Wikipedia:Reference desk/Archives/Mathematics/2014 April 22

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April 22

Likelihood two polling responses are different

I know I've asked s.t. similar before, but I can't find it in the archives.

A poll with error σ reports pro and con opinions of an issue to be relatively close. If the reported figures differ by Δ, what is the chance that the higher result is actually more popular? (We have a lot of reports of popular opinion on WP that I doubt can be justified by the polling we use as the ref.) — kwami (talk) 01:37, 22 April 2014 (UTC)

The problem of computing mean value and standard deviation for the result of polls is solved here: [[1]]. If the first poll shows X₁ ≈ μ₁ ± σ₁ and the second poll shows X₂ ≈ μ₂ ± σ₂ , then compute the difference X = X₁ − X₂ ≈ (μ₁ ± σ₁) − (μ₂ ± σ₂) = (μ₁ − μ₂) ± √(σ₁² + σ₂²) = μ ± σ . Compute the quotient U = μ/σ . If, say, U=2.7, then try www.wolframalpha.com with input CDF[NormalDistribution[0, 1], 2.7] , and get the probability 0.996533. Bo Jacoby (talk) 06:37, 22 April 2014 (UTC).

An excellent resource! Thanks.

You gave an example for two separate polls on the same question, where I was thinking of two responses to a question in the same poll. Will the same approach work? For example, we have a poll that reports 48% support a proposition, and 47% oppose, with a margin of error of ±4.6%. Obviously, that's not going to be significant, but is this how I'd determine that?: U = 1%/4.6% = 0.2174, and wolframalpha calculates CDF[NormalDistribution[0, 1], 0.2174] to be 0.586052.

— kwami (talk) 20:38, 22 April 2014 (UTC)

Often a poll just gives a "margin of error" or "sampling error", or words to that effect. However, sometimes they go on to state that they have 95% confidence in the results. Does this mean that the error is the range for p = 0.05 rather than σ? — kwami (talk) 05:34, 23 April 2014 (UTC)

160.129 gave you the correct answer there. That is, I assume it is a 95% c.i. for the true proportion of the population. If the question was "Do you think guns should be encouraged in children's sport?", then each answer is just scored 1 for "yes" and 0 for "no". Assuming most people think this is a great idea in principle, and it is only the expense that prevents it from happening, let's say you get 72% of people saying "yes". If the pollsters report 0.72 ± 0.04, then they are 95% sure that the true proportion is in that range. But can you state an actual case? Otherwise, I can only tell you I'm about 80% certain, give or take ;) IBE (talk) 04:31, 25 April 2014 (UTC)

I wish I could say I'm astounded, but I'm not. As so often happens with questions of probability and statistics, people are firing up impressive-looking mathematical machinery with apparently no idea what it does, confusing probability, confidence, likelihood, hypothesis testing, and any number of other things. No sample (except a complete, no-replacement census) can tell you the "chance" that an otherwise-unknown population parameter "is" such-and-such. EEng (talk) 08:09, 26 April 2014 (UTC)

x=2^n question

Non math guy here. x=2^n With N being higher than 0, smaller than 22 and a positive integer, what would be the value of N, where X would be closer to the power of 1.5? PS:No, this is not homework. 201.78.176.96 (talk) 18:49, 22 April 2014 (UTC)

So are you saying X=1^1.5? That would mean X=1, and then N=0.--Dreamahighway (talk) 20:42, 22 April 2014 (UTC)

No, he is saying out of the sequence 2^1, 2^2, 2^3, ... 2^21, which of those has the smallest difference with (1.5^k) where k is a natural number. Since (1.5^2) is 2.25 the question is whether there is a power of 1.5 which is closer than .25 to a power of 2. (with the bounds on the powers of 2 given above).Naraht (talk) 21:09, 22 April 2014 (UTC)

... and the answer, of course, is no in absolute terms, but yes as a proportion (1.5¹² is just over 1.3% more than 2⁷). Dbfirs 07:05, 23 April 2014 (UTC)

And that is why there are 12 semitones on an octave on the piano. 12 fifths ≈ 7 octaves. Bo Jacoby (talk) 09:44, 23 April 2014 (UTC).

