Appreciate that, thank you. 
Well, if you define pi as the ratio of a circle’s circumference to its diameter, you need to prove that this ratio is the same for every circle; but this can be done relatively easily. You also have to worry about whether such a ratio counts as a number: does it make sense to call it a “value”? what counts as a number? This can get you into fairly deep waters mathematically, but mathematicians have addressed such questions rigorously, though it took them a while to do so.
Okay, well what I mean is if you take a number and then divide it by another number and it divides perfectly without any remainder isn’t that perfectly exact?
Okay so what do I know? But no harm is asking - I might just learn something. :rolleyes:
Except 1/3 can be expressed as a finite sequence of digits in a non-decimal system. Pi cannot be.
On edit: unless the base is Pi or a multiple of it. But usually number systems are integer-based
From the context, it looks like by “number” you here mean “whole number” or “integer”; but you have to be careful because there are different kinds of numbers.
By definition, if you take an integer and divide it by another integer (that isn’t zero), the result is a rational number. If it “divides perfectly without any remainder,” it’s an integer; if not, it’s not. It can be proved relatively easily that the rational numbers (numbers you can get by dividing one integer by another) can be written in decimal form in such a way that they either terminate (like 3/4 = 0.75) or repeat (like 3/11 = 0.27272727…).
Irrational numbers, like pi and the square root of two, neither terminate nor repeat when you write them in decimal form. Since pi is irrational, this means that when you divide the circumference of a circle by its diameter, those numbers can’t both be whole numbers for the same circle; at least one of them must, itself, have an irrational value.
abashed writes:
> Well, why do we hear that the value of pi goes on forever? It means it can never
> reach an exact vale.
No, what’s true is that there is an infinite series that equals 1/3:
3/10 + 3/100 + 3/1000 + 3/10,000 + . . .
and there is an infinite series that equals pi:
4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - . . .)
The “value” doesn’t go on forever. In each case, the infinite series goes on forever. However, the infinite series in each case is exactly defined. (Do you understand what the entire infinite series would be in each case?) There is an exact value for both 1/3 and pi. If you asked what the value of 1/3 or of pi would be rounded to the tenth decimal place, we could tell you. If you asked what the value of 1/3 or of pi would be rounded to the thousandth decimal place, we could tell you. If you asked what the value of 1/3 or of pi would be rounded to the billionth decimal place, we could tell you. We can tell you what any decimal place of 1/3 or of pi is for any number, no matter how large that number is, although for pi you’re going to have to wait a while for us to tell you what the decimal place is. Any number that you can say that of has an exact value.
Yes, I understand that an irrational number never terminates or repeats, but I’m still having trouble with being told that pi has an exact value because you have already told me that an irrational number never terminates, which means it will never reach a conclusion so it can’t be theoretically exact, can it? I understand that for practical purposes it doesn’t matter that much but looking at it from a strictly theoretical point of view I can’t see how it can be literally exact. Or do people use the word ‘exact’ to mean ‘exact enough?’
But what is the difference? As you progress along the infinite series isn’t the value of pi constantly changing?
Well, okay, but I was specifically talking about the decimal system.
The reason why this seems an important question is the fuss created by mathematicians about the number of decimal places pi has been calculated to so far. If it can be resolved by another number system what is the big deal?
That the digits of pi do not end or repeat was first proven in the 18th century. Looking at the Wikipedia article none of the proofs seem very simple, especially when compared to the trivial proof that sqrt(2) is irrational — attributed to a student of the ancient Pythagoras.
Two in base 2 is ‘10’; Ten in base 10 is ‘10.’ My hexadecimal is rusty but IIRC, sixteen is ‘10’ in that base. Is there something special about base pi that pi is ‘1’ instead of ‘10’?
As answered above by several people, one question is whether you are cool with irrationals as numbers, or not. A lot of ancient mathematicians were not, but in the modern approach to analysis, including the decimal system taught to schoolchildren, irrational numbers are no problem.
