
A Simple Proof That Pi Is Irrational - mgdo
http://fermatslibrary.com/p/607b76ca
======
exacube
I feel really bad when I read stuff like this and don't understand, especially
when it's titled "Simple" (granted I didn't spend more than 5 minutes trying
to following the proof)

~~~
davvolun
Note: Not a mathematician, by any means

I'm not a fan of how the proof is explained, specifically why are we doing x
or y.

I would prefer this --

Let pi be a rational number thus ( * by the definition of rational numbers) pi
= a/b, the quotient of positive integers. We will show no such a and b can
exist, therefore pi cannot be rational.

( * is not completely necessary, since the definition of rationals is so
elementary)

<Next Paragraph, and so on>

I suspect a lot of mathematicians prefer the format given because it is more
obtuse...

~~~
jordigh
> I suspect a lot of mathematicians prefer the format given because it is more
> obtuse...

No, because as you say, defining what a rational number is seems pretty
pointless here, as why would you be reading the proof of something whose
definition you don't even know? And you also want an explanation of what a
proof by contradiction is, which also seems to be way too elementary. Proof by
contradiction is one of the most basic techniques.

Spivak's version of this proof explains it a bit more, but still requires work
from the reader. Any proof does. Mathematics cannot be a spectator sport.

~~~
Natsu
A lot of people struggle at math because they're missing things like that.
With today's technology, there's no reason things can't be broken down to
arbitrarily small steps, so that people can be filled in on the cracks in the
foundations of their understanding of math.

The usual reason for not including everything is that it becomes ridiculously
laborious. With mathematics being abstraction piled atop abstraction, this is
quite reasonable. Just look at how much work it is to prove 2+2=4 --
[http://us.metamath.org/mpegif/mmset.html#trivia](http://us.metamath.org/mpegif/mmset.html#trivia)

But, as you can see from the above, there's technology to simplify things. I
wonder if someday we'll be able to break things down in a friendly way so that
for any piece a student doesn't understand, they can get a proof in terms of
things they do understand?

~~~
chx
Erm, the linked metamath page asks what's the longest path one can take. Well,
geez, it's long. Proving that 2+2=4 from the usual Peano axioms isn't that
hard or long. We define addition by a+0=a,a+S(b)=S(a+b). Now what we mean by a
positive integer number is just a shorthand for that number of successive S().
So 2+2=4 is just S(S(0)) + S(S(0)) = S(S(S(S(0)))). To prove this, let's apply
the second part of the addition to S(S(0)) + S(S(0)) we get S(S(S(0)) + S(0)).
Let's repeat this "move the S() from the second to the outside" and you will
get S(S(S(S(0)) + 0)). Now we can use our first half of definition where +0
can be left out: S(S(S(S(0)))). Now, that wasn't that long, was it?

~~~
lokedhs
I'm not a mathematician, so please bear with me. I'm trying to understand your
definition of addition. Surely that can't be enough? At the very least a+b=0
satisfies your formulas (should I use the word axiom there?).

What I'm trying to understand is how I am supposed to think in order to
prove/disprove my statement above. Disproving something by contradiction is
easy enough, if you can find a contradiction but what if you can't? How can
you be sure that you've covered all variations?

This is something one struggles with as a developer too. There are cases where
you think that a fundamental function does the right thing for all inputs but
then you discover an edge case where that isn't true. The number of cases
where this has been the case even in high profile libraries suggests that I'm
not alone in this.

~~~
gizmo686
What is missing from chx's description is the definition of the natural
numbers (and the S (successor) function). The standard definition is:

    
    
        0 is a natural number
        S(n) is a natural number, if and only if n is a natural number
        x=y if and only if S(x)=S(y)
        for all x, S(x)!=0
    

These axioms are sufficient to show that the natural numbers form a line.

At this point, we can use the definition of addition that chx provided:

    
    
        a+0=a
        a+S(b)=S(a+b)
    

It is true that, without the restrictions on S, than defining a+b=0 would
satisfy these. However, because of the restrictions on S from our definition
of the natural numbers, it is impossible for S(a+b)=0.

------
jordigh
Oh, I get it. "Fermat's library," because we're writing on the margin, which,
for once, is large enough to contain our marginalia.

Cute.

