Pi

Factual Questions
101

Derleth said that they don’t exist in familiar number systems. You (and Indistinguishable) are certainly correct that there are number systems in which they do exist, but those number systems aren’t nearly as familiar as the ones where they do not. You can even do a large amount of computer programming without having to worry about the details of how your computer represents negative numbers.

That said, though, such unfamiliar number systems are still cool, and ultimately, that’s the point of mathematics anyway.

102

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.

103

I’d say “yes,” if you take the approach that the natural numbers are a subset of the rational numbers, which are subset of the algebraic numbers, which are a subset of the real numbers, which are a subset of the complex numbers…

But I think some mathematicians might instead say, at least in some contexts, that the real numbers map to the set of complex numbers, or that the set of rational numbers is really a set of equivalence classes of ordered pairs, or things like that that muddy the waters.

Nitpick: Zero is not one of the natural numbers—unless it is. Some people consider the natural numbers to be {1, 2, 3, …}; others, {0, 1, 2, 3, …}. It’s one of those things where there isn’t a universal standard, and if it’s going to be important you need to be careful to define your terms.

104

Pi is more popular than e ? Maybe what e needs is a better publicist! pi has circles, spheres, sine waves … who could resist?

But e ? If the Ponzi Bank offers 100% annual (simple) interest, at the end of a year your $1 has changed into $2.
But if they compound daily – no, compound it every millisecond – you’ll get $2.718281828459 back.

Does e have any better, “everyday” publicity stunts than that?

105

There is a unique zero element (additive identity) within any given ring. Also, if f: R → S is a ring homomorphism (map between rings), then f(0) = 0. So zero is always zero.

In an example like the surreal numbers mentioned above, there are infinitesimally small positive numbers, but those are still non-zero.

106

PLEASE do not show this guy the Cecil Adams column about .999… = 1 !! :eek::eek:

You persist in being shackled by the notion that decimal representations of numbers have some a priori special value. They don’t. We don’t care that the decimal representation of π can never be completely known. The value of π is quite well known: it’s C/d, where C = circumference of a circle and d = that circle’s diameter. We do a whole lot of math with π without ever worrying about the decimal representation. Indeed, one of the blessings of π is that we can do all sorts of math where we never even bother to try and use that representation.

For example: suppose you take a square with sides of length 2. Inscribe in that square a circle. Calculate the area of the square not included within the circle. The answer is easy: 4 - π. As a mathematician, I don’t need to do anything else with that answer, and, indeed, trying to reduce it to some decimal value may be counter-productive, as I may need to use it to complete further calculations. As an engineer, trying to build such a thing, I might need to come up with a decimal approximation, in which case I would decide how much tolerance for error I’m allowed, determine how many decimals of π I need to accomplish that, and calculate away.

The value of sqrt 2 is exact. It’s not the least bit unknown. It’s precisely the value that, when squared, produces 2. To say it’s not exact would mean that you would be in SERIOUS trouble when confronted with the concept of i (defined as being the sqrt (-1). That’s a value that decimals cannot even begin to handle. :rolleyes:

107

The ancient Greek point of view was indeed that sqrt(2) is obviously not a number, but that was then and this is now.

Compare, by the way, the following situation: you mathematically prove that an object, say a number, satisfying a certain property exists, but the proof is completely non-constructive. I can imagine some people have a philosophical problem with that.

108

I want to grab at this and see if abashed can’t benefit from a brief and from memory history/math lesson.

Math started out dealing only with the tangible, with the most tangible being counting numbers. You can have one sheep, two sheep, three sheep and so on.

You could also measure stuff and have one jug, two jugs, three jugs, one rod, two rods etc.

Now if you had less than a full jug, or a full rod, you had two options. You either used a different, and smaller, additional unit, like a thimble and a thumb, or you used fractions.

To some extent perfect fractions don’t always exist as tangible units. In practice you can’t get perfect halves, or thirds, or sevenths. But people still readily accept 1/2, 1/3 and 1/7 as just as real as the counting numbers.

But even back then people dealt with physical realities that defied these acceptably nice numbers.

Take two rods and stick them end to end at 90 degrees. What’s the distance between the ends? It’s sqrt(2). If we accept the reality of fractions, then we pretty much have to accept there really is a sqrt(2), which is equal to the length of the hypothenus in a 1 by 1 right triangle.

Pi is another such a number. If you take a string, create a circle with the length of the string as diameter, the string will go round 3 whole times, 1 tenth of a time, 4 100ths of a time and so on.

Now there is a circumference to such a circle, and there is a diameter, so although you can’t have both of those expressed as a fraction or whole number at the same time isn’t it just as accurate to say that the ratio between them exists as it is to say the square root exists?

109

Except that in the world world there exist neither perfect circles (nor straight lines nor right angles) nor arbitrarily precise measurements. The distinction between rational and irrational isn’t one that arises from real-world measurement but from deductive reasoning about the ideas behind such things.

110

I’m reminded of the fun little proof that it’s possible for an irrational number taken to an irrational power to be rational.

Consider A, defined as sqrt(2). We know that that’s irrational (proven upthread).
Now consider B, defined as (sqrt(2)^sqrt(2)). Is that irrational? I don’t know. We’ll get back to that.
And now consider C, defined as B^sqrt(2). Is that irrational? OK, this one I can answer. (sqrt(2)^sqrt(2))^sqrt(2) is, by properties of exponents, equal to (sqrt(2)^(sqrt(2)*sqrt(2)). We can simplify that to sqrt(2)^(2), and in turn to 2. C = 2, and we know that’s rational.

