Can you square pi (3.1415965...)? If so, what is pi squared?

Random question :smiley: please help me find the answer!


Fess up - is this homework?

Of course not, all good pi is round.

Of course you can square pi. You can square any real number and many others of course. You can’t get an exact answer, but then you can’t get an exact answer to pi itself either, but you can get as accurate approximation as you want. You know that pi ~ 3.14159 so 3.1415 < pi < 3.1416 making 3.1415[sup]2[/sup] = 9.8690225 < pi[sup]2[/sup] < 3.1416[sup]2[/sup] = 9.8696506.

You can square any number. Why should it be surprising that this includes pi?

As for what it’s used for, it shows up in the formula for the hypervolume of a sphere in 4 or 5 dimensions (pi^3 is in 6 or 7 dimensions, pi^4 in 8 or 9, etc.), and pi^2/6 is the sum of the reciprocals of the perfect squares (that is, 1 + 1/4 + 1/9 + 1/16 + …).

For something more trivial, if you had a measuring-wheel of diameter 1 meter, and used it to measure out the sides of a square, it’d have an area of pi^2 square meters.

Of course, even if it weren’t used for anything, it’d still be perfectly valid to square it. By comparison, the number e+pi (5.859874482…) doesn’t show up in any interesting formula, but it’s still a perfectly valid number.

Note that this is not the right value - should be 3.14159265…

Is there a proof that the square of pi is irrational? Is there a general proof that the square of any irrational number is irrational?

The square root of two is irrational, so no, there isn’t a general proof.

Pi are not square. Pi are round. Cornbread are square.

(I’m sorry, I couldn’t resist the straight line.)

No, because the square root of 2 (for example) is irrational. Root 2 squared is, of course, 2.

Pi, on the other hand, is trancendental (it can’t be expressed as the root of any polynomial equation with rational coefficients). That (I think) means its square has to be irrational, because if pi^2 was some rational number a/b the equation bx^2 - a = 0 would have root pi.

But there is a proof that the square of pi is irrational; here’s a sketch of the proof: pi has been proved to be transcendental, meaning that it is not the root of any polynomic equation with rational coefficients. All transcendental numbers are irrational. If pi^2 were not transcendental, it would be the root of some polynomic equation with rational coefficients - and it would be simple to convert that equation to one which had pi as a root; therefore pi^2 is transcendental (and thus irrational)

I was thinking along those lines, but I’m having trouble seeing exactly how that transformation works. I think if the polynomial that has pi^2 as a root has any odd degree terms that you could be in trouble.

This is beautiful.

You could be right about that.

Yes it is.

Let me WolframAlpha that for you.

Ah yes, now I remember that pi[sup]2[/sup]/6 is the sum of the infinite series
1/1 + 1/4 + 1/9 + 1/16 + … + 1/n[sup]2[/sup] + …
(so pi[sup]2[/sup] = 6/1 + 6/4 + 6/9 + 6/16 + …)

No, it’s not for homework :slight_smile: I’m one to ask unusual, random and/or factual questions either regarding theory, opinion or fact.

Sorry that I missed the two. I never really memorized pi. 3.14159265…
Since pi goes on forever and ever amen, you can never multiply itself times itself, correct? You can only round to the nearest thousandth.

I think OldGuy already answered this in Post #5. You can compute it to whatever (finite) degree of accuracy you need.

Okay, it’s obvious when you put it like that.

If the polynomial x |-> p(x) has pi^2 as a root, then the polynomial x |-> p(x^2) has pi as a root. It’s only the other direction that’s tricky (although it’s also true that if x is algebraic, so is x^2; it’s just not as immediate to see). The concern about odd degree terms is only for going in the other direction.

Infinite series are exact answers. Exact as in exact. It’s true that they can be computed only to a finite degree, but that’s a practical limitation, not a theoretical one. The infinite series that Thudlow Boink gave is the exact expression of pi[sup]2[/sup]/6. It’s as exact as the diagonal of a unit one square is exactly the square root of 2.

BTW, if you don’t already know, you can enter in any mathematical relationship into the Google box and get an answer. You don’t even need symbols. Just put in “pi squared” and you’ll get an answer to seven decimal places.