If the question is why calculating and storing billions of digits in the decimal expansion of pi is a big deal, with modern computers it’s not, really. What is more interesting today is the mathematics underlying all the different infinite series and algorithms that can be used to calculate the value. But merely computing a trillion digits of pi, without context, is not something very exciting or that you would ordinarily do.
Using “base pi” is kind of cheating. It’s kind of like saying pi can be expressed exactly as tau/2 (where tau = 2*pi).
Whether a number is rational or irrational doesn’t depend on the system used to write the number. Having an infinite decimal expansion is a consequence, or symptom, of pi’s irrationality.
The way some people talk about pi, they give the impression that the fact that its decimal expansion goes on forever is somehow special, and that that’s a big part of what makes pi such a remarkable number. But this is backwards: It’s not that pi is special or useful or famous because it is irrational (i.e. its digits go on forever); it’s that pi is one of the most special or useful or famous of the many, many numbers that have this property.
Another such number, that has been mentioned several times in this thread, is the square root of two. It has an exact value: it’s the number that, when multiplied by itself, gives 2. The fact that you can’t write down the exact value in decimal form doesn’t mean it doesn’t have an exact value. And similarly for pi.
I’m tempted to say that any number has an exact value, because the number is its exact value. Otherwise (and I think this may be confusing abashed), what’s the distinction between an number and the value of that number?
But there is no specialness to decimal system. It’s just an arbitrary base that was picked most probably because humans (usually) have 10 fingers.
Having a non-integer-based number system is theoretically possible, but completely impractical. Integers are important. You can’t buy Pi cars, or use 10*Pi nails.
Having a Pi-based numbering system just so that you can represent Pi with a finite number (1) is basically tautological. Of course in a transcendent-based system you can represent that particular transcendent number simply. But no other.
abashed writes:
> As you progress along the infinite series isn’t the value of pi constantly changing?
No, the sum is changing, but that doesn’t mean the value of pi is changing. Furthermore, the sum is getting closer and closer to a particular value. Do you understand the difference between a convergent and a divergent series? It’s possible to show that the following two infinite series are convergent:
3/10 + 3/100 + 3/1000 + 3/10,000 + . . .
4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - . . .)
This means that as you take the sum of the series, it slowly gets closer and closer to a single value. If you ask us what the sum of all of the infinite terms of the series is rounded to a particular decimal place, we can tell you. On the other hand, there are divergent series. For instance, it’s possible to show that the following series is divergent:
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + . . .
Given any number, no matter how high, it’s possible to show that if you add enough of the terms of this series, the sum will be higher than that number. Thus
4 * (1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11 + 1/13 - . . .)
is an exact value just as much as
3/10 + 3/100 + 3/1000 + 3/10,000 + . . .
is an exact value, but
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + . . .
is not an exact value, since the series (as it’s possible to prove) is divergent.
Some unfocused observations incorporating, I believe the discontinuity in representations discussed above:
1: I ask you to remove .3333… of the billiard balls in what you say, and I agree–to humor you only, for the time being-- is a bag of three. You say, “sure, you ‘mean’ 1/3, here’s your 1 ball.” I then say “no, I want ‘that same amount’ – [NB: no representation] of the “billiard ball[hood]” as a non - integer-ball-based amount of whatever the hell is in your bag” -and if you come back to me and say “well, all I have is ballhood in units, and this case 3 of that representation,” I can say two things:
-
That’s cool, just take 1/3 away from each–shouldn’t be any problem, right, because you said you were cool with 1/3. But I want to see that counted out (proven) that way, and we’ll be here forever; or
-
You forget about that representation that worked so well when we agreed on how that amount of stuff/billiard-ballhood in your bag, call that your “unit” (no relation to any of ours) and when I ask for this bizarre amount of .3333… of it, you slice it easy as pie (heh) and hand it over no problem. And when I say "how am I sure this is my [or anybody’s] system which is not predicated on the “amount of billiardball(hood) which fits in that bag” you will say: yup. You can’t.
