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

**URL:** <https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454>\
**Category:** Factual Questions\
**Created:** [April 30, 2012, 11:04pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454 "2012-04-30T23:04:28Z")\
**Posts on this page:** 20\
**Page:** 2

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**Author:** ![Hail\_Ants](https://avatars.discourse-cdn.com/v4/letter/h/dc4da7/32.png) [@Hail\_Ants](https://boards.straightdope.com/u/Hail_Ants)\
**Post date:** [May 1, 2012, 3:14am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/21 "2012-05-01T03:14:58Z")

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> [@sheriru](#):
>
> Can you square pi (3.1415965…)? If so, what is pi squared?

**π²**

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**Author:** ![zoid](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/zoid/32/336_2.png) [@zoid](https://boards.straightdope.com/u/zoid)\
**Post date:** [May 1, 2012, 3:35am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/22 "2012-05-01T03:35:12Z")

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[Pie squared](http://www.google.com/imgres?hl=en&safe=off&sa=X&biw=1536&bih=712&tbm=isch&prmd=imvns&tbnid=WUidRQ307r8mRM:&imgrefurl=http://scienceblogs.com/tfk/2010/03/pi_day.php&docid=RGsHChFP7lC9fM&imgurl=http://scienceblogs.com/tfk/images/squarepear.jpg&w=454&h=300&ei=glmfT5jzIKTv6AGElb2OAg&zoom=1&iact=hc&vpx=1218&vpy=203&dur=5004&hovh=182&hovw=276&tx=130&ty=104&sig=102923980558450143824&page=1&tbnh=163&tbnw=244&start=0&ndsp=21&ved=1t:429,r:6,s:0,i:98)

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**Author:** ![RadicalPi](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/radicalpi/32/9806_2.png) [@RadicalPi](https://boards.straightdope.com/u/RadicalPi)\
**Post date:** [May 1, 2012, 4:48am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/23 "2012-05-01T04:48:26Z")

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I feel I should chime in here, but I have nothing to say.

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**Author:** ![shijinn](https://avatars.discourse-cdn.com/v4/letter/s/cc9497/32.png) [@shijinn](https://boards.straightdope.com/u/shijinn)\
**Post date:** [May 1, 2012, 5:01am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/24 "2012-05-01T05:01:44Z")

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i thought all pies are squared, at least that’s how it is in my circles.

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**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\
**Post date:** [May 1, 2012, 6:11am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/25 "2012-05-01T06:11:26Z")

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> [@Sheriru](#):
>
> Random question 😃 please help me find the answer!

You can actually go to Google and type:

pi^2

and get your answer.

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**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\
**Post date:** [May 1, 2012, 6:22am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/26 "2012-05-01T06:22:27Z")

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> [@Indistinguishable](#):
>
> 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.

For what it’s worth, here’s the argument that if x\_1, …, x\_n are roots of [nontrivial] polynomials [with coefficients from the rationals, or whatever base field] of degree m\_1, …, m\_n respectively, then any rational function of x\_1, …, x\_n is a root of a polynomial of degree at most m\_1 \* … \* m\_n.

Note that, as x\_i is a root of a polynomial of degree m\_i, we have that x\_i^m\_i is equal to a polynomial function of x\_i of degree \< m\_i. By repeatedly invoking this fact, we can in fact reduce any power of x\_i to a polynomial function of x\_i of degree \< m\_i, and thus we can in fact reduce any polynomial function of the various x\_i to one in which each term’s power of each x\_i is \< the corresponding m\_i.

Thus, the space of values which result from polynomial functions of x\_1, …, x\_n is at most m\_1 \* … \* m\_n-dimensional.

Accordingly, if z is some value in this space, the sequence {1, z, z^2, z^3, …, z^(m\_1 \* … \* m\_n)} cannot be linearly independent. And any linear combination of these values which demonstrates linear dependence amounts to a polynomial of degree \<= m\_1 \* … \* m\_n of which z is a root.

This establishes our result for polynomial functions of x\_1, …, x\_n. To generalize to rational functions, observe that such a z as in the last paragraph must, if non-zero, also have a reciprocal in the same space, simply by its algebraicity [take any polynomial of which z is the root, normalize it by dividing out the lowest term so that the lowest term becomes -1, then set this normalized polynomial of z equal to zero (as we know it is), add 1 to both sides, and factor out the z on the polynomial’s side to see z’s reciprocal as a polynomial function of z]. As reciprocation does not leave this space, neither do rational functions, and thus we automatically generalize our result from polynomial to rational functions of x\_1, …, x\_n.

