Yes, thank you for that Musicat. Maybe I was allowing myself to become too immersed in the pure mathematics of it. 
We know that pi has an infinite number of decimal digits without repeats, and that it can’t be exactly expressed as a ratio of two integers, because we’ve mathematically proven it. That’s what math is all about: Proving things.
How does the proof that pi is irrational go? To be honest, I’m not entirely sure: I’m not a mathematician (though if you want, you can look it up). But pi isn’t the only irrational number, either, and some of them are a lot easier to work with. Another example is the square root of two. And it’s actually fairly easy to prove that the square root of 2 is irrational: The ancient Greeks were able to do it. Here’s how it goes:
Suppose that the square root of 2 were rational. That is to say, there are a couple of integers a and b, such that a/b = sqrt(2). And let’s further suppose that we picked the fraction that’s in lowest terms, so we can’t reduce it (like, you wouldn’t write a half as 2/4 or 3/6, you’d write it as 1/2). If sqrt(2) is rational, we should be able to do that, right?
Now let’s see what we can figure out about a and b. Well, a bit of simple algebra will show that b^2 = 2**a*^2 . That means that b^2 must be even, and hence b must also be even.
Well, if b is even, then we can define c as b/2, and c will still be an integer. And so we can re-write b^2 = 2a*^2 as (2c*)^2 = 2**a*^2 .
But now let’s do a little bit of algebra again. We can expand out (2c*)^2 , so now we have 4c*^2 = 2a*^2 . Or, canceling out a 2, 2c*^2 = a^2 .
Which means that a^2, and hence also a, is even. And we already proved the same for b. So both of them are even. But we started by assuming that a and b were reduced as far as possible, which contradicts that. So our initial assumption must have been wrong, and sqrt(2) cannot be expressed as a ratio of two integers, after all.
You see how that worked? We didn’t go computing any digits of sqrt(2) at all, and yet, we proved something about all of them. We learned something about something infinite, and yet it only took a single post on a message board to do it. That’s the beauty of mathematics.
That was wonderfully argued Chronos. It shows me that philosophy is a powerful way to develop knowledge. Thank you.
ETA: pretty thoroughly beaten by Chronos’ fine post. But I tackle the problem from a different direction, so I’ll leave it here FWIW.
As you said, math isn’t your strong suit. Right now it looks like you’re lacking the background to understand the various darts being thrown towards your question.
The issue is not with the decimal system as such. The decimal system is just a way of writing down values. There are other systems, such as the base 2, 8, and 16 systems used in various computers. There are lots of others.
In a base 3 system it’s easy to write a precise value for “one third”. But it’s impossible to write a precise value for “one tenth”. That’s the opposite of the case in the decimal system. The system of writing does not change the value of the number. One third is always one third and one tenth is always one tenth. It just changes how we write it down. And the ease of doing so.
Big idea: A “number” is an idea. Not a string of digits. How you write it doesn’t change what it is.
Numbers can be divided into categories based on the *nature *of their value. Like how we can divide animals between mammals, birds, fishes, etc. There are many different types of categorization that are more or less useful depending on what you’re trying to accomplish. Just like we can divide animals up by number of legs, which lumps some mammals, amphibians, and reptiles together, while splitting off some other mammals such as people and reptiles such as snakes.
Simplifying mightily from here out to keep this around a beginning high school level. Please nitpickers, help keep this simple enough for this guy. …
The simplest numbers are the positive integers: 1, 2, 3, etc.
The next larger category are all the integers: 1, 2, 3, etc. as above, plus zero and the negative integers like -1, -2, -3, etc.
Notice that there’s no way to use the positive integers to describe = write down the rest of the integers. Can’t be done. Not because -4 doesn’t exist or is weird, but because it’s simply in a different category from the “1, 2, 3, etc.” numbers.
It’s kinda like asking “what color is 6?” The idea of six has no color. The idea of the “1, 2, 3, etc.” numbers has no -4. The only way to get from positive integers to -4 is to introduce some math. We can’t write -4, but we can write a specification for a process that yields the value -4. e.g. “3 minus 7”.
big idea: Positive integers plus some certain math yields all integers.
So far so good?
The next “bigger” category of numbers are the rational numbers. Which are all the fractions composed of two integers. Such as 22/7 or 1/3 or 1/2. These include all the integers since any integer, e.g. 12, can also be expressed as 24/2, 36/3, etc.
The rationals are all the values that *can *be written as a fraction of two integers. It doesn’t matter whether we write in the familiar base 10 or some other such as base 9 or base 17. The roster of rational numbers is the same set of values no matter how we choose to write them.
What we cannot do with the rationals is write them simply as integers. You can’t write the value “one half” without using more than just an integer. It takes the combo of 2 integers and the slash and the math the slash stands for to make that value.
