Post by: alancalverd on 01/09/2021 14:25:18
Post by: Eternal Student on 01/09/2021 17:14:02
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.
Post by: Origin on 01/09/2021 17:21:24
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.
Post by: alancalverd on 01/09/2021 18:06:41
I have an idea for demonstrating "why" based on Pythagoras but it needs a bit of fleshing out.
Post by: Petrochemicals on 01/09/2021 19:03:32
Post by: Eternal Student on 01/09/2021 22:06:08
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.
Post by: Bored chemist on 01/09/2021 22:54:57
Post by: Eternal Student on 02/09/2021 00:15:37
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 eiπ = -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.
Post by: Bored chemist on 02/09/2021 07:32:29
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.
Post by: Eternal Student on 02/09/2021 19:43:49
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.
Post by: Bored chemist on 02/09/2021 21:37:22
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.
Post by: alancalverd on 02/09/2021 23:34:47
because ........ it cannot be rationalis 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.
Post by: evan_au on 02/09/2021 23:47:45
Quote from: Eternal Student
It is much quicker to show them that √2 is irrationalFirst 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.
Quote from: Bored Chemist
formulae for sums of seriesThat 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...
Post by: Eternal Student on 03/09/2021 10:19:48
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.
Post by: alancalverd on 03/09/2021 11:25:10
Post by: Eternal Student on 03/09/2021 15:41:13
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.
Post by: Eternal Student on 03/09/2021 16:55:14
Best Wishes.
Post by: alancalverd on 04/09/2021 00:13:36
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.
Post by: Eternal Student on 04/09/2021 03:23:08
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.
Post by: alancalverd on 05/09/2021 00:39:51
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.
Post by: Eternal Student on 05/09/2021 03:35:46
I'm happy to have the nits picked off it, but it might just convince a schoolkid.OK, let's look at two issues:
1. Vague approximations to straight lines by reference to locus:
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.I can draw a line which is a dense series of zig-zags (or tight curves if I want a smooth line). If I put enough zig-zags or tight turns in then I can get it to any length I want. I can do this while keeping the locus of the line "close" to the chord. It can all be kept inside of the escribed figure on your diagram but still be longer than the escribed tangents.
2. Sadly, we can always find a rational between two rationals:
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.which is presumably based on something you said in an earlier post:
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.I think what you're trying to say is that there aren't any proper fractions between
and 
because there aren't any whole numbers between 4 and 5. Is that right?After dinner I can understand that you might think this is true. However, in the morning you might recognise that the fraction
is a perfectly good proper fraction and it does lie between and .The key is that we don't have to keep the same denominator, we have an infinite set of choices.
Theorem: If A and B are rational with A < B then there always exists another rational C which is between them ( we have A < C < B).
Proof:
Let C =
.It should be apparent that C is the mid-point between A and B and so we will have that A < C < B as required.
We need only show that C is rational.
A is rational and B is rational, so A+B is rational (we can add fractions and we always produce a fraction).
Then
is also rational. (dividing by 2 is just multiplying by and we will produce a fraction)Corollary
If s (the arc length) lies between two rational fractions of r (the radius), then, I'm sorry to say, s can be a rational fraction of r.
Proof
We have Ar < s < Br where A,B are rational
Find C as above, C =
.Then A < C < B => Ar < Cr < Br
Hence Cr is a perfectly good candidate for s. It is a rational fraction of r and it is between Ar and Br.
Best Wishes.
Post by: alancalverd on 05/09/2021 13:16:03
Vague approximations to straight lines by reference to locus:But I didn't approximate a straight line by zigzags! My construction was straight lines and a continuous curve.
Quote from: alancalverd on Today at 00:39:51
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.
I can draw a line which is a dense series of zig-zags (or tight curves if I want a smooth line). If I put enough zig-zags or tight turns in then I can get it to any length I want. I can do this while keeping the locus of the line "close" to the chord. It can all be kept inside of the escribed figure on your diagram but still be longer than the escribed tangents.
It is obvious that the curve is longer than the inscribed line, by Euclid's definition.
It is also clear that the escribed zigzag will be longer than the curve because inside every step in the staircase you can draw tangents to the curve, each of which is closer to the curve.
So what we are considering now is the number of infinities in an interval. We can define an irrational number as one that sits between two rationals, however small the difference. So the number of irrationals is a larger infinity than the number of rationals, which is interesting, but it also suggests that whilst your zigzag may converge towards the curve, since it cannot match the curve, the defining equation for the curve involves an irrational.
And now another interesting question rears its head! There being more irrationals than rationals, why are so few of them important? Apart from e, π, √2 and an occasional visit from √3, which dominate my working hours, I can't think of another that has swum into the ken of this ancient physicist in the last 60 years!
