What if $\pi$ was an algebraic number? (significance of algebraic numbers)

Question

To be honest, I never really understood the importance of algebraic numbers. If we lived in an universe where $\pi$ was algebraic, would there be a palpable difference between that universe and ours?

My choice of $\pi$ for this question isn't really that important, any other ''famous'' transcendental number (i.e. $e$) could work. I'm aware there are a lot of open problems about deciding whether some number is transcendental or algebraic (for an example Apery's constant, Euler-Mascheroni constant and even $\pi + e$).

Are those problems important only because they are hard to tackle? Are they important at all? If tomorrow was published a proof of algebraicity of those numbers, what would we gain from it?

EDIT: OK, maybe I took too much ''artistic freedom'' with the title of my question. I wasn't really curious about alternate universes. Bottom line was: why are those proofs important? Why is ''being a transcendental number'' important property of a number?

Answer 1

No such universe is possible, it would be a universe in which $1$ is equal to $2$.

That said, a rational approximation to $\pi$ with error $\lt 10^{-200}$ is undoubtedly good enough for all practical purposes.

Lindemann's proof that $\pi$ is transcendental was a great achievement, but knowing the result has no consequences outside mathematics.

Answer 2

You have to understand that although $\pi$ is a real number, it's not actually a real number. That is, it's in $\mathbb{R}$, but that set does not exist in the physical universe. It's an abstraction, just like the imaginary number $i$ is an abstraction, and one that has found significant use in physics (quantum mechanics and electrical engineering among others). Just like the idea of a number at all is an abstraction: the abstraction of assigning the same description to different quantities that are not directly related.

My point is that the quality of the universe that allows such abstractions to be imagined by intelligent creatures seems not to be separable from the quality that allows intelligent, imaginative creatures to exist at all. It requires only a sufficiently descriptive language, such as the kind considered in mathematical logic, to write down the formal definition of $\pi$ and indeed, of all of our mathematics, which implies all the algebraic and analytic properties of $\pi$ that we have proven because we wrote the proofs in that language!

So no such alternate physical universe can exist. On the other hand, one could imagine basing the definition of $\pi$ on alternate axioms, such as those specifying a particular non-Euclidean geometry, in some of which one does have $\pi = 3$, say. At least for some circles.

Answer 3

Interesting that nobody has mentioned: A practical consequence is that you cannot construct $\pi$ using a compass and a straightedge. This has saved so-so many man-hours; if Lindemann hasn't proved $\pi$ were transcendental we wouldn't have, e.g. caramel macchiato (or more significantly, aircraft).

Answer 4

Just to expand a bit on trb456's answer.

If $\pi$ was algebraic then that could mean there is some reasonably simple polynomial $p(x)$ for which it was a zero. This would be really nice because then all I'd have to do with a mystery number $\xi$ to check if it was $\pi$ is to check if $p(\xi)=0$ and then perhaps do a bit more book-keeping to verify that $\xi$ was really the real $\pi$.

For example, imagine we had some mystery number $\eta$ and we want to check that $\eta = \sqrt{2}$. How to do this? (for the purposes of this hypothetical, suppose calculators are all controlled by evil, self-aware, robot masters, they cannot be trusted, we have to do calculations by pencil and paper to be safe) $p(x)=x^2-2$ has $p(\sqrt{2}) = 0$, but $p(-\sqrt{2}) = 0$ so as a check on the number being $\sqrt{2}$ I'd also need to check on the sign of the number by some method.

So, perhaps you can see the utility of a number being algebraic. There is some finite number of algebraic operators we can perform on a potential candidate to ascertain if it is in fact the algebraic number in question.

In contrast, to show some potential number is $\pi$ we'll have to resort to a deeper mathematics. Some analysis, series, etc... We have convenient notations to hide the sophsitication, but $\sqrt{2}$ is much easier to define than $\pi$.

In any event, it probably should be agreed that there is some sufficiently precise rational number which captures the concept of $\pi$ for the need of any physical problem which involves $\pi$, so the absence of the polynomial check it of little concern. Most of the time $p(x)=x-22/7$ will do just fine.

Answer 5

This probably doesn't answer what you really want to know. It seems like you are more interested in knowing about the significance of algebraic numbers.

That said, This paper by Ivan Niven provides a proof that $\pi$ is transcendental. The proof is a proof by contradiction. That is, Niven assumes that $\pi$ is algebraic and derives a contradiction. So when you ask what would happen if $\pi$ was algebraic, then Ivan Niven actually has something very concrete. You could try to take a look at the paper to figure out what contradiction he derives.

And the question now becomes: what other statements can you derive from this? The fact is that you can prove anything from a false statement. So, as mentioned in the other answers, you can prove that any statement is true. And this is (at least one place) where the "significance" is.

(I found the link to the article in this answer.)

Answer 6

Being an algebraic number is just a property, like being an even integer. Not all integers are even, and not all real numbers are algebraic. No big deal. The algebraic numbers happen to be the zero of some polynomial in one variable over the integers. That's all. But it's a nice property, and it's easier to identify algebraic numbers versus non-algebraic (transcendental) numbers. We know very little about individual transcendentals, but we know lots about algebraics!

Answer 7

In his answer to this question on Math-Overflow, Francois G Dorais considers such a possibility, in a weaker system of arithmetic:

Shepherdson presented a simple method for constructing such models, I will present such a model where π is rational!

I admit that I cannot follow the entire proof, but his closing remarks can be understood by those with only essential maths.