The natural numbers

We all think the counting numbers that we know and love $(1,2,3,4,5,6,7,8,9,10,andsoon...)$ are the natural numbers. However, higher level mathematics suggests that the more natural definition of the natural numbers is $0,1,e,pi,i$ . It would be more natural to call the numbers we call natural the common numbers. Any thoughts on this statement?? Georgia guy (talk) 19:02, 22 April 2014 (UTC)

What "higher level mathematics"? I'm guessing you're talking about

$e^{i\pi }+1=0$

is that correct? Personally I think that equation is way over-hyped. Sure, it's pretty and all, but it's not that big a deal.

On a slight tangent, I think the reason a lot of people are so impressed with it is that they misinterpret it. It would be mind-bending, if it meant something like "if you multiply iπ copies of e together, you get −1". But it doesn't mean anything of the sort. The exponentiation here referenced is a different sort of exponentiation from the one where you raise one natural number to the power of another. By the way, many, perhaps most, mathematicians now count 0 as a natural number as well. --Trovatore (talk) 19:42, 22 April 2014 (UTC)

I always thought it was the same operation (exponentiation,) only with the domain of the operation extended to other kinds of numbers. Why isn't this description right?? Georgia guy (talk) 19:50, 22 April 2014 (UTC)

Look at the definitions; it's a very different idea, not intensionally the same. Natural-to-natural exponentiation n^m is just taking the product of m copies of n. The exponential function e^x is defined in various ways (by a solution to a differential equation, or as the inverse of the natural logarithm defined as an integral, or as a power series), none of which have anything to do with multiplying together copies of e.

It happens that there's a commutative diagram lying around; if x is the image of n under the natural embedding from N into C, then e^x is the product of n copies of e (or rather the complex number e+i0). But that doesn't change the fact that it's a very different idea, and should be thought of as a separate function. --Trovatore (talk) 19:57, 22 April 2014 (UTC)

Your statement of natural-to-natural exponentiation is still true of integer-to-natural exponentiation, rational-to-natural exponentiation, real-to-natural exponentiation, and complex-to-natural exponentiation. $e^{4}$ can be defined as $e*e*e*e$ the same way $2^{4}$ is $2*2*2*2*$ . Do you deny this?? Georgia guy (talk) 20:12, 22 April 2014 (UTC)

Depends on what you mean. You can define real-to-natural or complex-to-natural exponentiation in terms of a product of <power> copies of <base>, and in that sense, sure, e⁴ means eeee. However, no, exp(4) definitely does not mean eeee. It happens to equal it, once you do the appropriate type-casting, but that is no part whatsoever of its definition. --Trovatore (talk) 20:20, 22 April 2014 (UTC)

You must be saying that $e^{x}$ and $exp(x)$ are not the same by definition, but that they happen to be the same, just like an irrational number and a decimal that doesn't terminate or repeat. Georgia guy (talk) 20:24, 22 April 2014 (UTC)

Well, actually, if I write e^x, I probably ordinarily do mean exp(x), because x is usually used for a real or complex variable, not a natural-number variable. But if I write eⁿ, I probably mean the product of n copies of e, because n is usually used for a natural number.

And right, exp(n) and eⁿ are definitely not defined to be the same. They slightly more than "happen" to be the same; it's a theorem. But it's not the meaning of the notation. --Trovatore (talk) 20:27, 22 April 2014 (UTC)

Yes, I understand that the $x$ in $e^{x}$ is not necessary a natural number, so this is not always real-to-natural exponentiation. However, we can make related definitions for negative and fractional values of $x$ . $e^{x}$ for integer values of $x$ less than 1 can be defined as follows: $e^{x}$ = $e^{x+1}/e$ . For fractional values of $x$, we can define $e^{x}$ so that $e^{ax}$ is equal to $(e^{x})^{a}$ . Values of $e^{x}$ for irrational values of $x$ can be defined as limits. Georgia guy (talk) 20:34, 22 April 2014 (UTC)

