Pi

[watchwolf49]

The main problem with using pi in decimal form is punching into an electronic calculator … it’s just a tick mark on my slide rule …

Sorry for interrupting again …

[Alan_Smithee]

I can’t believe we’re on the second page (What? That’s how my settings are!) and no one has posted this yet: Saturday Morning Breakfast Cereal - How Math Works

I haven’t totally kept up with the thread, but maybe ambushed could consider this: We know irrational numbers have an exact value because because we have series that get closer and closer to the actual value. If pi weren’t an exact value, what would increasingly accurate approximations be getting closer to?

If pi were not exact, eventually there would be a number so close to pi we couldn’t tell it was wrong. But we can. For every approximation of pi (efor very number, in fact) we can not only tell whether it is closer than any other number, we can also tell how close we are! And even whether pi is greater or less than the approximation.

What else could “exact value” mean than that?

[Chronos]

Quoth Chessic Sense:

Pi doesn’t “go on” anymore than the Taj Mahal is constructed during and by my flight to India.

I love this analogy, and I’m going to steal it the next chance I get.

[abashed]

@Leo_Bloom

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:

1. 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

2. 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.

Well, it seems a strange thing, because in the first case where we are regarding each ball as exactly 1/3 of the total there seems to be no problem in removing a ball. In the second case, where we are defining each of the 3 balls as 0.333333’, we cannot ever remove any of the balls because we are never able to completely define what a ‘ball’ is and would take infinity to do so. We could become incredibly close, yes, but we never actually get there. This seems to be where the logic takes me or am I wrong?

[septimus]

There’s one interesting point about the digits of pi (or at least the digits in base 2 or base 16 and perhaps base 9) that hasn’t been mentioned yet.

It would seem like computing the trillionth bit in the binary expansion of pi would be a mammoth project — after all, wouldn’t you need to first compute the preceding 999.99+ billion bits?
***No! ***There is a formula for computing the trillionth bit or sextillionth bit or umpteenth bit without computing the earlier bits!

Apparently the special expression for pi that allows this fast calculation wasn’t discovered directly by a human: it was discovered by a computer program!

People are surprised to learn how many digits of pi are now known:

[QUOTE=http://crd-legacy.lbl.gov/~dhbailey/dhbpapers/bbp-bluegene.pdf]

*n 1989, famous British physicist Roger Penrose, in the first edition of his best-selling book The Emperor’s New Mind, declared that humankind likely will never know if a string of ten consecutive sevens occurs in the decimal expansion of pi. This string was found just eight years later, in 1997, by Kanada, beginning at position 22,869,046,249. After being advised of this fact by one of the present authors, Penrose revised his second edition to specify twenty consecutive sevens. :smack:

[/QUOTE]

[abashed]

@Thudlow_Boink

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.

Right, but doesn’t that mean our mathematical systems are imperfect?

[abashed]

@Chronos

I love this analogy, and I’m going to steal it the next chance I get.

No, pi does not go on because, as said, it has a definite value, but the point I was making was that the approach to this value goes on and on. I know this has nothing to do with the* real* world but there it is.

[abashed]

@Alan_Smithee

I can’t believe we’re on the second page (What? That’s how my settings are!) and no one has posted this yet: Saturday Morning Breakfast Cereal - How Math Works

I haven’t totally kept up with the thread, but maybe ambushed could consider this: We know irrational numbers have an exact value because because we have series that get closer and closer to the actual value. If pi weren’t an exact value, what would increasingly accurate approximations be getting closer to?

If pi were not exact, eventually there would be a number so close to pi we couldn’t tell it was wrong. But we can. For every approximation of pi (efor very number, in fact) we can not only tell whether it is closer than any other number, we can also tell how close we are! And even whether pi is greater or less than the approximation.

What else could “exact value” mean than that?

Well, the difference is, it seems to me, that a rational number does not allow us to approach its value since there is no necessity to do this, whereas an irrational number gives us the scope to do it. Obviously, I’m no mathematician and don’t think like one but this is how it seems to me. In fact, irrational numbers have aided the development of super-computing of endless digits which improves our computational powers. Aren’t irrational numbers a God-send, therefore? What would we do without them?

