Is the rationality of Pi known
Debate Rounds (4)
Please note: I have not studied maths beyond high school
Let it begin
I hope that you don't mind my using Round 1 to get started; if you're not happy, just ask and I will forfeit the last round in lieu.
It strikes me as an almost done deal that pi is known to be irrational. I say almost because I can see a very, very tiny room for debate about what the word "know" means.
Many independent proofs of the irrationality of pi have been provided over the years that mathematicians have been studying it; these proofs have all been subject to scrutiny from other mathematicians in a process of peer review... I would say that one can trust any one of these proofs with almost complete certainty; when there are so many of them, even if they are too complex for your current understanding of maths to confirm for yourself, that almost complete certainty approaches perfect certainty as close as I feel merits the description that we know it's irrational. I think that in order for you to argue that we don't know, you would have to tighten the definition of knowledge so far as to become a solipsist. Either that or you would have to suspect a conspiracy of mathematicians from all over the World who are all lying about their professional opinion that the matter is proven beyond contention; this seems unlikely in the extreme because I cannot fathom what their motive could possibly be!
But it goes further than that... pi is one of the rare examples of a known transcendental number; all transcendental numbers are irrational and many proofs have been given that pi is transcendental. To question pi's irrationality is to question it's transcendence. It seems to me quite mean to wish to strip everybody's favourite mathematical constant of it's special properties!
I've had a look at some proofs and have to concede that most are too complicated for my current level of mathematics for me to be able to confirm them easily myself, although I am most of the way working through Niven's and will have confirmed it to my own satisfaction soon.
I'd like to draw your attention to the fact that we have calculated pi to literally trillions of decimal places; nobody has yet spotted a pattern in that sequence of digits (but not for lack of trying!). And that leads me onto the point which I'd like to finish on: with the irrationality of pi being a mathematically accepted fact, imagine the kudos that would be awarded to anybody who could show otherwise! I am personally totally convinced (and will be more so when I get to the end of Niven's proof) that pi cannot be expressed as a fraction... but would not the failed attempts of thousands of years of trying also count as extremely suggestive?
Finally, I'd like to say thank you for bringing up such an interesting question that has to a degree challenged my own assumptions, trust, knowledge and confidence in the field of mathematics!
Over to you...
I have watched https://www.youtube.com... which is Ivan Niven's proof but even this seems too complicated for me. However if it still makes sense after I have made my points then I will trust you on this.
The thousands of years people have been trying to work out pi is partly why I believe it's rationality is unknown, and when people say that pi's trillionth decimal place has been calculated it doesn't make me think we are any closer in finding out whether Pi is rational or irrational.
I have worked out how to estimate pi without string by "straightening" the circumference so it can be measured with a ruler but if you draw an ever smaller section of an ever larger circle it will never completely straighten the circumference.
Pi is most commonly calculated by using Archimedes method with polygons . A value which is called "Pi" can be produced to an infinite number of decimal places since you can place any sided polygon inside a circle, and obviously there are an infinite number of numbers.
Pi = circumference/diameter
A circle with 1 side is not the same as a polygon with infinite sides even if it appears to our eyes it is. What we call "Pi" is actually equal to an approximation of the circumference divided by diameter OR is the exact circumference divided by an approximation of the diameter. This makes "Pi" or 3.14.... an accurate approximation, nothing else, even if we calculate Pi to infinity, at least in my opinion anyway.
If the true value of Pi is unknown then how can anyone know whether it is rational or irrational?
I look forward to hearing your reply.
I totally understand your correct position that "it doesn't matter how many trillion digits you determine, you haven't demonstrated irrationality". I offered the idea only as suggestive evidence, not as a serious route to demonstrating rigorous proof.
Reading between the lines in what you've written about polygons approximating circles, I conclude that you have deep suspicions about infinite series. I understand this; Infinities and how to handle them have long been hotly contested by professional mathematicians... there is now almost universal consensus, though, and in cases such as the determination of the rationality of pi I doubt that you'll find a single professional mathematician in the World who would admit to even the slightest doubt. I myself, not being a professional mathematician, admit to areas of personal doubt in mathematics (such as Cantor's Diagonal Argument) where I have to deal with the cognitive dissonance of trusting the mathematical consensus whilst harbouring secret reservations. As it happens, I suspect that the handling of infinite series is exactly the nub of your doubt; I've spent a long time soul-searching the questions that arise around this topic and getting comfortable with some infinite series, so perhaps I am in a position of being able to convince you of those things which I have satisfied myself about; I'm gonna try, at least!
