[alancalverd (OP)]

One for the mathematicians! Please provide an explanation suitable for whatever age we introduce π into school mathematics.

[Eternal Student]

Hi.

   At school level you just wouldn't prove that π is irrational.   The standard proofs are either complicated and require too much prior knowledge or else just exceedingly laborious.  You run the significant risk of scaring the children and/or boring them to death.  This is the opposite of what you normally want to be doing as a maths teacher, you're supposed to let them see how much fun and how interesting mathematics can be.  It is much quicker to show them that √2  is irrational and this establishes that at least some irrational numbers exist.

   If you must have a go at proving that π is irrational, see this YouTube video:     "Pi is IRRATIONAL: animation of a gorgeous proof"  by Mathologer.   

   That video takes almost half-an-hour and the presenter was using plenty of pre-prepared animations to do this.  There are some bits that are rushed and you would want to stop the video and grab your own pencil and paper.   If you had a sixth form class (age 16+) and a 1 hour lesson, then you might be able to show the video and fill out the bits that were overly forced to get close enough to a complete formal proof.

Best Wishes.

[Origin]

I don't really understand the question.  I mean is there even a 'why' to ask about for pi being irrational?  I am definitely not a mathematician... 

edit:  Cross posted with ES.  (I am not going to watch a half hour video on pi!)
So I guess the question is what is the proof that pi is irrational as opposed to why pi is irrational.

[alancalverd (OP)]

In a formal mathematical proof, there's no difference: you begin with stated axioms and demonstrate the consequence of those axioms: "why" is "because it follows inevitably from our definitions". 

I have an idea for demonstrating "why" based on Pythagoras but it needs a bit of fleshing out.

[Petrochemicals]

Because it's never ending, basically a histerical number you can't get any sense out of.

[Eternal Student]

Hi all,

   It is stated in the You Tube video that many professional mathematicians have NOT seen a proof that Pi is irrational.  Just to make it clear, they know that it is irrational but haven't bothered to study the actual proof.  I would agree with this.  Even if you are going to specialise in Number Theory, it is the parent technique which inspired this proof (Euler's techniques using continued fractions) that will be more useful and transferable to other situations.

I have an idea for demonstrating "why" based on Pythagoras but it needs a bit of fleshing out.

  We look forward to seeing it.   

Best Wishes.

[Bored chemist]

I don't know if that video answered the OP's question, but it's an interesting proof.

[Eternal Student]

Hi Bored_Chemist,
  Thanks for your comment.  The video seemed suitable for a school level presentation, that's the only reason I suggested it.
   Alancalverd was the OP.  I'm guessing as to what the OP was hoping to find.  The question in the title is "Why is Pi irrational?",   the basic answer is   "because it's a Real number but it cannot be rational - so by definition it is irrational".
   The broader context of the OP's question might be  "why is it important to know that Pi is irrational at school level?".   There are many good reasons for this:

1.  It might be used as question.  For example, "Show that the decimal expansion of Pi does not terminate or repeat".

2.  It has some real world significance:  Explain why your calculator cannot determine the exact area of a circle with a radius of  1cm.

3.  Philosophical and historical interest:    Why did people think that the formula   e^iπ = -1   proved the existence of god?

4.  It's just interesting.  The number falls out quite naturally as the ratio of a circumference to the diameter, it's not artificial or deliberately constructed to be irrational.  It forces everyone to recognise that irrational numbers are unavoidable if you want to have geometry available.  However, the √2 and Pythagoras' theorem is usually an easier way into that.

Best Wishes.

[Bored chemist]

It's interesting to think about introducing a proof of the irrationality of pi for fairly young school children- I seem to think I would have been about 8 when I first learned about the diameter, circumference and areas of circles.

School maths isn't about proof.
I gather that it's typically some time in the second year of a university degree that mathematicians actually prove that 1+1=2.

The only proofs I recall doing at school were the formulae for sums of series, though my memory may be letting me down.

We also did some derivations of results in calculus.

[Eternal Student]

Hi again.

School maths isn't about proof.