Q.E.D.

In particular, this means sums, differences, products, and quotients of algebraic numbers are algebraic.

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**Author:** ![septimus](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/septimus/32/410_2.png) [@septimus](https://boards.straightdope.com/u/septimus)\
**Post date:** [May 1, 2012, 6:53am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/27 "2012-05-01T06:53:10Z")

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> [@merrick](#):
>
> 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 …

But using the fact of pi’s transcendence to prove irrationality is overkill. Adrien-Marie Legendre proved that π² was irrational in 1794, even writing  
[QUOTE=Legendre]  
It is probable that the number π is not even contained among the algebraic irrationalities, i.e., that it cannot be the root of an algebraic equation with a finite number of terms, whose coefficients are rational. But it seems to be very difficult to prove this strictly. [and indeed it wasn’t proven for almost another century.]  
[/QUOTE]

Googling a bit, I found two pages that claim to summarize Legendre’s 218-year old proof:

[http://www.usenetmessages.com/view.php?c=science&g=453&id=487165&p=0](http://www.usenetmessages.com/view.php?c=science&g=453&id=487165&p=0)  
[http://stephenpi.blogspot.com/2010/05/pi-squared-is-irrational.html](http://stephenpi.blogspot.com/2010/05/pi-squared-is-irrational.html)

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**Author:** ![bienville](https://avatars.discourse-cdn.com/v4/letter/b/4bbf92/32.png) [@bienville](https://boards.straightdope.com/u/bienville)\
**Post date:** [May 1, 2012, 11:18am UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/28 "2012-05-01T11:18:10Z")

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Sicilian

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**Author:** ![Great\_Antibob](https://avatars.discourse-cdn.com/v4/letter/g/e47c2d/32.png) [@Great\_Antibob](https://boards.straightdope.com/u/Great_Antibob)\
**Post date:** [May 1, 2012, 2:19pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/29 "2012-05-01T14:19:14Z")

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Another way of looking at it:

Consider 1/3. The decimal representation is 0.33333…

This decimal representation is infinite. Yet, we can easily multiply it by 2, without worrying about the infinite decimal representation. It ends up being 2/3, or 0.66666…

It doesn’t really matter if we can or cannot multiply 2 by all those 3s in a finite amount of time. We can still find the end result pretty easily without doing all that work.

Likewise, multiplying 1/3 by 1/3 doesn’t require an infinite number of decimal operations, either.

I realize pi is a bit different, since 1/3 is a rational number and pi is transcendental, but it serves as a demonstration that an infinite decimal representation is not a prime consideration for how easy it is to perform arithmetic with a number.

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**Author:** ![Jake](https://avatars.discourse-cdn.com/v4/letter/j/f475e1/32.png) [@Jake](https://boards.straightdope.com/u/Jake)\
**Post date:** [May 1, 2012, 6:11pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/30 "2012-05-01T18:11:35Z")

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> [@Shagnasty](#):
>
> Of course not, all good pi is round.

Pie aren’t squared, pie is round… Cornbread are square!

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**Author:** ![Chessic\_Sense](https://avatars.discourse-cdn.com/v4/letter/c/7c8e57/32.png) [@Chessic\_Sense](https://boards.straightdope.com/u/Chessic_Sense)\
**Post date:** [May 1, 2012, 6:14pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/31 "2012-05-01T18:14:21Z")

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One of the things about math and irrational numbers that humans have a hard time grasping is that they don’t “go on forever”. They’re “_already at_ forever”. So .999… doesn’t “get closer and closer to 1,” as my father once fervently argued; it already _is_ 1. The ellipsis is not a stand-in for a bunch of other digits. It’s its own, independent symbol.

Hopefully this helps you understand why it’s possible to square an irrational number.

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**Author:** ![Jake](https://avatars.discourse-cdn.com/v4/letter/j/f475e1/32.png) [@Jake](https://boards.straightdope.com/u/Jake)\
**Post date:** [May 1, 2012, 6:14pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/32 "2012-05-01T18:14:50Z")

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> [@Chronos](#):
>
> 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.

Er… Thanks for clearing that up. I think.  
Or maybe not.  
Either way you are one of my favorite posters!

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**Author:** ![Thudlow\_Boink](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/thudlow_boink/32/320_2.png) [@Thudlow\_Boink](https://boards.straightdope.com/u/Thudlow_Boink)\
**Post date:** [May 1, 2012, 6:54pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/33 "2012-05-01T18:54:47Z")

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> [@Chessic\_Sense](#):
>
> One of the things about math and irrational numbers that humans have a hard time grasping is that they don’t “go on forever”. They’re “_already at_ forever”.

