take the ordinary integers $$\mathbb{Z}$$ and adjoin the number $$i = \sqrt{-1}$$. Unfortunately, this conflicts with the definition of ring of integers.

Definition The ring of integers $$\mathcal{O}_K$$ is the set of solutions $$x \in K$$ of monic irreducible polynomials $$x^n + a_0 x^{n-1} + \dots + a_{n-1}$$ with coefficients in $$\mathbb{Q}$$

And then it's not at all obvious that $$\mathcal{O}_{\mathbb{Q}(i)} = \mathbb{Z}[i]$$. Would it be safe to skip this definition entirely and just say here's an interesting mathematical object:

$$\mathbb{Z}[i] = \big\{ a + bi : a , b \in \mathbb{Z} \big\}$$

and discuss whatever topic of interest, e.g. unique factorization.

• Why do you believe it is "obvious"? The naive approach fails in other cases, adjoining $\sqrt{-3}$ to $\Bbb Z$ yields a ring $R = \Bbb Z[\sqrt{-3}]$ that is not integrally closed since the proper fraction $\omega = (-1+\sqrt{-3})/2$ over $R$ is a root of $x^2+x+1$. Thus RRT = Rational Root Test fails, so the ring lacks unique factorization (because UFD $\Rightarrow$ RRT). So we need to enlarge the "naive" ring of integers to be integrally closed if there is any hope that it will enjoy unique factorization. Sep 25 '18 at 14:38
• Mathematicians use the word "obvious" in a bizarre way. I'll fix the wording in a bit. Sep 25 '18 at 14:41
• @paul If you can make sense of the question then please do elaborate. It is far too vague as it stands now. Sep 26 '18 at 13:29
• @paul So the question is about things that are "more subtle than they appear at first glance", or something more specific? Sep 26 '18 at 22:23
• perhaps it's not even a complete thought yet. I thought i had a question Sep 27 '18 at 1:56

The Jordan Curve Theorem is the canonical example of this.

Also, the fact that the sum of the first n odd numbers is n squared is obvious from the well-known ‘wrapping’ diagram, but the actual proof is via mathematical induction, which is a challenge for many students.

Also: The theorem on sphere-packing, long time in coming, merely confirmed what what obvious to everyone. (“While the Kepler conjecture is intuitively obvious, the proof remained surprisingly elusive.” – Wolfram MathWorld)

Also: The maximum possible area bounded by a curve of fixed length is a circle. (Dido’s problem)

Also: There are 3 regular tessellations.

Also: The base angles of an isosceles triangle are equal.

Also: A cannon ball travels farthest when fired at a 45-degree angle (ignoring air resistance).

and also perhaps: Every bounded infinite set of numbers has an accumulation point. (Bolzano-Weierstrass theorem)

and also perhaps: If there is an injective mapping from A to B, and an injective mapping from B to A, then there is a bijective mapping between them. (Schrӧder-Bernstein theorem)

and also perhaps: A continuous function defined on a closed interval attains a maximum value. (high-point theorem)

Belatedly:

It is my considered opinion that a reasonably refined "common sense" is rarely truly wrong, with regard to meaningful mathematical questions. Perhaps wrong on some technical disqualifications... that's easy enough.

To my observation, a main reason that mathematics is useful, and clarifying about (human interaction with) the world, is that it is a good narrative for accurate narratives about the world...

The absolutely amazing thing is that in many cases we can somehow see that, in the language of that narrative, no other outcome was possible. The language of the narrative constrained reality? Whoa. How often does that happen?

:)

• Your answer calls to mind two adages: “Calculus is an inspired response to the world.” and “Constraint produces better art.” Jun 20 '21 at 8:09
• Your answer also calls to mind the remark by Eugene Wigner about the ‘unreasonable effectiveness’ of mathematics in the natural sciences. Jun 20 '21 at 10:34