And the damn thing is (how do you say that in Ancient Greek) we know that something is in there, after all. It’s just got two different representations (names) suitable for an agreed conversation (thought representation) of that amount.
The unmeasurable diameter mentioned up thread, like the famous third leg of the unit right triangle, is there, right in front of your eyes. Calm and serene in plane geometry, unspeakable in rational/ration_ing_ process of amounthood.
I would have to say you have it almost exactly backwards. But I see where you are coming from (I think.)
What we have been describing here is number theory, which is pretty much a branch of Pure mathematics. As opposed to Applied mathematics, which is what you might consider as the useful mathematics. Engineers and physicists study applied mathematics, and they learn a huge set of tools that allow them to do all manner of useful things, from designing bridges to working out how the universe works. To a very large extent they could not care less about the nature of pi, the difference between rationals and irrationals, and in reality are perfectly happy with anything that has about 7 decimal places of accuracy. (Apart from numerical analysis where the core thing you study is how to keep your calculations accurate, and when you really really need a few more decimal places.)
The nature of numbers is almost entirely a philosophical problem. Or one that concerns some pure mathematicians. But, there is core problem with simple discussions, one that seems to be rooted in poor understanding of the use of terms.
The most important thing to recognise in the preceding discussions is that decimal notation is a way of writing down a rational number. 3.14159 is shorthand for the rational number 314159/100000. 3.141592653589793238462643383279502884197 is the rational number 3141592653589793238462643383279502884197/10[sup]38[/sup]. Writing down a number like this is intrinsically limited. It isn’t the most natural or “pure” way of representing numbers. It is however what people get taught a school, and what many people think numbers are. But it isn’t numbers. It is a very limited subset.
Making a statement like “Pi goes on forever” is highly misleading, and misses the point badly. We know is that pi is not a rational number. What that tells us is that there does not exist any number of the form 31459…/10000… that equals pi. None. It simply does not exist. However, what we can do is get arbitrarily close. How close would you like? As close as you want, we can write down a rational number that is closer to pi than that. But you can’t get exact. Ever. There will always be a tiny (infinitesimal) gap. The closer you make that gap, the bigger and nastier the representation of your rational number approximation will become. And since you can specify as arbitrary closeness, you should expect and arbitrarily long rational number (or decimal) representation. That is the only sense in which pi goes on forever. It goes on forever when you use the wrong tool for the job. Use the right tool and the idea is just silly or meaningless.
The point about transendentals is that rather than this being something special about pi, and maybe a few other numbers, it is actually the reality with “almost all” numbers. However the easy, simple set of numbers we use in ordinary life are chosen using a set of rules that makes them easy for us to use, and we generally have exact decimal representations for them. (Because that is how we defined then in the first place.) It is this set of numbers (nowadays pretty much the set of rational numbers representable on a computer) that are useful tools.
Pi as it is defined isn’t a tool. It is what it is. It is the name for a mathematical constant. One that appears in a great deal of mathematics. Mathematics itself is a tool, and the practice of mathematics involves the application of mathematical tools to the manipulation of mathematical definitions and descriptions. Sometimes these systems describe useful earthbound things, like fluid flow, or stresses in beams. Othertimes they express imponderables, like the nature of infinity.
[Off-topic nitpick]
I might have phrased this the opposite way! Non-integer bases are impossible (according to reasonable definitions of place-value number systems), but a tiny number of them (with terminology abused somewhat) are of practical use in the engineering of codes!
The “Fibonacci” counting sequence 1, 10, 100, 101, 1000, 1001, 1010, 10000, 10001, 10010, … is frequently used by code designers and sometimes called base-φ representation, or base 1.61803399… Similarly, base 1.46557123… may be useful.