   Agreed.  Overwhelming it is NOT about formal mathematical proof, it is about solving little problems here and there with some Mathematics. 
   However, just to keep things balanced we should mention that it does now feature on the newer GCSE Mathematics syllabuses and again on the A level syllabuses.  Just to be clear - children do NOT prove that Pi is irrational because that is complicated and tedious.  It's just that they should be introduced to the idea of "mathematical proof" somewhere and with some example(s).   
   
Speaking from personal bias:    "Proof" is a fundamental part of Mathematics.  Any good teacher should have been introducing the idea to their pupils since forever.  It doesn't matter whether it was on the syllabus or not.  It is an essential part of Mathematics but you balance this with keeping things interesting and not too frightening.

I gather that it's typically some time in the second year of a university degree that mathematicians actually prove that 1+1=2.

    I would say that in the first year of a Mathematics degree (in the UK) people will study algebraic structures like Rings and Fields at which point they would believe they can establish 1+1 =2 from quite elementary axioms.
    Like many things, you can always go one level deeper.  This may happen if the student chooses some more set theory in later years but Applied Mathematicians may be quite happy without ever knowing about Peano's axioms or abstract set theory.

Best Wishes.

[Bored chemist]

  I would say that in the first year of a Mathematics degree (in the UK) people will study algebraic structures like Rings and Fields at which point they would believe they can establish 1+1 =2 from quite elementary axioms.

The irony of my guess being 2 rather than 1 in this context is remarkable.
:-)

It doesn't matter whether it was on the syllabus or not.  It is an essential part of Mathematics but you balance this with keeping things interesting and not too frightening.

I think "useful" might figure there somewhere too.
I might actually need to know how many cans of paint it takes to cover a circular patio; but I will never need to prove that the area is pi r squared.

On the other hand, if I don't know that proof, (or lots of others), I miss out on the beauty of maths.

[alancalverd (OP)]

because ........ it cannot be rational

is called Proof by Assertion - definitely not tolerated in this forum!

I recall a teacher saying "Please accept for the moment that everything I say three times is true - but I promise we'll challenge it next year." Whilst P by A is intolerable, I find Teaching by Cliffhanger admirable!

My Pythagorean idea is to look at a segment of a circle with an inscribed (hypotenuse = r) and an escribed (adjacent = r) right-angled triangle. Now we can calculate the opposite  in either case as o = √(h² - a²) or, if you like, r sin θ or
r tan θ, but it is clear that the arc length s always lies somewhere between those two values.

So whatever value we choose for θ, we can't calculate s exactly as a multiple of r. But we have defined  s = πrθ, so we cannot write down an exact value for π.

[evan_au]

It is much quicker to show them that √2  is irrational

First you have to identify what defines an irrational number: "irrational numbers cannot be expressed as the ratio of two integers.". That immediately restricts the discussion to school kids who are comfortable with dealing with fractions, finding lowest common denominators, etc.
https://en.wikipedia.org/wiki/Irrational_number

The Pythagoreans knew of one proof that √2 is irrational - and apparently executed one of their members for revealing this disturbing little secret (an anecdote to spice up the lesson!). Today we would talk of the numerator and denominator being odd and even, rather than male or female.

formulae for sums of series

That may be one way in... We know (but school students will need to take it on faith until they get to Taylor series in university) that

π/4 = 1 -1/3 + 1/5 -1/7 +....

You can convert the first n terms to a fraction, but the next term will cause the denominator to get larger every time,  and you never get to the end of this process, so it can't be represented as a fraction.

Of course, some bright student may point of that 1 + 1/2 + 1/4 +1/8... = 2, which is another infinite series which has a rational answer

For younger students, an experimental approach may be more successful: As far as we can tell, the decimal expansion of Pi contains every possible sequence of digits. So you can search to find your name in the digits Pi:
https://www.atractor.pt/mat/fromPI/PIalphasearch-_en.html
This latter approach requires students who are familiar with decimal notation, and a bit of hand-waving about how they represent decimal digits as letters...

[Eternal Student]

Hi.  I hope everyone is well.