That's way over-complicated, and horribly inelegant. Definitions-by-cases are to be avoided when possible. The better way is to consider the notations to be overloaded, but elide the distinctions whenever it is safe and convenient. --Trovatore (talk) 20:37, 22 April 2014 (UTC)

The use of the term "natural number" by working mathematicians is largely conventional. The origin is presumably that it is (in some sense) "natural" to count that way. Note that it is "natural" to start our counting with "1", because that's the first thing we count. By all means think of them as the "common numbers" in your head. Similarly, if you start referring to well-known transcendental numbers as "natural numbers" in conversation with working mathematicians, you will merely cause a communications breakdown. RomanSpa (talk) 20:44, 22 April 2014 (UTC)

But, again I want to raise this point, very often these days "working mathematicians" start the natural numbers with 0. --Trovatore (talk) 21:18, 22 April 2014 (UTC)

In my experience we often use the set of non-negative integers, but at least on this side of the Atlantic we tend to simply call them the "non-negative integers". As I've already said, by convention the term "natural number" strongly tends to be used as a synonym for the positive integers. Since this is a matter of pure convention I imagine you can find exceptions, though I have the strong impression that these are not particularly common. By all means use the term "natural numbers" to refer to the non-negative integers in your work, if you so wish; just remember that since this is an unconventional usage it may lead to misunderstanding. RomanSpa (talk) 06:15, 23 April 2014 (UTC)

I know some mathematicians on this side of the Atlantic, too, who start the natural numbers at zero. Dbfirs 06:39, 23 April 2014 (UTC)

Yeah, sorry RomanSpa, I really think you're just factually wrong on this. It's not an unconventional usage, on either side of the pond. There do tend to be differences based on area of specialization, though; our natural number article touches on this if I recall correctly. --Trovatore (talk) 06:42, 23 April 2014 (UTC)

It probably depends what value is taken by the variable "factually"! As you say, there may be differences depending on what area of maths you're in. Most of my "work" is in statistics, where it's generally very (ahem) natural to start our indices at 1. My main recreational interests are in abstract algebra. Presumably we are both exposed to sampling bias! RomanSpa (talk) 11:46, 23 April 2014 (UTC)

Reading the above I see that e^x and exp(x) refer to different definitions (regardless of the domain of the function). I wouldn't rely too much on that convention. You often see them used interchangeably. I personally thought that the exp (which is ugly) came around because of poor typesetting capabilities. YohanN7 (talk) 21:41, 22 April 2014 (UTC)

I think e^x is mostly used to mean exp(x). However, the latter form is more specific. It's possible that, in some weird context, e^x might mean exp(x log e), for a different branch of the complex logarithm (for example e^2πi could come out to be exp(−4π²) ). So exp is not just for typesetting. If you have an extreme fear of being misunderstood, then exp is the thing to use if that's what you mean. --Trovatore (talk) 21:51, 22 April 2014 (UTC)

What I meant is that what you call exp(x) is sometimes introduced as e^x, without mention of exp, or with mention of exp as an alternative notation. So using exp doesn't automatically make anything completely unambiguous (because it might be unknown to the reader). I see what you mean by e^x possibly having another meaning, but I have never seen it used that way. I still think the historical reason for using exp(X) at all has to do with (sound) laziness and/or typesetting problems. YohanN7 (talk) 22:51, 22 April 2014 (UTC)

Hmm. I disagree (mainly with your last, small comment). However anything is introduced (by a lazy author), any rigorous introduction will involve a clear set of definitions. The notation e^x is often ambiguously interpreted especially in pedagogical settings, and indeed I think that it is almost impossible to define a clearcut convention where there will never be confusion as to what is meant. If you have even half followed some of the (extensive and heated) debates about this sort of topic on WP, you'll realize that the value of having the notation exp(x) to hand serves a needed purpose. It is much shorter than saying "... and I don't mean the number e raised to the x. —Quondum 23:22, 22 April 2014 (UTC)