First up, I'm going to point out something that might seem, at first glance, to throw into question how much we can trust decimal expansions. It is simply this:
0.999... is strictly equal to 1
Now, that's not approximately equal to... that is identical to. Fortunately, since both representations (0.999... and 1.000...) are infinitely recurring decimal expansions, they share the property of being rational... but... what would happen if there were such an equality in which one side was recurring and the other was not? Is such a thing possible? Well, how about we go the other way and consider the possibility of a number that differs from 1.000... by being just one digit larger in the "last" digit? Such a thing, were it possible, would clearly not be a recurring decimal expansion and it would be as close to 1.000... as is 0.999... (literally no difference!) - but, since the focus here is working out what ratio we could use to arrive at our desired number, we can establish that we would need the ratio of (infinity plus one) to (infinity) to produce this number... and since (infinity plus one) is equal to (infinity), we simply get 1... or, you could just say "you can't have infinity plus one" and let the trail go cold. I cannot offer you a rigorous proof that you'll easily grasp at this point, but I hope that you can accept in your heart and mind that constructing an equality with a recurring decimal and a non-recurring decimal is impossible. This is, I contend, one way of looking at the concept that rationality and irrationality are "real" concepts! This is just one way of looking at the limit of the definition of what a recurring decimal *is* (since we can never list all of the digits of any number).
The take home idea that I'd like you to consider from this example is loosely expressible as "the infinitieth digit really doesn't matter".
Now, I'd like to take a little time to explore how one can convert any recurring decimal expansion into a ratio. I suspect you may have seen the method, but here it is being used to work out what 0.345345345... is as a ratio (I honestly just made that number up off the top of my head):
x = 0.345345345...
1000x = 345.345345345...
999x = 345
x = 345/999
x = 115/333
Looking at that third step, we are asked to accept that when we shift the digits three to the left and subtract, *all* of the remaining expansion of the decimal cancels out... even the final three digits, which intuitively don't exist in the 1000x line. I'm hoping that you accept the validity of the logic despite the slightly jarring counter-intuitiveness. Our intuition is sometimes wrong!
The take-home idea that I'd like you to consider from this example is loosely expressible as "any arbitrarily large number of digits from the extreme end of a decimal expansion really don't matter"!
Now, if the final digit, or indeed the final million digits don't matter, why is that? It doesn't sit well with our intuition! Simply put, though, we can say that differing by an infinitely small amount IS the same thing as being equal. This is not just a "for all practicable purposes" argument... this is a statement of absolute equality. I'd be interested to know where you stand at this point in my exposition!
Now to extend into slightly more abstract infinite cancellations...
Consider S = 1 + 1/2 + 1/4 + 1/8 ... ad infinitum
Then S/2 = 1/2 + 1/4 + 1/8 ... ad infinitum
Therefore S - S/2 = 1
Therefore S = 2
Hopefully you are still comfortable that we are on reasonable grounds. There is danger here, to be noted... this is not necessarily going to hold if the terms of our infinite sequence are increasing in magnitude!
I'm going to end this round, soon, a little lamely (I'm running out of time this evening and characters to use, and I'd like to know where you stand before carrying on any further).
You alluded to polygonal approximations of pi... we can construct a triangle, square, pentagon, hexagon, heptagon, etc, of unit "radius" and get closer and closer approximations of the area of the circle (and, thence, pi) by this method. Since this is an infinite series in which successive terms strictly decrease, we can in theory get a good handle on what the devil pi is!
In the light of what you said about multifaceted polygons approximating but never equalling circles, I'd point out that the nature of talking about series that approach the limits of infinity are such that, in fact, an infinitely-sided polygon is identical to a circle... and that's the very nature of calculus in a nutshell... put another way, there is no more than an infinitesimal difference between the two, which should be sufficient. But, even if you are not happy with the accuracy of the physical model in Euclidian space, I hope that I have persuaded you of the truth contained in the infinite algebraic expansions... essentially, I appeal to your reason to accept that a convergent series that approaches the answer we desire at infinity is not just "sufficient for practical purposes" but is, in fact, absolutely, completely, truthfully, mathematically accurately the right answer... and that this level of precision is absolutely sufficient for determining whether a decimal expansion is recurring and whether a given constant is, in fact, irrational.