In short, rationals are some of the values that lie between the integers. They’re the mathematically simplest values that live in the gaps between 1 and 2, and between 2 and 3, etc.
Big idea: All integers plus some certain math yields all rationals.
So far so good?
The next “bigger” category is the irrational numbers. Those are values that *can’t *be written as a fraction of two integers. I said that rationals fill in some of the gaps between integers. The irrationals fill in the gaps between the rationals. And, just like before, you can’t use any combination of rationals to directly write the value of an irrational. You have to apply more math to a pair of rationals to get to an irrational value.
Big idea: All rationals plus some certain math yields all irrationals.
pi is an irrational number.
A consequence of which is that you can’t write it using any string of digits in any numbering scheme. That does not mean it doesn’t have a definite specific value. It just means we can’t write that down as a straightforward typical number. But we can write it down as a specification: “pi is the ratio of the circumference to the diameter of a perfect circle.”
The specification takes just a few words. Writing the value would take all the bits in the Universe for all the time in Time to just begin to scratch the surface of getting started.
This is true of every irrational number. Of which there are an infinite quantity just between the integers 1 and 2.
Ain’t math fun?
Thanks for the maths lesson, even though I’m not quite that dumb, but I think the penny has dropped.
Pi has to have a precise value since it is a ratio of something. In this case the ratio of the circumference of a circle to its diameter, right?
I think the confusion arises (at least to me) where people say it’s not possible to perfectly calculate it using any known mathematical system. It kind of suggests that pi itself is not precise, yet I guess it must be (I think) because…now wait a minute. How do we know philosophically that pi does have a precise value as does, for example, one side of a triangle with equal sides? In the latter case we can deduce logically that it is the case but what logic can we apply to pi to show the same thing?
And it gets even worse 
That there were numbers that could not be represented as a rational number was deeply shocking to the ancient Greeks. The notion flew in the face of all the mathematical philosophy that had come before. The simple idea that you could construct a triangle with a right angle between two sides with a length of one, but that there was no possible finite way of representing the length of the other side took them to an abyss.
As noted, the square root of two is thus a member of the irrational numbers. But it can be simply expressed as the solution to the equation x[sup]2[/sup] = 2. IE, the (positive) number that when squared yields the number two. This is a simple “root” of a polynomial. In general a polynomial is an expression that takes a single unbound value (ie it doesn’t have a specified value before we start) and applies a formula of the form a + b * x + c x[sup]2[/sup] + d * x[sup]3[/sup] … + zzz * x[sup]n[/sup]. (Where the coefficients a,b,c…zzz are all rational numbers.) The question is for what values this expression equals zero. These are the roots of the equation. In general there are as many roots as there are terms of the equation. Now, what sort of numbers can we get from this sort of definition of a number? Well, you can easily see that you can get all the integers, and rational, and a whole slew of irrational numbers, including every square root of every rational. But can you get Pi? Well the answer is still no. There is yet another division of the numbers.
The numbers we so define as roots of a rational polynomial are the algebraic numbers. And there are an infinite number of them. (But it is possible to count them !?) There are still more numbers, of which Pi is one. These are the transcendentals. They are numbers that a not algebraic. And there are infinitely more of them than algebraic numbers. “Almost all” irrational numbers are transcendental. Which mean that “Almost all” numbers are transcendental.
The philosophical nature of the discourse is amazing. We can start with an extraordinarily small amount of preconceived notions. We can start with just small number of assumed truths. These are Peano’s axioms. Basically nothing more than defining that there is at least one number, defining the nature of equality of numbers, and defining a successor (“next number in sequence”) operation appropriately. That gets you the Natural numbers. You don’t need much more than these, plus some reasonably obvious ways of extending the operations you want (addition, multiplication, their inverse operations) to get the entire toolkit of things you need to get to the conclusions described above. That and some plane geometry, which is similarly based. On the way you can take diversions with prime numbers, and a great deal more. It is something of astounding beauty that such is possible.
Pi equals exactly 4 times (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - . . .).
That’s as good as saying that 1/3 equals exactly .3333333333333333…, which means that 1/3 equals exactly 3/10 + 3/100 + 3/100 + 3/1000 + . . .
If you object to the formula given above for pi being called “exact” (whatever you mean by that term), then you should also object to the formula above for 1/3 being called exact. We can say what 1/3 is exactly in decimal notation by saying that it equals the sum of an infinite set of fractions (which we can simply specify), each of which has a denominator which is an integer power of 10. Similarly we can say what pi is exactly as 4 times the sum of an infinite set of fractions (which we can simply specify).
The exact value of pi is 1, in a base-pi system … not useful, but you asked for pi’s exact value …
You mean, how do we know that the ratio circumference : diameter is the same, regardless of the size of the circle? I believe this scale invariance is actually one of the defining axioms of Euclidean geometry. As a matter of fact, the same statement is not true in, for example, spherical geometry (think of circles drawn on a globe, where every measurement is made along the surface).