I'm not trying to establish a convention. From where do you get that? What exactly do you disagree with? All I say is that I think the exp-notation origins in laziness and typesetting problems because it is natural (though ambiguous) to denote exp(X) with e^x because of the coincidental identities alluded to above. Mathematicians don't mind ambiguity or notational abuse (at least not as long as it appears in their own subfields) and as long as everything can be gleaned out from the context. In fact, they love it. In Wulf Rossmanns Lie Groups, An introduction through linear groups you can find the same symbol used three times, denoting three different entities, in one single equation, without all unnecessary and without most necessary parentheses.YohanN7 (talk) 00:17, 23 April 2014 (UTC)

You may be right, I just find that mindset difficult to identify with and tend to assume that at least some others share my preferences. My inclination is to avoid ambiguity as much as possible because of the energy it absorbs; the sheer wastage of time on WP talk pages due to this and closely related notational ambiguities is testament to the misinterpretation it produces. It is not only ambiguity – people are prone to fuzzy thinking especially when fuzzy notation is used. I find it especially painful when people remain blind to this when the distinction between interpretations is pointed out to them, to the extent that I think of Orwell as having a point with the concept of Newspeak in his novel Nineteen Eighty-Four. —Quondum 01:28, 23 April 2014 (UTC)

I think you two are talking past each other a bit. Yohan is talking about what (he thinks) did happen; Quondum is focusing more on what (he thinks) should happen.

That said — on balance I have to go with Yohan, to the extent that there's an actual difference here. I would normally be unembarrassed to define e^x as, say, the sum of a power series, and then when expanding e^e, and I got to the e² term, well, that would mean e·e, with no need to belabor the point. It's only when I really have to worry about the distinction (as, for example, in a discussion like the current one) that I would bother about it. --Trovatore (talk) 01:37, 23 April 2014 (UTC)

Spot on Trovatore. I'm not trying to convince anyone about a certain style, though I prefer the pretty version. But then when I refer to the function, I always write exp, unless I have the physicist hat on, in which case it's e^x again. (Not a mathematician, not a physicist, but I have all sorts of hats to put on:D) YohanN7 (talk) 02:34, 23 April 2014 (UTC)

Returning to the original question: each of the numbers you specify is capable of being specified. For example π can be specified as the sum of and infinite series and i as a solution of a polynomial. The specifications can each be written as a string of symbols. Of all possible strings of symbols, consider just those that are meaningful and identify a number that isn't specified by any shorter string. These strings can be ordered by length (number of symbols), and strings of the same length can be ordered lexicographically. Having ordered the strings and hence the numbers they specify, we can put them into one-to-one correspondence with the natural numbers. Hence the specifiable numbers are countable. Neither the complex nor real numbers are countable (apparently), so almost all of them must be impossible to specify by any means. So I reckon your set of numbers is the countable numbers - but this is of course OR. --catslash (talk) 01:34, 23 April 2014 (UTC)

Or did you want only {0, 1, e, π, i} and not {2, 3, 4,... }?--catslash (talk) 01:39, 23 April 2014 (UTC)

See Computable number. Count Iblis (talk) 01:53, 23 April 2014 (UTC)

Trovatore is badly mistaken in believing that equality between functions depends on how the functions are defined. Functions are equal iff (1) they have the same domain, and (2) they have the same function value for every argument in the domain. Example: If n is a natural number and f(n) is defined by repeated multiplication by e: f(0)=1, f(n+1)=e⋅f(n), and g(n) is defined by the power series ∑_i nⁱ/i! , then f=g. Bo Jacoby (talk) 06:53, 23 April 2014 (UTC).

It is true that functions are extensionally equal given the conditions that you mention. However, first of all, I was talking about intensionality, and second of all, the conditions you give do not in fact obtain here. --Trovatore (talk) 06:56, 23 April 2014 (UTC)

I don't know what you mean by extensionally and intensionally, nor what "conditions do not obtain" means. Bo Jacoby (talk) 09:21, 23 April 2014 (UTC).