[abashed]

@Pleonast

Actually, that brings up an more fundamental question: when is zero zero? For example, is the zero in the set of all natural numbers the same as the zero in the set of all rational numbers? Or the zero of algebraic numbers, or the zero of real numbers? As a physicist, my answer is of course “yes, within these error bounds”. As a computer programmer, my answer is “no, but it’s easy to change type”. I’ll let mathematicians answer for themselves.

So, perhaps it is strictly accurate to say that “0.999…” is not the same as one, despite having the same value.

I’m going to naively make the observation that zero must always be zero because it can be nothing else. Right? Or wrong?

[Francis_Vaughan]

@abashed

Right, but doesn’t that mean our mathematical systems are imperfect?

So now the question is: what do you mean by imperfect? We are talking mathematics here, so you need to define what the imperfection you think exists is. Or define the property you think a perfect mathematics has.

What has been described above is a set of properties that has been deduced via a highly developed and strict logic. It hasn’t been designed, and you can get into all sorts of philosophical debates about the intrinsic “truth” of the system derived. Any many of the properties were surprising (at best) to the discoverers. As has been alluded to, you can build up an infinitude of other possible systems and definitions of number, and the properties that they have. However where you get into trouble is that almost all of these have no useful properties, such simple things as being able to count stuff, or do basic arithmetic. The moment you start to demand what you might call useful properties you discover that these properties make slightly inconsistent demands upon a “perfect” notion of numbers. Perhaps the most obvious demand we might place upon a system of numbers is that it is closed. That is, under the operations we define on our numbers, you always get another number. The Natural numbers are closed under addition. Add any two Natural numbers together, and you get another Natural number. But they are not closed under subtraction. Subtract one number from another, and you might get another Natural number, but sometimes you get a result that can’t be held within the Naturals - the result is negative. Now it might be perfectly reasonable to define such results as simply illegal. After all, what is minus one apple? Or a size of minus one foot? It is meaningless. It has no physical manifestation and clearly has no part to play in our universe. Until you want to somehow keep track of deficits. Which is what was done in some systems (accounting placing a deficit in parentheses for instance). But it becomes obvious pretty quickly that your entire system of numbers make much more sense and is generally more consistent if you allow negative numbers, which gets us the Integers. Which is great. Of course you can then decide that you might like to include multiplication in our system, and it is an obvious extension to the idea of addition. And, great, our system is still closed. Then someone decides that they would like the inverse operation of multiplication - division. After all subtraction was useful. And, gosh darn, the system isn’t closed again. But fractions are a useful idea. We can simply retain fractions as notation, and carry the unevaluated division around, sort of like keeping the deficit in parentheses. But again, it becomes pretty obvious that the system makes a lost more sense if we actually allow those fractions to denote fully paid up members of our numbers, and we get the rations. Except for one particularly nasty case. Division by zero makes no sense. You an argue all sorts of questions and ideas about it, but in the end, division by zero can’t denote a useful number whilst still keeping the nice properties we like about our number system. So we have to make a special rule, that division by zero is undefined.

And on it goes. But each step involves additional interactions and edge cases. And sometimes surprising results. The best you can do is be like the constructivists, they simple say that certain “numbers” don’t actually exist. Like Pi. There is some very deep mathematics and logic at play here.

And we have not made it to complex numbers, (or quaterions and octinons.)

Back back to the start - when you define your “perfect” properties on numbers, you are invariably placing formal constraints on the way the system must behave, and you will discover that these can come back to bite you. Creating a system that is able to match every possible constraint on “perfect” is intrinsically not possible. This is deep.

And we can then get onto Gödel and his incompleteness theorem, for even more deep questions. His result came as a serious (and in many places highly unwelcome) surprise.

[Francis_Vaughan]

@abashed

Well, the difference is, it seems to me, that a rational number does not allow us to approach its value since there is no necessity to do this, whereas an irrational number gives us the scope to do it. Obviously, I’m no mathematician and don’t think like one but this is how it seems to me. In fact, irrational numbers have aided the development of super-computing of endless digits which improves our computational powers. Aren’t irrational numbers a God-send, therefore? What would we do without them?