Post by: Eternal Student on 05/09/2021 16:05:40
And now another interesting question rears its head! There being more irrationals than rationals, why are so few of them important? Apart from e, π, √2 and an occasional visit from √3, which dominate my working hours, I can't think of another that has swum into the ken of this ancient physicist in the last 60 years!
This is probably because we don't tend to use them as an input. For example, if you were making an aeroplane wing you probably wouldn't make it or measure it to be √2 metres long, you would have it specified to a few decimal places. Anyway, after that the rationals are closed under arithmetic manipulation involving other rationals (so that'll be any polynomial expression). So your ouptut will also be rational.
In reality, an aeroplane wing is overwhelming more likely to be an irrational number of metres than a rational number of them - but tape measures aren't callibrated finely enough to measure irrational lengths. That's something I'm going to invent - a tape measure that shows all the irrational numbers. It shouldn't be too difficult.
Best Wishes.
Post by: alancalverd on 06/09/2021 16:57:29
tape measures aren't callibrated finely enough to measure irrational lengths.Oh yes they are! If I use lathe and a calibrated ruler to make a 0.5m radius wheel, I can use it to measure exactly π meters of cloth!
Post by: Bored chemist on 06/09/2021 17:27:46
This is probably because we don't tend to use them as an input. For example, if you were making an aeroplane wing you probably wouldn't make it or measure it to be √2 metres long,If you were specifying a loudspeaker cabinet, you might (and √3 metres high).
That way the simple resonances of the sound bouncing round will not coincide.
Post by: Halc on 06/09/2021 18:28:25
In reality, an aeroplane wing is overwhelming more likely to be an irrational number of metres than a rational number of them - but tape measures aren't callibrated finely enough to measure irrational lengths.In reality, no object is any particular length at all. It is impossible to measure both ends at once, and the length (distance between the two most separated particles) is constantly changing.
Tape measures aren't calibrated finely enough to measure infinite precision, and at any finite precision, both rational and irrational numbers lie withing the margin of error.
In other words, if it were true that tape measures aren't calibrated finely enough to measure irrational lengths, then you could prove that a rod was a rational length because you used a tape measure to measure it.
Perhaps it would be better worded as 'enough to distinguish' one from the other, but then we're back to my first point, especially since the tool used to do the measuring is continuously changing dimensions as much as is the thing being measured.
Post by: Bored chemist on 06/09/2021 18:45:14
In reality, no object is any particular length at all.For a while, this one was.
https://en.wikipedia.org/wiki/Metre#/media/File:US_National_Length_Meter.JPG
Post by: Eternal Student on 07/09/2021 00:42:31
@Halc
OK.
@Bored chemist
Using the old standard for length was quite clever.
- - - - - -
Hopefully, everyone knows the tape measure with markings to show all irrational fractions of a metre is not something I seriously intended to produce.
Best Wishes.
Post by: alancalverd on 07/09/2021 10:27:45
Quote
One of the oldest yard-rods in existence is the clothyard of the Worshipful Company of Merchant Taylors. It consists of a hexagonal iron rod 5⁄8 inch in diameter and 1⁄100 inch short of a yard, encased within a silver rod bearing the hallmark 1445.....The US survey yard is very slightly longer....Probably a relativistic effect due to the rotation of he earth.
.But horologists have attempted at least two irrational and one completely pointless measures.....
Post by: Bored chemist on 07/09/2021 11:48:24
Hopefully, everyone knows the tape measure with markings to show all irrational fractions of a metre is not something I seriously intended to produce.Indeed, just including 1 in 10 would be enough
Post by: Bored chemist on 07/09/2021 11:50:18
Using the old standard for length was quite clever.I was reminded of a quote from a guy who worked at the BIPM something like:
"If I scratch the prototype kilogram and remove some of the metal, it is still a kilogram and everything in the universe suddenly gains mass.".
Post by: evan_au on 07/09/2021 11:57:28
Quote from: alancalverd
There being more irrationals than rationals, why are so few of them important? Apart from e, π, √2 and an occasional visit from √3I think we are visited by a lot of irrationals, but the ones you mentioned are the ones that keep coming back.
Other irrational numbers you may encounter are:
- when working in radians, π/2 and π/4 crop up fairly often (when working in degrees, these are nice round numbers)
- Trig functions (eg sin, etc) of nearly any angle: There are some angles where sin is rational, but most are irrational
- Probability functions (eg erf ): How many standard deviations do you need for 99% confidence on a Normal distribution?
- Anything to do with friction - even the pure mathematics solutions include log functions
Post by: alancalverd on 07/09/2021 12:27:01
The Director of the UK National Physical Laboratory used to welcome new recruits with "How long is a piece of string? Legally, it is as long as I say it is. Your job is to tell me what to say."Using the old standard for length was quite clever.I was reminded of a quote from a guy who worked at the BIPM something like:
"If I scratch the prototype kilogram and remove some of the metal, it is still a kilogram and everything in the universe suddenly gains mass.".