Extensional functions are arbitrary associations between points. By default, this is what we mean by "function", but intensional functions, which are given by some rule, are also of interest. In this case, we were talking about the equation

$e^{i\pi }+1=0$

which surely would not startle anyone if interpreted as "the point (iπ,−1) is in this here collection which I'm calling the exponential function". The reason, I conjecture, that it catches people's imagination, is that they try to understand it as being intensionally analogous to the claim, say, 2³=8, which just means the product of 3 copies of 2 is 8.

But it isn't analogous to that. The claim "the product of iπ copies of e is −1" is not some deep mathematical truth. It's just gibberish.

As for "the conditions do not obtain", that means that the domains of the functions being considered are not the same, and in at least one case the values aren't either, even if you allow automatic type promotion. --Trovatore (talk) 09:50, 23 April 2014 (UTC)

Using my physicist hat, I'll refer to a function given an argument as a function in this post. Isn't it true that the function eⁿ, n ∈ N (repeated multiplication) and the function exp(z), z ∈ Z determine each other in a more or less canonical fashion? eⁿ is obtained by restriction of exp, and eⁿ can be generalized to e^q, q ∈ Q. The set of rationals Q ⊂ R is dense and determines a unique continuous function e^r, r ∈ R which can be expressed as a power series with infinite radius of convergence. (I don't think every function of the rationals determine a continuous function of the reals, but this one certainly does, and it is unique.) Analytic continuation then uniquely gives e^z, z ∈ C ≡ exp(z). Using this as a definition of exp, it is no coincidence at all that there is agreement on the common domain. YohanN7 (talk) 12:53, 23 April 2014 (UTC)

This does still not, of course, mean that e multiplied iπ times with itself equals −1, just as little as the values of the gamma function is some sort of factorial everywhere. But it lends support to the idea that e^z is a perfectly acceptable notation, even unambiguous in this particular sense. YohanN7 (talk) 13:05, 23 April 2014 (UTC)

I'd go even further and say that

$e^{i\pi }+1=0$

is a "pretty much canonical" consequence of the properties of exponentiation mⁿ of natural numbers. As such it is "pretty much remarkable". The definition of exponentiation of natural numbers isn't taken from thin air either. It's "pretty much canonical". In fact, the peano axioms, together with our everyday experience of what m + n ought to equal, "pretty much dictate" that Euler's identity holds. As a non-mathematician, I can use this vague terminology without blushing, but I think it's clear what I mean mathematically anyway. It is no coincidence that the Euler identity holds. At no point in the generalization of the arithmetic operations is there much of a choice if "naturality" is to be maintained. YohanN7 (talk) 13:47, 23 April 2014 (UTC)

I'd say no, your repeated multiplication function mⁿ, n ∈ N in no way uniquely determines e^z. Even if you regard a bunch of arbitrary choices as "canonical", they are not natural (depending on how you define this). —Quondum 02:00, 24 April 2014 (UTC)

And Quondum (as usual) says "no you're wrong" just as a matter of principle. Where do I say that "repeated multiplication mⁿ, n ∈ N uniquely determines e^z."? You have a tendency of misreading close to 100% of my posts, not only here, because you are so eager to say "no, you are wrong". I'm right. Else, where is the error? (And don't misquote me.)YohanN7 (talk) 13:51, 24 April 2014 (UTC)

Sorry, I'm not trying to be combative. I am also not saying you are wrong, only saying that that your use of "canonical" implies quite a few choices being made, resulting in the specific function exp(z). For example, that the function f : R⁺ × C → C be everywhere complex-analytic in the second argument and must embed f(m,n) = mⁿ is a very strong requirement, yet a whole family of functions satisfies this. Alternatively, one can demand that the identity f(m,x+y) = f(m,x)f(m,y). In both cases, this is only possible by restricting m substantially. But my disagreement is really only mild: to me it sort of seems that the "pretty much remarkable" is tempered by the feeling that the result is "engineered" by the number of specific choices made during the generalization of mⁿ, n ∈ R, n ∈ N. —Quondum 19:55, 24 April 2014 (UTC)

There is no choice of function f : R⁺ × C → C (we are focusing on e). But if there were, there would be no choices at all. A continuous function is determined by its values on a dense set. We'd want the function continuous when restricted to real arguments. Likewise, analytic continuation of a function from the reals to the complex plane yields unique results, especially when the radius of convergence is infinite as here.

