How can tan(30°) be irrational and at the same time be a ratio within a triangle?

Question

Dumb question... if tan(30°) is irrational (what I believe it is, it should be $\sqrt 3$), how can it be that, at the same time, is is describing the ratio of two sides within a triangle?

The triangle I have in mind looks like this (sorry for the cheap sketch): So, $a$ is the angle (in this case 30 degrees), and AFAIK it should be possible to describe that angle by the ratio of the opposite and adjacent sides, $A$ and $B$, so: $a = \frac{A}{B}$... and there certainly exists a real-world triangle for $a=30°$, where the other two angles are $90°$ and $60°$... and the two lengths, $A$ and $B$, are just some real values, so their ratio should be a rational number... or not? :-)

It seems I'm missing something here...

Oh ... bummer... I should go back to school, I guess :-D Thanks! — mlimper, Apr 28 '18 at 02:21
$\frac{\sqrt{2}}1$ is a ratio but it's still irrational. Actually all numbers are ratios! But not ratios of integers. — Stack Exchange Broke The Law, Apr 28 '18 at 08:28

score 8 · Accepted Answer · answered Apr 28 '18 at 02:34

the two lengths, $A$ and $B$, are just some real values

That's correct, but you cannot assume that those values must be integers (or rationals).

so their ratio should be a rational number... or not?

Not so, a rational number is defined as the ratio of two integers, not of arbitrary reals. (Of course, every real can be written as the ratio of two reals e.g. $x = \dfrac{x}{1}$, so that doesn't define any additional or interesting property for such $x$.).

Incidentally, the fact that $\,\tan 30^\circ\,$ is irrational is (one) reason why an equilateral triangle cannot be drawn of graph paper (so that the coordinates of all vertices would be integers).

yes, that thing about the graph paper is actually what came to my mind when you mentioned integers... nice to see that there is even an article on that. marking this as the accepted answer since you were first. — mlimper, Apr 28 '18 at 04:52

score 3 · Answer 2 · edited Apr 28 '18 at 04:12

Given a $\Delta{ABC}$ with $\widehat{BAC}=90^\circ$, then $BC$ is the hypotenuse. Let $BC=a;CA=b;AB=c$ then $\tan{ABC}=\dfrac{b}{c}$. If $\widehat{ABC}=30^\circ$ then $\dfrac{b}{c}=\sqrt{3}$.

Obviously, a number can only be rational if it can be expressed in the form of $\dfrac{a}{b}$ with $\operatorname{GCD}(a,b)=1$ and $a,b\in\mathbb{Z}$ and the sides of the triangle do not have to be rational.

How can tan(30°) be irrational and at the same time be a ratio within a triangle?

2 Answers2