Certainly I have not tried to outline a proof of pi's irrationality here; all I have tried to do is to lift Con's apparent mental block towards the possibility that mathematics has the authority to even talk to the rationality of pi.
Depending on Con's response to these musings, I may continue on this path or I may return to trying to win the debate for the next round : )
I accept the number calculated using Archemede's method known as Pi is an irrational number, because not seeing any numbers repeating in a sequence in one trillion digits is good enough evidence as any.
You have hit the nail on the head about what I disagree on i.e. infinity, and having read your arguments I will accept that if 0.9999... equals 1 then Pi has been calculated exactly, and is "known" even if we can't ever write down all of the digits.
I didn't realize there were so many ways to show 0.999.. equals 1, see below
The simplest is: 0.1111... = 1/9
multiply both sides by 9
0.9999... = 1
Another is: S = 0.999...
Then 10S = 9.999...
10S - S = 9.000...
S = 1
And then there is your method too, which is equally good.
However I believe the decimal system is made more for practical purposes
Let's look at the first one, my instinct is not to type 0.999... when I multiply 0.111... by 9. I'd simply put 1.
It just seems like a clever trick to show 0.999 and 1 are the same thing, when really when we say 0.111... is 1/9 it is because this is the best way to show what one ninth is in a decimal format.
You say there is no number between 0.999... and 1, however I believe there is. This number is 0.00000.....1 which is basically an infinitely small number and is the same as 0.0000... 2 and 0.0000...3 etc.
Take a look at the below to see where my idea is from:
0.9 + 0.1 = 1
0.999 + 0.001 = 1
0.999999 + 0.000001 = 1
if I have 1 million 9's after 0, I need 1 million minus 1 zeros before 1. I thus need to make an addition to make a whole number..
You are saying that 0.999999... = 1 ,meaning that 0.00...1 is equal to EXACTLY ZERO
If this is true then 0.9999.... + (0.000...1 x 0.9999...) should equal 1
Consider another equation
0.9999.... + (0.000...1 to the power of 0.999....) = _______
The point I'm trying to make is that an infinitely small number is still a number.
Mathematicians solve the above by saying that anything multiplied by zero is equal to zero, and you have showed that 0.000...1 is equal to zero so you would fill the blank space with either 1 or 0.999...
I have a deep mistrust of maths, e.g. answer the following,
How many degrees are there in a square?
How many degrees are there in a triangle?
How many degrees are there in a circle?
If you have remembered what you are taught you will have said that a square and circle both have 360 degrees when a circle is NOT a square.
If we accept that a triangle has 180 degrees and make the infinitely small number be known as K then the internal angle of a circle should be (180 - K) x any reoccurring whole number.
I'd also like you to consider the following
Time can be continually split in half. Imagine having a sophisticated timer and lamp. The lamp is on, but after exactly 1 minute the lamp is turned off. When 30 seconds pass it is turned on. After 15 seconds the lamp is turned off. It will be turned on after 7.5 seconds, and so on. This demonstrates the existence of 0.000....1 i.e. an infinitely small number. The same can be done with space too, though we like to think space is limited and this is why I think mathematicians say that 0.999... is equal to 1.
If 0.999... equals 1, and thus 0.000...1 doesn't matter since it is nothing then you should in my opinion be able to tell me whether the light will be on or off in exactly 2 minutes of the above experiment.
I welcome any new arguments.
"having read your arguments I will accept that if 0.9999... equals 1 then Pi has been calculated exactly, and is "known" even if we can't ever write down all of the digits."
But Con then offers two proofs and accepts mine that 0.9999... equals 1!
I'm a little confused. However, I can only assume that Con does not accept any of the presented proofs that nought point nine recurring equals one, since Con later rejects them.
Con also clearly understands something important when they say that 0.00...1 is the same as 0.00...2 which is the same as 0.00...3 which is the same as an infinitely small number (which is the same as 0).
Or, does Con still disagree with this?
Okay, let's look in more detail at what the representation "0.999..." means. It is simply the sum of the following infinite series:
9/10, 9/100, 9/1000, 9/10000, 9/100000, 9/1000000, etc.
It's basically like starting with 0.9 and then repeating the step "now get 90% closer to 1" an infinite number of times. Infinite! That means you never stop getting 90% closer. Yes, when you start at 0.9 and get 90% closer to 1 an infinite number of times you arrive at 1, despite the false intuition that there must therefore always be 10% left. 10% of what? 10% of nothing is what! Not convinced, still? Well, how far away from 0 is it? It's less than a tenth, less than a hundredth, less than a thousandth, less than an infinitieth.. and that really is 0.