I hope you weren’t insulted. Often failures to understand a high level complicated idea start from a pretty humble misunderstanding down near the roots of one’s knowledge. So I wanted to start at the bottom and work up from there.
Pi is indeed a ratio. But it is not a ratio of integers, nor of rational numbers. No matter what unit of measure you use for a diameter that gives you a nice rational value for the diameter, you can’t express the circumference as a rational number in those same units. Nor vice versa.
You’re using sloppy terminology or more likely, reading people who’re using sloppy terminology. Math is crystal clear and perfectly sharp-edged. English is a muddy mess. Any time we try to use words to describe math we’re threading a needle whilst wearing boxing gloves. Confusion often arises right here.
“…perfectly calculate it using any known mathematical system …” is an attempt to say in words “write out the numeric value of pi”. Which is a flat impossibility for an irrational number like pi.
More than that, it’s the very definition of irrational numbers: “those numbers which cannot be written down exactly in any form; only described as the result of certain classes* of math process.”
But again, a number is not a string of digits. Its an idea. The idea is perfectly precise. The digits are not and cannot be. The confusion arises when you explicitly or implicitly conflate the idea of a number’s *value *with its representation.
You claim you’re confident about the definite length of some triangle but you’re somehow unconfident about the definite length of the circumference of a circle. Examine why you have that difference in confidence. Remembering that being able to write down a number for the length is not what gives it definiteness. With a ruler of the correct calibration, the length of anything, anything at all, is exactly 1. Just choose the right ruler and Bob’s your uncle. There’s got to be more to the objection than that. Otherwise we’ve “proven” (for very sloppy values of proven) that the glitch is in the objection, not in the numbers.
Hope this helps.
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- I’m leaving out at least one layer of complexity on purpose. We’ll see if that can of worms needs to be opened and eaten later or isn’t relevant to your misconnect.
Mini addendum:
You’re floating right at the boundary where a sizeable fraction of humanity says “That math’s too abstract for me” and throw up their hands. Don’t feel bad if looking over the border makes you a little dizzy. You’re far from alone. Whether you want to turn back or charge ahead is 100% up to you.
When I wanted to completely confuse my Geometry students, all I had to do was note that, if we knew the length of the diameter of a circle was a definite value expressible as an integer, we could not determine the exact decimal value of the circumference, and if we know the circumference as an exact integer value, we cannot determine the exact length of the diameter (in decimal form).
But what I find more interesting in this thread is the fact that the OP has confusion about the issue of “exactness” of decimal values, but doesn’t seem to have a problem with the bare assertion that all circles have the same ratio of circumference to diameter, which value we call π. Yes, it’s actually a simple proof, but if it wasn’t proven, the whole bit about π being a constant value would be bollocks.
As for proving that π is irrational, well, that’s actually a pretty intense mathematical proof, probably beyond the scope of this thread. You can, however, see several different proofs of it at Wiki’s entry on π.
Moderator Note
No accusations of trolling outside of the Pit, please.
Isn’t this all because the original concept of a number was indivisible, yet because it was found we had to subdivide this indivisible entity we had to compromise and just be satisfied with approximation? Originally all people wanted was a system to count objects in the real world and it was only people like the Greeks and others who began to think about numbers and philosophise about them that new ideas arose. I think this is why, despite numbers being useful in representing reality, they are not reality, just ideas and, therefore, can only provide us with rough representations of the universe. Numbers are based on a few axioms which must be adhered to, obviously, but that means (it seems to me at least) that we have to compromise in maintaining mathematical integrity. It’s like having a tool which we want to use for many purposes so that it will never be the ‘perfect’ tool but a general purpose way of ‘shaping’ things’ For example, pi is not actually the ratio of the circumference of a circle to its diameter but is useful way of mapping it. The point is pi is a practical tool.
Well, why do we hear that the value of pi goes on forever? It means it can never reach an exact vale.
Well I suppose so but is that helpful?
There are things here that could be nitpicked, and things that some people would dispute depending on their basic philosophy (e.g. a Platonist might say something like, ideas are ultimate reality)—but I think you’ve basically got it.
It’s not the number, or value, of pi itself that goes on forever; it’s the decimal representation. A lot of confusion arises when you don’t distinguish between a number itself and the way that number is written.
What I was getting at is, sure, intuitively we tend to think that since pi is a ratio it must have an exact value but that seems to be based more on commonsense than on any mathematical proof which I have yet to see. We can’t just use our intuition with mathematics since logic has to be used to underpin its ideas.
May I just say, no I am not trolling, and if I come over that way I apologize. My ardent interest is genuine, I assure you. I’m not just asking question to stir up trouble.