The calculation of Pi on computers is really little more than a sideshow. It was something of a tradition in the early days of computing (where every new computer was super). But apart from providing a single test case and bragging rights for nerds, it has very little value.

Computational science for almost all useful intents is served by floating point numbers, and usually by only 32 bit float, although the ubiquity of 64 bit makes them pretty much the default. (This ubiquitousness leads to some level of laziness on the part of scientists. Many never run into numerical instability problems, and so are oblivious to it when it does occur, and may not realise their results have become meaningless. This is not a good thing.)

The counterpoint are symbolic mathematical tools - Mathematica, Maple, which manipulate the formulae as symbols. They can be very powerful, and really form a critical part of much scientific work. You may never actually evaluate a numerical result with these tools. Although their ability to produce graphical representations of complicated systems does involve evaluation.

Complex numbers are a special case. Languages like Fortran and tools like Numpy provide complex numbers as an intrinsic type. But they are simply a pair of floating point numbers managed internally by the system.

The distinction between rational and irrational numbers is mostly the province of Pure mathematicians, but it had its roots in the fundamental logic of mathematics. For millennia mathematics has not been nearly as rigorous as we now know it. That came late. But the rigour has been of huge value. A big part of the problem of mathematics without the rigour is that whilst progress was made, and important discoveries made, there was an uneasy question about the absolute reliability of the work. Nobody could say there were not important corner cases. You asked how we knew that some given irrational had a single “value”. That is an important question. If you could not say so, it would call into question other results, and deeper ideas. Coming up with a theory of number that included irrationals was important in closing off some of these gaps. Once the gaps closed they become something of an academic thing, for most practical purposes. Until someone starts asking questions on the Dope that is.

[abashed]

@Francis_Vaughan

So now the question is: what do you mean by imperfect? We are talking mathematics here, so you need to define what the imperfection you think exists is. Or define the property you think a perfect mathematics has.

What has been described above is a set of properties that has been deduced via a highly developed and strict logic. It hasn’t been designed, and you can get into all sorts of philosophical debates about the intrinsic “truth” of the system derived. Any many of the properties were surprising (at best) to the discoverers. As has been alluded to, you can build up an infinitude of other possible systems and definitions of number, and the properties that they have. However where you get into trouble is that almost all of these have no useful properties, such simple things as being able to count stuff, or do basic arithmetic. The moment you start to demand what you might call useful properties you discover that these properties make slightly inconsistent demands upon a “perfect” notion of numbers. Perhaps the most obvious demand we might place upon a system of numbers is that it is closed. That is, under the operations we define on our numbers, you always get another number. The Natural numbers are closed under addition. Add any two Natural numbers together, and you get another Natural number. But they are not closed under subtraction. Subtract one number from another, and you might get another Natural number, but sometimes you get a result that can’t be held within the Naturals - the result is negative. Now it might be perfectly reasonable to define such results as simply illegal. After all, what is minus one apple? Or a size of minus one foot? It is meaningless. It has no physical manifestation and clearly has no part to play in our universe. Until you want to somehow keep track of deficits. Which is what was done in some systems (accounting placing a deficit in parentheses for instance). But it becomes obvious pretty quickly that your entire system of numbers make much more sense and is generally more consistent if you allow negative numbers, which gets us the Integers. Which is great. Of course you can then decide that you might like to include multiplication in our system, and it is an obvious extension to the idea of addition. And, great, our system is still closed. Then someone decides that they would like the inverse operation of multiplication - division. After all subtraction was useful. And, gosh darn, the system isn’t closed again. But fractions are a useful idea. We can simply retain fractions as notation, and carry the unevaluated division around, sort of like keeping the deficit in parentheses. But again, it becomes pretty obvious that the system makes a lost more sense if we actually allow those fractions to denote fully paid up members of our numbers, and we get the rations. Except for one particularly nasty case. Division by zero makes no sense. You an argue all sorts of questions and ideas about it, but in the end, division by zero can’t denote a useful number whilst still keeping the nice properties we like about our number system. So we have to make a special rule, that division by zero is undefined.

