Yet again, you seem to be using the wrong words. Pi never changes. The approximation we use for pi may be changed at any time, but that doesn’t make pi less real, it just makes the approximation less useful.
Example: You want to build a 10 ft wide tower, out of 1 foot long blocks.
You use the approximation pi=3. (This is a silly example, quoted as biblical, usually to laugh at bible literalists.) You will not have enough.
If you use pi=3.14, you will still be low, but less low than you would otherwise be.
If you use 22/7, you will use too many bricks by a tiny amount. What the heck, you can break one and build your tower.
If you use an approximation of pi to 44 decimal places, you can build a tower, of hydrogen atoms around the entire universe.
Well, well, what if … I wanna build a tower, around 2.5 universes, otta … uh … uh … quarks? How many decimal places do we need then? Nobody knows, because that’s silly. However, the ratio stays the same, and it is pi.
It’s actually not that complicated (though a bit moreso than for sqrt(2)). Here you go:
Let f[sub]N/sub be the N-th derivative of cos(sqrt(x)).
Note that f[sub]N/sub = P(y)f[sub]0[/sub] + Q(y)f[sub]1[/sub], where y = 1/(4x) and P and Q are integer coefficient polynomials of degree growing at rate O(N). [This follows inductively from f[sub]2[/sub] having this form, which is just the fact that cos’’ = -cos, translated to this reparametrization.]
But, for any given x, we have that |f[sub]N/sub| ~ |f[sub]N/sub| = N!/(2N)! as N grows large. [This can be seen from the easy Taylor series expansion of f_N(x); the = is immediate, and the ~ follows from each term dominating the next as N grows large (if you’re worried about commuting limits here, it’s justified by dominated convergence).]
Given this super-exponential rate of decay, it cannot be that both y is rational and f[sub]0[/sub] and f[sub]1[/sub] are in rational ratio. [If y were rational with lowest-terms denominator d, then the smallest nonzero value integer polynomials of degree O(N) in y could produce is 1/d[sup]O(N)[/sup], which is a merely exponential decay rate; any fixed rational combination of them would also therefore have a merely exponential decay rate.]
In particular, plug in sqrt(x) = π/2, at which f[sub]0[/sub] vanishes, to conclude y = 1/π[sup]2[/sup] is irrational. More generally, this establishes that wherever tan(z)/z is rational or infinite, 1/z[sup]2[/sup] is irrational (including, for what it’s worth, the case of complex z; indeed, the whole thing might as well have been phrased in terms of cosh, but people find cos more familiar).
Of course, the corollary people talk about most is that thus π itself is irrational.
I should have written “f[sub]N/sub = P(y)f[sub]0/sub + Q(y)f[sub]1/sub”, I suppose. Or perhaps just “f[sub]N[/sub] = P(y)f[sub]0[/sub] + Q(y)f[sub]1[/sub]”, which I mish-mashed it with. Well, whichever gets the idea across most clearly.
Well yeah, sure, there is a distinction between pi and our best approximation of it but the fact remains there has been quite a bit of discussion over the years about how many decimal places pi has been calculated to. This is really what I’m addressing and I apologize if I’ve been vague about this. I suppose you could compare this to estimating the distance between some point on the earth and some point on the moon; it depends how much accuracy you want. Actually, I can’t believe these calculations are real since when you get down to the level of sub-atomic particles it’s impossible to measure with any accuracy due to the indeterminacy principle. Still, that’s pure mathematics for you!
I’m not so sure about that. Some people involved in pushing the envelope in calculating pi seem to very enthusiastic about it. I suppose this may be because it’s something that has not been achieved before.
Again though, in my naivety, I cannot see the logic of saying that the square root of two has an exact value when it can never, ever be resolved, since its calculated value goes on forever. In other words, to a laymen like myself, you can argue that you never find out the true value of the square root of two because whatever point you have reached in its calculation there will always be more calculations to do. In fact, how could you possibly prove it has an exact value when it is impossible to calculate it? It must always be an* approximation*, surely. Or are you saying you can precisely calculate the square root of two in other numbers systems, in which case it does have a precise value but may only be dealt with using an irrational number when using the decimal system?
Yes, so in this set-up the square root of pi would be the square root of 1, i.e.1, right? Okay this makes sense but I think the fascination for me is about how we have to adapt the decimal system in terms of using irrational numbers in order to calculate pi.
In practice I don’t think the circumference/diameter ratio is pi at all, in the real world. Einsteinian gravitation renders space non-Euclidean. Perhaps the board’s physicists will comment on the expected deviation from a perfect 3.1415926535… depending on where you draw the circle.
Of course you are. I was addressing Mr Wolf.