It's just a different way of writing 1, when all is said and done. A bit like 1/2 is another way of writing 0.5. The underlying value that we are using this odd human notation to denote is the same thing! 0.999... is not just equal to one, it's the same thing!
and I did say:
"having read your arguments I will accept that if 0.9999... equals 1 then Pi has been calculated exactly, and is "known" even if we can't ever write down all of the digits".
this is because it makes sense to say Pi is irrational "IF" Pi is a number which can have two different representations, I'm not going to ask for the 2 trillionth digit and declare the rationality is unknown, I wouldn't have a debate where I can't learn anything.
When I offered two proofs showing 0.999.. equals 1, it is not because I believe any are correct. In fact I believe they are examples of BAD MATHS (only when dealing with a crazy amount of precision though)
Let's look at the second proof that I showed in round 3, with my corrections, see below:
S = 0.999.. i.e. 1 - k
10S = 10 - 10k
(10 - 10k) - (1 - k) = 9 - 9k
S = 1 - k
At this point you may be thinking 'hang on I've said 0.00...1 and 0.00...2 etc are the same, so surely I should write 10k as 1k' and you may tell me it should look like the below:
S = 1 - k
10S = 10 - k
(10 - k) - (1 - k) = 9
9S = 9
S = 1
Though the end result of 10k and k is the same, they are not the same, and therefore should not be treated the same! Doing so is BAD MATHS in my opinion.
Infinitely small numbers are much like infinitely large numbers, imagine you can live forever and can press a button to double your money, you will be infinitely rich, pressing a button which can quadruple your money each time will not change how wealthy you can be, however with the second button you can get rich faster.
Alternatively imagine if I lost half of my money every second and someone else lost a quarter of their money each time, it is natural to say we will both end up with "nothing" but the pattern actually continues for eternity. Applying this maths someone can have an infinite amount less than me. But it is possible to make it so that a person always has 10 times less than me.
Let's move on now..
0.999... = 0.9 + 0.09 + 0.009 + 0.0009 etc
1 = 1 + 0...
So, basically 0.999.. is ALWAYS less than 1, and by an infinite amount i.e. 0.00...1
Onto your comments, why is the internal angle of a circle 179.99...?
How to calculate internal angles of polygons - let's take an octagon which as you know has 8 sides,
360/8 = 45 degrees
There are 180 degrees in a triangle, so the other angles add up to 180 - 45 = 135
There are 8 sides, so an octagon has a total of (8 x 135) degrees = 1080 degrees
If we have a 16 sides shape now
360/16 = 22.5 degrees
There are 180 degrees in a triangle, so the other angles add up to 180 - 22.5 = 157.5
It will therefore have total of (16 x 157.5) = 2520 degrees
If we have a million sided shape now
360/1 million = 0.00036
There are 180 degrees in a triangle, so the other angles add up to 180 - 0.00036 = 179.99964
It will therefore have (1 million x 179.99964) = 179999640 degrees
As you can see the more sides there are in a polygon the closer the angle "OF ONE SIDE" becomes to 180. A circle only has one side! I can place any numbered side polygon inside a circle and the angle will never reach 180, because only a straight line has 180 degrees. The internal angle of a circle is therefore 179.99... degrees
Having written that I have come to realize that an infinite sided polygon perfectly represents a one sided circle, and concede to defeat.
I would like to thank you for having this debate. You have helped to convince myself that the rationality of Pi is known, and that Pi (circumference/diameter) is without doubt an irrational number, at least in my mind anyway, but I will let voters decide
Hope you have found this debate interesting, best regards, Kirk
I will note, however, that it is interesting how the infinite can come into a discussion about something so seemingly finite as the area of a circle. Understanding infinities is one of the great steps that one must take when learning maths - it is a bit of a leap of faith at points, at first... but it's a rewarding study. Bear in mind that when 0 was first proposed, people (even mathematicians of the day) baulked. Then for negative numbers. Then for imaginary numbers (do they exist? No, but then neither do numbers exist - they are darned useful for getting real answers that are otherwise unattainable, though). People doubted that irrational numbers really existed even when the Greeks had a proof that the square root of 2 was irrational.
Maths is often at odds with our fundamental intuitions. But that's why we love it!
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