And on it goes. But each step involves additional interactions and edge cases. And sometimes surprising results. The best you can do is be like the constructivists, they simple say that certain “numbers” don’t actually exist. Like Pi. There is some very deep mathematics and logic at play here.

And we have not made it to complex numbers, (or quaterions and octinons.)

Back back to the start - when you define your “perfect” properties on numbers, you are invariably placing formal constraints on the way the system must behave, and you will discover that these can come back to bite you. Creating a system that is able to match every possible constraint on “perfect” is intrinsically not possible. This is deep.

And we can then get onto Gödel and his incompleteness theorem, for even more deep questions. His result came as a serious (and in many places highly unwelcome) surprise.

So I suppose what you are really saying is that mathematics is made up of many tools which have been developed for specific purposes and that the notion of ‘perfection’ has no meaningful definition. Thank you for enlightening me, I’m learning so much here.

[Francis_Vaughan]

@abashed

So I suppose what you are really saying is that mathematics is made up of many tools which have been developed for specific purposes and that the notion of ‘perfection’ has no meaningful definition. Thank you for enlightening me, I’m learning so much here.

Rather the contrary - for a pure mathematician at least, perfection has a great deal of meaning. As indeed it does for many theoretical physicists.
Perhaps the point is that “perfection” isn’t a simple single concept. Not unless you want something trivial. Again, if you want to talk about “perfection” you need to define it. Used without it is a meaningless term.

You will get the word “beautiful” used in maths and physics. That is perhaps closer to what is looked for. Perfection is a slippery idea.

Heck, you might enjoy this book: Why Beauty Is Truth: A History of Symmetry: Stewart, Ian: 9780465082377: Amazon.com: Books

[DPRK]

Perfection and beauty are important concepts. Just not mathematical concepts.

[abashed]

@Francis_Vaughan

Rather the contrary - for a pure mathematician at least, perfection has a great deal of meaning. As indeed it does for many theoretical physicists.
Perhaps the point is that “perfection” isn’t a simple single concept. Not unless you want something trivial. Again, if you want to talk about “perfection” you need to define it. Used without it is a meaningless term.

You will get the word “beautiful” used in maths and physics. That is perhaps closer to what is looked for. Perfection is a slippery idea.

Heck, you might enjoy this book: Why Beauty Is Truth: A History of Symmetry: Stewart, Ian: 9780465082377: Amazon.com: Books

So would it be true to say that mathematics is not just one but many things? (Thanks for the reference BTW).

[abashed]

@DPRK

Perfection and beauty are important concepts. Just not mathematical concepts.

In the eye of the beholder?

[DPRK]

I did not say that; I just meant that books like Kritik der reinen Vernunft are not considered books of mathematics, while books like Disquisitiones Arithmeticae and Principia Mathematica are.

[CurtC]

@abashed

In the second case, where we are defining each of the 3 balls as 0.333333’, we cannot ever remove any of the balls because we are never able to completely define what a ‘ball’ is and would take infinity to do so. We could become incredibly close, yes, but we never actually get there. This seems to be where the logic takes me or am I wrong?

It might be a good time to consider the old chestnut about the number 0.9999999…

Is it exactly equal to one? Not to start up a hundred-page thread again; I think the board has settled on the answer for the most part. And that answer is that 0.9999999… is EXACTLY equal to one. It doesn’t approach one, is already exactly one.

@abashed

Well, the difference is, it seems to me, that a rational number does not allow us to approach its value since there is no necessity to do this, whereas an irrational number gives us the scope to do it.

What about the series 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + … ? Does it approach the number one?

[DPRK]

Decimal notation expresses the (exact) value of any real number in terms of digits; it is just that you need an infinite number of digits (1/5 = 0.2000… or 0.199… for this purpose, e = 2.71828…, etc.)

And you do really need the concept of infinitely many digits, otherwise you are not working with real numbers any more. (I got the impression the OP has accepted this.)

There is a philosophy called finitism, which accepts the existence only of finite mathematical objects, but we really do not want to go there.

[Chronos]

The series is one. The sequence of partial sums approaches one.