Would you say the same thing about 1/7 = .14285714285714285714… ?
That post was aimed at Chronos instead of you. Were I aiming it at you, I would go much more slowly through building up the relevant concepts you are likely not already aware of.
Oh, and for those reading carefully, yeah, there’s an f_N I forgot to turn into an f[sub]N[/sub]. Give me an unlimited edit window, and these screw-ups will not linger to my permanent embarrassment as they do… Also, give me LaTeX math and a pony.
Yes, it’s an infinity of zeroes, but that’s still an infinite number of terms. Numbers don’t cease to exist simply because they’re zero: There’s a distinct and useful difference between an equation which doesn’t have a solution and an equation whose solution is zero, after all.
So, on that footing, the idea of an infinite decimal is less of a special case: All numbers can be written with an infinite number of terms out to the right of the decimal point, but some have an infinite number of nonzero terms out to the right. And even more numbers can be written with an infinite number of non-repeating terms to the right. But the underlying point is, that infinity of terms is not special.
(Now, as for an infinite number of nonzero terms to the left of the decimal point, those numbers don’t exist in any of the familiar number systems, since infinite values are not especially useful or intuitive to work with.)
Okay, but in these cases there exist recurring zeros which is not the case with the kind of irrational numbers we are considering. If you divide 2 by 1 for example you would not need to express it as 2.0000000000000… Simply 2 would be accurate enough.
[P]eople have worked strenuously to compute π to thousands and millions of digits. This effort may be partly ascribed to the human compulsion to break records, and such achievements with π often make headlines around the world. They also have practical benefits, such as testing supercomputers, testing numerical analysis algorithms (including high-precision multiplication algorithms); and within pure mathematics itself, providing data for evaluating the randomness of the digits of π.
Your use of the word “calculated” is the core problem. What do you think you mean by this word? It seems you mean “expressible as a decimal number.” Which as I noted above is identical to saying “expressible as a rational number.” So why? Why insist that calculation of a value requires that it be expressed as a rational number? Apart from being taught in school that numbers are written as decimals there is absolutely no reason to insist on this. Decimal numbers, or any fractions (ie rationals) are just a convenience that are easy to use in many applications. There absolutely no reason to insist that these are somehow the “right” form of number, or that they are anything other than an accident of history.
Now we get to the core problem. What do you mean buy true? You seem to again believe that the only true values are those that are expressible as decimals. Why?
We have discussed earlier, almost all numbers are not so expressible, and only an infinitesimally small set of numbers are so expressible. By your logic almost no number has a true value.
By calculation you mean express as a decimal value. Most mathematicians would not regard this as calculating its value. We might determine a quick approximation to its decimal representation for engineering purposes, but the value of the square root of two is just that - the square root of two. Anyone doing mathematics (as opposed to arithmetic) will carry that value around in expressions until they (might) be able to combine terms in some manner to make it vanish (or not, as the case may be.) There may be a point where some arithmetic is performed to determine a useful expression of the value for some purpose. But that isn’t mathematics.
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In fact, how could you possibly prove it has an exact value when it is impossible to calculate it? It must always be an* approximation*, surely.
Easy. You have already seen proof by reductio-ad-absurdum. Typically you assume that there is more than one (different) value as a solution and show that if you assume this, some simple algebraic manipulation leads to a contradiction. Indeed proving the: existence of, number of, or non-existence of, such solutions is a core part of mathematics. This is a vastly more powerful a concept.
Again, this problem with calculate. The answer to the above is trivially yes, you can use non-rational bases. But it doesn’t actually help your main issue. Such a base cannot represent all other numbers, so it is just moving the deckchairs.
The resolution is to let go of the idea that decimal numbers (or any other form of rational numbers) are the basic true representation of numbers. They are not. It is sad in a way that some schools seem to ingrain people with this notion, and it is hard to unlearn, and perhaps counter intuitive. Numbers are much more interesting than this. Sure the Greeks had problems with this, but they also realised that the rationals were not the basic true form one they discovered the square root of two. They had a lot of soul searching to do, and it took a long time (centuries) for a solid understanding of numbers to be built, but it has been built, and the Greeks are regarded as a bit of history, not as having some deep knowledge.
If I understood you correctly, you are considering two, or possibly three, sets of real numbers.
One (the largest one!) consists of numbers which cannot be computed.
Two, numbers which can be computed in the sense that a program running forever can print out more and more decimal digits, but it cannot terminate. This class includes numbers like 1/3 and π.
Three, rational numbers with a finite decimal expansion, like 1.23345.