Answer 8

$\pi$ is a pretty important real number which has been studied since antiquity. It is natural to ask every question you can about such numbers to see what you can find out; that's how mathematics progresses. First, is $\pi$ an integer? The answer, of course, is no, and this has been known probably as long as the concept of $\pi$ existed. Next, you might as if $\pi$ is the ratio of two integers; that is, if it's the solution to an equation of the form $p \pi + q = 0$ for integers $p,q$. That was answered by Lambert negatively in the 18th century.

Now that you've considered that, it's a natural extension to ask whether $\pi$ is a root of any polynomial equation, rather than just linear equations. If it is, we can probably say a lot about it with number theoretic methods. The fact that this is impossible was established in 1882 by Lindemann, who proved that $\pi$ is transcendental. So in that sense, it's more of a no-go sort of theorem which says that $\pi$ can't be described purely algebraically based on $\mathbb Q$.

So this is just an example of a natural question about a natural object. The answer turned out to be the less interesting of two possibilities for the purpose of practical work. The fact that $\pi$ is transcendental isn't really that important from a modern perspective, and certainly not for anything as applied as physics or engineering. Ultimately, the more important thing is the proof itself. Proving basically anything in transcendence theory is hard because it requires both algebraic and analytic tools to be used in conjunction, and combining them is always tricky.

Also, it's worth pointing out that the Lindemann–Weierstrass theorem proves more than just the transcendence of $\pi$. It's actually quite a far-reaching result with a number of applications and significant importance. $\pi$ being transcendental just happens to be one of them. This continues to be one of the few nontrivial facts with proofs in transcendence theory, and a lot of modern research is related to various generalizations and related conjectures.

Answer 9

If $\pi$ were algebraic, it could be the limit of the ratio of consecutive terms in an integer sequence satisfying a linear recurrence! A famous property of $\phi$, the golden ratio, is that it is the limit of the ratio of consecutive Fibonacci numbers. Since $\pi$ is transcendental, there is no similar corresponding sequence. Here are some details:

Suppose a sequence of integers $\{a_n\}$ satisfies a linear recurrence

$$a_n + c_{n-1} a_{n-1} + c_{n-2}a_{n-2} + \cdots + c_{n-d}a_{n-d} = 0,$$

where each $c_i$ is rational. Such sequences often (*) behave like a geometric sequence in the long run:

$$a_n \sim c r^n,$$

where $r$ is an algebraic number. The root $r$ with maximum modulus of the polynomial

$$p(x) = x^d + c_{n-1} x^{d-1} + c_{n-2} x^{d-2} + \cdots + c_{n-d}$$

is the limit of the ratios of consecutive terms of the sequence (*) provided there is only one root with that maximum modulus.

For example, the Fibonacci sequence satisfies

$$a_n - a_{n-1} - a_{n-2} = 0,$$

though this is usually written as $a_n = a_{n-1} + a_{n-2}$. The associated polynomial is $p(x) = x^2 - x - 1$. Its largest root is $\phi$, so the ratio of consecutive terms tends to $\phi = \frac{1+\sqrt{5}}{2}$ as $n$ gets large. We will never see a similar result for $\pi$ or $e$.

Linear recurrences like this arise frequently, for example, in counting paths in a digraph or counting words of size $n$ in a regular language.

Answer 10

Lots of good answers up there, but what strikes me is that we do have to have transcendental numbers. The algebraic numbers (roots of polynomials with rational coefficients) are countable; yet we know the set of real numbers is uncountable. We thus need a lot more numbers than the algebraics to make up the real line.

That in itself doesn't make any particular number transcendental. The fact that π is transcendental has to do with the metric in which we are measuring. If we use the supremum norm where the "length" of a vector x is |x_i| where x_i is the component having largest absolute value, the unit circle winds up looking to our eyes like a square, and the ratio of its diameter to its circumference is 4. That is a nice integer number and avoids the inconvenience of having to approximate π for any practical applications.

So why do we use the Euclidean metric where |x| = $\sqrt{\sum x_i^2}$ leading to a transcendental π? Well, because it is everywhere differentiable, indeed analytic, and so we can apply the power of calculus to any questions that we might have.

Answer 11

If tomorrow was published a proof of algebraicity of those numbers, what would we gain from it?

New theory that applies where those numbers appear, which for $\pi$ is most of mathematics (physics, engineering, ...).

When a transcendentally constructed number is algebraic or rational, it can be a sign of additional structure and a theory waiting to be built. In some cases, a very substantial theory, such as "arithmetic geometry" coming from observations about numerical values of zeta- and L-functions. If a number as ubiquitous as $\pi$ had previously undetected algebraic structure, it would probably be so for a systematic reason, such as hidden symmetries that once discovered can apply to all kinds of problems where $\pi$ appears.

If $\pi$ were rational this would potentially be more dramatic. For example, $\pi = 22/7$ might imply some secret $11$-fold symmetry of the circle, or the existence of a previously unsuspected category of spaces in which the circle participates in a $22$- or $7$-fold covering with some interesting new structures. And the notion of geometry would be updated to reflect these new possibilities.

This is not that far from what really happened for numbers that generalize $\pi$ (periods of algebraic varieties): there is a compelling point of view that such numbers are just the numerical manifestation of more structured "upgraded" objects, and statements about transcendence and irrationality of the numbers then acquire the meaningful and practical interpretation that all

relations between the numbers are accounted for by relations between the structures.

The highlighted sentence (stripped of "the"'s) is the real answer to the question.