My Pythagorean idea is to look at a segment of a circle with an inscribed (hypotenuse = r) and an escribed (adjacent = r) right-angled triangle.

   Can we have a diagram or sketch?   I'm not sure what you're describing here.

@evan_au     You answered yourself perfectly well. 
- - - - - - - - - -
On a minor note, did you (anyone) know that Pi is also a transcendental number?  Just thought I'd mention it.  Someone probably cares and I don't have any baseball facts to share.   Anyway, the ancient Greeks with their passion for Geometry stumbled on a few properties of numbers that they were really going to hate and refuse to believe.
 
Best Wishes.

[alancalverd (OP)]

I think this is what I had in mind - but it was very late and after a good dinner!

[Eternal Student]

Hi.
 
  Thanks Alanacalverd.  You didn't have to produce a diagram and I appreciate the time it must have taken.  That is interesting.

   I've looked at your diagram but I can't see why the curved red line  (the arc) can't be longer than all of the blue straight lines that hold it prisoner inside a trapezium.   Don't get me wrong, it is and I could prove it by drawing diagrams of trigonometric functions and finding gradients (derivatives) but that seems to be asking a lot from children.  Just glancing at the diagram, one thing is a curve and the other thing is a straight line and nothing comes to mind which prevents the curve being longer than that straight lines.

But we have defined  arc length s = πrθ,

   This could have been a typing error:      s =  rθ     with θ in radians,  so there's no  Pi  that appears in it.
You could use degrees  and nothing prevents you from defining your own angle measure if you want to.   However, the last part of your argument seems a bit woolly anyway.

The last part of your argument seems to be that  both straight lines take irrational values and our number Pi will always be sandwiched between these two values:         Irrational < Pi < irrational       is that right?
   An interesting feature of the Real numbers is that although the Rationals are only a small subset of them, they are DENSE in the Reals.    Between any two distinct irrational numbers we can always find a rational number.  So the number Pi could always have been a rational number.

Best Wishes.

[Eternal Student]

   Anyway..... the main thing is that you (Alanclaverd) and we (everyone else) thought about proving that Pi is irrational,  which is obviously a good thing.  I'm sure our lives will be enriched by the experience.

Best Wishes.

[alancalverd (OP)]

I apologise for screwing up the punch line of my proof! It was very late and I'd had at least half a bottle of dinner after a long day's work and 250 miles of British traffic.

Tonight, after just a gig and a pint, maybe I can be a bit more rational. What I was trying to get at is that however small we make θ, if the uprights are rational fractions of r, s lies between them and must therefore be something other than a rational fraction of r. If we can't describe even the tiniest bit of a circle as a rational fraction of its radius, then the whole circumference can't be a rational multiple of r. 

Okay, this stinks a bit because in another thread I pointed out the dangers of computer simulations in geometry, which can lead kids to thinking that a circle is a large number (usually 360) of straight lines. But at least it doesn't involve 30 minutes of mathematical elegance for the cognoscenti only.
     

[Eternal Student]

Hi Alancalverd,

   You've done well.  Not the drinking but the maths.  I'm not going to point out a minor problem today, it's just too late.  Anyway, it is just a minor issue and it's the effort that counts.   (You have no idea how difficult it is to let a piece of corrupt Mathematics go without a word about it - but there we go).

Thanks and best wishes to you.

[alancalverd (OP)]

Another diagram, another thought, which I think addresses your point that the length of the arc in my previous effort is not demonstrably between those of the two uprights.

I can approximate to the perimeter by inscribing lots of chords, or escribing lots of tangents.

We have defined (or at least Euclid did) the straight line as the shortest distance between two points, so the arc (which is not straight) must be longer than the chord. However the locus of the arc is closer to the chord than the "two tangents" construction, so it must be shorter than the locus of the escribed figure.

Therefore however closely we approximate the circumference with rational fractions of r, the arc will lie between them    and cannot be a rational fraction of r.

It's still untidy and I'm happy to have the nits picked off it, but it might just convince a schoolkid. And if it doesn't, then we will have set another skeptical mathematician on her career path, which is no bad thing.