Btw, this is the same proof that is in Spivak's Calculus, but Niven explains
it a little less than Spivak does.

~~~
semi-extrinsic
Since the linked proof was published in a mathematical journal [Bull. Amer.
Math. Soc. Volume 53, Number 6 (1947), 509], I think it's safe to assume that
Niven was first.

------
throwaway1967
Socrates (in Meno) helped a slave kid prove to himself that the square root of
2 is irrational, using nothing but a stick and the sand on the ground.

I don't think that kid would understand this "simple proof".

------
hebdo
Neat! I remember that presenting (from memory) a slightly longer proof that pi
is irrational was the final question on my Analysis oral exam.

It is surprising how many people know that pi is not rational, yet how non
obvious is the proof. Like the first one ever presented. This one is also a
nice piece of cake, pulls this polynomial f(x) from a hat.

------
tzs
If the annotations are not working right when you expand them, particularly
the second one, and you are using Safari, try reading in Chrome or Firefox.
Those appear to work fine.

In Safari, it looks like when the height of the expanded annotation display
region is calculated for dealing with scrolling, it is including the area
under the sign up doohickey. It thinks you can see more than you can. This
means the scrolling limits are set too small, and so you cannot scroll the
bottom of the annotation up into view.

If your window size and text size make it so that the second annotation fits
in the visible region, this won't matter.

------
hasenj
Somehow, the "original proof" seems simpler.

> The first rigorous proof that π is irrational is from Johann Heinrich
> Lambert in 1761. He proved that if x≠0 is rational, then tan(x) must be
> irrational. Since tan(π/4)=1 is rational, then π must be irrational.

Of course, the proof for (tan(x) is irrational for rational x != 0) might be
complicated, but at least it's easy to see why this proves that PI is
irrational.

This one, on the other hand, I can't even follow in principle what it's trying
to say.

That's of course because I don't understand integrals and differentials.

~~~
jordigh
That just wraps up a mystery inside another. Why would tan(x) be irrational if
x is rational? That seems like an even bigger mystery than why π is
irrational.

~~~
dnautics
Thats easy if you remember arctan(y/x) is the angle from x axis to (x,y)

~~~
hebdo
I fail to see how this helps. Actual proof is more complicated, it is based on
continued fractions.

------
pdpi
It's worth noting that there's a number of annotations added that make the
proof much easier to follow for someone whose calculus is not quite as...
hum... fresh as the proof expects.

------
amelius
It would be interesting to see this as a formal proof, and to compare the
complexity and lengths of both approaches.

~~~
jordigh
Most mathematicians would consider this a formal proof. Do you mean a
computer-checkable proof?

~~~
amelius
Of course :)

(Natural language is not a formal language.)

------
izzydata
Every time I tried studying from a math textbook back in school I got the
impression that examples and explanations were purposefully obscure to make it
look more sophisticated. Then a friend comes along and explains it in 10
seconds and it makes perfect sense.

------
selestify
Finally, no more "The proof is left as an exercise for the reader."

------
jackmaney
I started reading it, and then there was a nagging interstitial, begging for
my email address so that I could be sent spam.

Closing that garbage, various buttons and unnecessary margin bits ("Click here
to see more!" [ _bounce_ ] [ _bounce_ ] [ _bounce_ ]...) kept jittering for my
attention.

So, I closed the tab. I have no idea who is responsible for the design of
fermatslibrary.com, but they should feel ashamed of what they've done. This is
one of the most infuriatingly infantile designs I've ever seen. The old "punch
the monkey!" ads have nothing on this.

~~~
selestify
Weird, I didn't experience any of that when I visited it.

~~~
jackmaney
I don't know what to tell you, except that when I viewed the page, there was a
reasonably well-formatted PDF in the center of the screen surrounded by
unnecessary, fiddly, bouncing bullshit "click on me!" buttons.

~~~
mgdo
I think these are simply the paper annotations. It's not that obtrusive.

~~~
jackmaney
There are sites that allow for annotations and manage to do so without
bouncing, distracting "click on me!" buttons. Medium does a great job of this.
I have no idea what the site's creator was thinking when they decided not to
emulate the established, non-distracting examples instead of doing what they
did.

------
thams
Love this, want more papers though, there are a few computer science papers
that I'm going to suggest

------
Houshalter
This isn't simple at all. Is there an explanation anywhere of this proof?

------
wolfgke
Now I wish a simple proof that pi is transcendental (i.e. not algebraic).

------
phkahler
This doesn't work. The term (a-bx) is zero since bx = a. That turns the whole
equation into a big fat zero.

~~~
omaranto
(a-bx) is not zero, x is a variable. when x=pi, a-bx is zero, otherwise not.
a-bx=0 is an equation whose solution is x=pi. The function f(x) is not
identically zero, it is however zero when evaluated at x=pi: f(pi)=0.

~~~
phkahler
And he does talk about f(pi) quite a bit. It's not like there are limits being
discussed where we should consider something in the neighborhood of a 0/0
situation. He evaluates it at pi, and in doing so could just put a zero in
there.

------
itistoday2
Commenting requires signing up for Facebook or G+? That's a shame, would have
liked to use this site.