Do you have a philosophical problem with a simple fraction like 1/3 being in class 2 rather than 3 and therefore decimal notation sucks? Some people agree with you, which is why one pound used to be divided into 240 pennies instead of 100 cents. On the other hand, your computer probably uses binary floating-point arithmetic, which means that it cannot precisely represent a number like 0.1 any more precisely than it can an irrational number. Does that mean your computer sucks?
Both of these statements come back to the point that you seem to believe, mistakenly IMO, that a value is only definite when you can write it down as a finitely long list of digits.
That is simply wrong. It is exactly equivalent to saying “I accept that integers and rationals are actual factual numbers, but irrationals are somehow phony or fuzzy or otherwise defective in a way I can’t quite articulate in English.”
IMO at bottom, you have absorbed the idea that “number” and “finite list of digits” are the same idea. They’re not. To make progress you need a bigger, more inclusive definition of “number”.
Your intuition that irrationals are *different *from integers & rationals is spot on. They *are *a more abstract concept that takes greater comfort with abstraction to accept. As said above, generations of mathematicians in Ancient Greece simply couldn’t abide this idea. You can be like them if you choose. But you don’t have to be.
Everything you’ve said in this thread amounts to “I’m uncomfortable with irrationals; make me comfortable, but don’t challenge any of my assumptions, vocabulary, or logic.” We’re all succeeding at talking about irrationals, but failing at the rest.
Because you won’t make progress in understanding (or accepting) irrationals until you release the defective notion that writing down a string of digits is a prereq to definiteness or somehow is the very definition of what a “number” is.
Ref this tidbit “Or are you saying you can precisely calculate the square root of two in other numbers systems, in which case it does have a precise value but may only be dealt with using an irrational number when using the decimal system?” …
The number-writing system doesn’t change the irrational nature of an irrational number. Its irrational because it is, not because we can’t write it down in base-10. The fact we can’t write it down is an inescapable consequence of irrationality, not a cause.
As long as we’re using a number writing system that’s using the sum of powers of an integer, such as familiar base 10 or 2 or 3 or whatever, an irrational cannot be written down. At a very advanced level, there are alternative writing systems that work for those irrationals that are not also transcendental. But trying to describe that writing system would be a couple semesters beyond the place your confusion begins. In other words, it will add to your confusion, not reduce it. IMO we’re better off not going there.
Going back to my first post, the so called “math lesson” you said you didn’t need, I explained that each increasingly complicated and abstract set of numbers couldn’t be written in the earlier kind of number. Except as the output of a math process using the simpler numbers. e.g. you can only describe the number -4 using positive integers by arranging them in a math formula such as “3 minus 7”. You can only describe the rational number “7 tenths” using integers by arranging them in a math formula such as “7 divided by 10”. You can only describe an irrational number such as “square root of two” using rationals by arranging them in a math formula such as “SQRT(2)”.
The thing to notice here is that as we ascend the complexity scale, the math operations needed get harder. Subtraction was all we needed to create zero and negative integers. It took division to make rationals. It takes SQRT and/or other more complex operations to create irrationals. At some point these fancier operations become complicated enough that they’re not intuitive and it takes a certain leap of faith to accept that SQRT is just as real as is subtraction.
Hope this helps; I’m not trying to beat you over the head. If you feel I am, just say so.
Your post reminds me tangentially of ‘surreal numbers’, which is a big class of numbers used to represent the values of certain games in game theory.
One way to construct them is inductively, starting with 0, so that earlier steps produce simpler numbers and later steps produce increasingly complicated ones. For example, next after 0 appear 1 and -1, then the next step introduces -2, -1/2, 1/2, and 2, and so on. Any finite number of steps will produce rational numbers whose denominator is a power of 2.
The cute thing is, if you mathematically proceed for an infinite number of steps, you come up with all the real numbers, including 1/10, the square root of 2, and π. As well as infinite numbers (surreal numbers greater than all positive real numbers) and such things.
The point of this digression was to suggest that defining irrational numbers seems associated with concepts like infinite constructions, and it does not go without saying that people are intuitively comfortable with those. (Witness people signing into this board seeking various proofs that .9999… = 1)
ETA try constructing real numbers in an introductory analysis class without any notion of infinity like infinite sets or infinite sequences??
Not at all, no. I feel lucky that you are prepared to be patient with me and are allowing me to share your expertise and I thank you.
But just to clarify then, my misconception is to assume irrational numbers are inferior
to rational numbers (i.e. a numbers that can be written as a fraction) because they go on forever? And that pi can only be expressed as an irrational number? I think I’m beginning to see…
But, one further point: how can we be certain we will never arrive at a termination with an irrational number? I know it will never happen but to the uninitiated it seems mysterious.
