π²
I feel I should chime in here, but I have nothing to say.
i thought all pies are squared, at least that’s how it is in my circles.
You can actually go to Google and type:
pi^2
and get your answer.
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.
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
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://stephenpi.blogspot.com/2010/05/pi-squared-is-irrational.html
Sicilian
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.
Pie aren’t squared, pie is round… Cornbread are square!
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.
Er… Thanks for clearing that up. I think.
Or maybe not.
Either way you are one of my favorite posters!
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.
Is there an echo in here?
(see post #10)
We need to get you some LaTeX support on this board.
NO no no nope 
Well, Indistinguishable could have used the sup/sub tags: z[sup]2[/sup] and m[sub]1[/sub]. There was 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.
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.
The word you’re looking for is cobbler.
Sir, I bear a rhyme excelling
In mystic verse and magic spelling
Celestial spirits elucidate
All my own striving can’t relate