OK, so let’s look at B again. Is it rational? Well, it either is or it isn’t. If it is, then sqrt(2)^sqrt(2) is a rational number, and thus it is possible for an irrational number to an irrational power to be rational. Alternately, if it’s irrational, then B^sqrt(2) is again an irrational number to an irrational power, and we can again conclude that it is possible for an irrational number to an irrational power to be rational. So no matter whether the number B is rational or not, it still leads to the same conclusion.

(actually, it is known: Our number B is not only irrational but transcendental, but it’s still neat that we don’t need to know that)

111

I’m not trying to argue rational or irrational, I was trying to illustrate that without accepting some level of “perfect abstraction” fractions aren’t real either. Abashed appears in some posts to distrust pi and other irrationals due to an attachment with the obviously countable and divisible, and I thought this path might lead to the realization that anything between counting requires abstracting from the physical world to the abstract world of math.

112

Before I read your fun little proof, my thought was “of course it is.” If you take the “πth” root of, say, 0.85, it would be really really surprising to find out that it’s a rational number.

113

Aye, and so in intuitionistic logic, this would not count as a complete proof. But there is, albeit less fun for illustrating the same principle, a very simple constructive proof that an irrational raised to an irrational can be rational: consider, say, X = sqrt(2) and Y = log(3)/log(sqrt(2)). Both X and Y are irrational by considering prime factorizations, but X^Y (i.e., 3) is clearly rational.

[Of course, many other examples could be constructed along the same lines; nothing specific to 2, 3, sqrt, etc., here]

114

There’s also the simple cardinality argument (there’s too many irrationals for every irrational root of a fixed positive non-1 rational to be a distinct rational, and so many irrational powers of irrationals are said fixed rational), which is also ultimately constructive, but analytic rather than algebraic.

115

abashed may already be satisfied but in case he isn’t or someone else is curious here is another way to think of irrational numbers like pi or sqrt of two.

Think of them as points on the number line. Now if I have a point X on the number line, I can define the set M as being all numbers larger than X and a set N as being all numbers smaller than X. Conversely if I have two sets M and N such that all the points in M are larger than all the points in N, and there is no interval between M and N, then they must meet at a precise number. So I can in effect define a number in terms of the sets of numbers bigger than it and the set of numbers smaller than it. Further, since the rational numbers are dense, there is no gap that doesn’t contain at least one rational. So all that is necessary to define a number is to be able to take any rational number define whether it is larger or smaller than X.

For the sqrt of 2 this is easy.

M is the set of a/b for which a^2 > 2b^2
N is the set of a/b for which a^2 < 2b^2.

and the sqrt(2) is where those two sets meet.

For pi the sets may be a bit more complicated to define, but the theory is the same.

116

It’s also possible to use decimal expansion to define such sets. For pi, for instance, the set M would be {4, 3.2, 3.15, 3.142, 3.1416, 3.141593, 3.1415927, …}, while the set N would be {3, 3.1, 3.14, 3.141, 3.1415, … }

117

First, yes, I’m aware of the p-adic numbers. I didn’t mention them specifically because I was talking to a poster who still exhibits difficulty separating a number from its base representation.

So let’s get back to that.

abashed, there are numbers which are repeating “decimals” in one base but not in another, and I use the term “decimal” here because there’s no better term for the concept: 0.1 in base 2, meaning 1/2, is a fraction, yes, but so is the string “1/2”, and confusing them is… confusing.

So. One-tenth is a repeating decimal in base two. This has actual real-world consequences for people who do arithmetic on computers, since they represent floating-point values in base two, and it’s surprising when dividing by ten has the same problems in base two that dividing by three does in base ten.

Does that make one-tenth any less precise? Is one-tenth precise in base ten and fuzzy in base two? Because that’s nonsensical. Numbers are independent of their representation.

So the value of one divided by ten is one-tenth. It can be written very compactly in base ten, and not at all in base two. The value is one-tenth regardless.

118

(For what it’s worth, when these threads arise, I’m often more interested in the conversations they spur among everyone, rather than just narrowly tailored to the OP. Hence most of my posts in this thread have not been OP-directed. Other people seem to have the OP covered. Anyway, thought I’d explain that to excuse my whatever I’m doing in this thread.)

119

Numbers don’t have “ends” you can “come to.” They don’t “go on,” they don’t “progress”, and they don’t “change.”

Pi never changes. Its value is always and forevermore constant, like all numbers. It is you who changes your approximation of it, like a traveler moving toward a still object. People don’t calculate pi, they calculate digits of pi, a completely different thing. Pi doesn’t “go on” anymore than the Taj Mahal is constructed during and by my flight to India.

While we’re on the subject, I insist there is a base-1 number system. It’s that tick-mark writing system you use to keep track of points in a card game or victories at the pool hall. Y’know, the one where each fifth mark is a diagonal line across the other four. The first tick mark is 1^0, the second, 1^1, and so on. You don’t need symbol for “null” when we already agree it’s the empty space, and if you think about it, what purpose could a 0-value have anyhow, if each digit is 1^n?

120

Yes, many people use “base 1” or “unary” to mean this same thing (as do I!). Of course, it is distinguished from how people use “base b” for whole b > 1 in that instead of restricting to natural number digits less than b, we are now allowing b itself as a digit (and none others, except perhaps implicitly for the leading zeros!).