Yeah, I was tempted to say something similar when **Sheriru** wrote that “pi goes on forever and ever amen.”

The “going on forever” isn’t a property of the number pi itself; it’s a property of the _decimal representation_ of pi. Pi, or the square root of 2, or any other irrational number, can be thought of as a specific point on the real number line, or a specific length/distance (e.g. in the case of pi, the circumference of a circle with diameter 1)—with no “foreverness” about it.

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**Author:** ![Scarlett67](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/scarlett67/32/7387_2.png) [@Scarlett67](https://boards.straightdope.com/u/Scarlett67)\
**Post date:** [May 1, 2012, 7:20pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/34 "2012-05-01T19:20:18Z")

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> [@Jake](#):
>
> Pie aren’t squared, pie is round… Cornbread are square!

Is there an echo in here?

(see post #10)

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**Author:** ![Bytegeist](https://avatars.discourse-cdn.com/v4/letter/b/9dc877/32.png) [@Bytegeist](https://boards.straightdope.com/u/Bytegeist)\
**Post date:** [May 1, 2012, 7:50pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/35 "2012-05-01T19:50:25Z")

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[QUOTE=Indistinguishable]

If the polynomial x |-\> p(x) has pi^2 as a root, then the polynomial x |-\> p(x^2) has pi as a root.

… if x\_1, …, x\_n are roots of [nontrivial] polynomials [with coefficients from the rationals, or whatever base field] of degree m\_1, …, m\_n respectively, then any rational function of x\_1, …, x\_n is a root of a polynomial of degree at most m\_1 \* … \* m\_n.

… Accordingly, if z is some value in this space, the sequence {1, z, z^2, z^3, …, z^(m\_1 \* … \* m\_n)} cannot be linearly independent. And any linear combination of these values which demonstrates linear dependence amounts to a polynomial of degree \<= m\_1 \* … \* m\_n of which z is a root.

[/QUOTE]

We need to get you some LaTeX support on this board.

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**Author:** ![Jake](https://avatars.discourse-cdn.com/v4/letter/j/f475e1/32.png) [@Jake](https://boards.straightdope.com/u/Jake)\
**Post date:** [May 1, 2012, 7:57pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/36 "2012-05-01T19:57:42Z")

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> [@Scarlett67](#):
>
> Is there an echo in here?
> 
> (see post #10)

**NO** no no nope 🙂

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**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\
**Post date:** [May 1, 2012, 8:23pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/37 "2012-05-01T20:23:58Z")

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> [@Bytegeist](#):
>
> We need to get you some LaTeX support on this board.

Well, **Indistinguishable** could have used the sup/sub tags: z[sup]2[/sup] and m[sub]1[/sub]. There [was](http://boards.straightdope.com/sdmb/showthread.php?t=90639) supposed to be a symbol font but it doesn’t work for me. I just copy and paste html symbols from somewhere else. E.g., π, ⊕, and →.

It’s not that we can’t enter basic Math notation, it’s just so freakin’ hard to do.

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**Author:** ![ZenBeam](https://avatars.discourse-cdn.com/v4/letter/z/3ab097/32.png) [@ZenBeam](https://boards.straightdope.com/u/ZenBeam)\
**Post date:** [May 1, 2012, 9:32pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/38 "2012-05-01T21:32:35Z")

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> [@Bytegeist](#):
>
> We need to get you some LaTeX support on this board.

Could this be done with a GreaseMonkey script? That would be useful, although if it were overdone, the threads wouldn’t be accessible to everyone.

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**Author:** ![Nars\_Glinley](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/nars_glinley/32/8177_2.png) [@Nars\_Glinley](https://boards.straightdope.com/u/Nars_Glinley)\
**Post date:** [May 1, 2012, 9:44pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/39 "2012-05-01T21:44:13Z")

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The word you’re looking for is cobbler.

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**Author:** ![Apollyon](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/apollyon/32/4238_2.png) [@Apollyon](https://boards.straightdope.com/u/Apollyon)\
**Post date:** [May 1, 2012, 9:52pm UTC](https://boards.straightdope.com/t/can-you-square-pi-3-1415965-if-so-what-is-pi-squared/620454/40 "2012-05-01T21:52:34Z")

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> [@Sheriru](#):
>
> Sorry that I missed the two. I never really memorized pi. 3.14159265…

Sir, I bear a rhyme excelling  
In mystic verse and magic spelling  
Celestial spirits elucidate  
All my own striving can’t relate

🙂

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