Given e^q, q ∈Q, you would have to make decidedly unnatural choices either resulting in either a discontinuous exponential function for the reals, or with continuous exponential function for the reals, but not analytic for complex arguments. Your only way out is to define e^q, q ∈Q differently, violating our experience with how rational numbers, multiplication and division works. There is little leeway to come up with something different in each step. At least this is so if you want your function to be useful. And, finally, yes, "canonical" means something else in category theory. But this is just as canonical using the English sense of the word, because there is no other natural choice. (The same remark applies to "natural as applies to "canonical".) YohanN7 (talk) 20:56, 24 April 2014 (UTC)

Just to clarify what I think your point is, defining $x^{n}$ as the product of n copies of x for a positive integer n and positive real number x forces one and only one way of defining $x^{1/n}$ so that the usual rules for exponentials are satisfied. This then forces $x^{q}$ for rational q and positive x. Then $e^{x}$ is defined for real x by requiring continuity, and $e^{z}$ for complex z by imposing analyticity. Sławomir Biały (talk) 21:45, 24 April 2014 (UTC)

Yes, that is my point. YohanN7 (talk) 07:58, 25 April 2014 (UTC)

I understand how exp(x) is arrived at as a "nice" function related to xⁿ, including the as-yet unmentioned choice of the specific value of e, via any of a number of routes, including the one mentioned here. Since I have failed to express what I was getting at such that it is even understood, I'll leave it at that. —Quondum 22:28, 24 April 2014 (UTC)

Give it one more try. Is it f : R⁺ × C → C that you see undecided even when demanding analyticity in the second argument? And, from where do we get e? YohanN7 (talk) 07:58, 25 April 2014 (UTC)

Demanding that the function f : R⁺ × C → C be everywhere complex-analytic in the second argument while embedding f(m,n) = mⁿ further restricts the first argument. However, accepting that restriction still gives multiple solutions, e.g. (but not only) exp(kn) with m = exp(k) with integer k. The value e (or nearly equivalently e⁻¹) arises from "niceness" considerations, which will depend on the generalization+restriction of mⁿ. For example, in the real context, where f : R⁺ × R → R can be analytic in both parameters, considerations of differential calculus and/or simplicity of formulae seems to be the obvious motivation for its choice. If I may point out, this is a tangent from both your initial point and what I had tried to communicate. —Quondum 15:13, 25 April 2014 (UTC)

There are two kinds of natural numbers: ordinal numbers and cardinal numbers. Ordinal numbers, the positive integers {1,2,3,...} = {first, second, third, ... }, are for counting, such that the element in a set are identified by ordinal numbers. No element has number zero. The ordinal number of the last element in the set is the cardinal number of the set. Cardinal numbers are nonnegative integers {0,1,2,...} = {zero, one, two, ... }. As the empty set has no last element, it's cardinal number is zero. Cardinal numbers are the subject to arithmetic operations such as

a + b, a ⋅ b, a^b, ${\tbinom {a}{b}}$, a! ,

and to logical comparisons such as

a = b, a ≠ b, a < b, a > b .

Ordinal numbers may be subject to comparison but not to arithmetic. However, in computer programming it is useful to compute addresses, which are ordinal numbers, and so many programming languages use zero-based numbering. In order to reach element number n you have to skip n−1 elements, and so the cardinal number n−1 is used instead of the ordinal number n. Bo Jacoby (talk) 08:35, 24 April 2014 (UTC).

This seems to be your own personal formulation, and you may be able to get something workable out of it, but it is not the usual way that mathematicians look at it. In fact, the finite ordinal numbers are, in the usual (von Neumann) realization, identical to the finite cardinal numbers. You may be able to argue that there is an "intensional" sort of difference between them, but honestly, I have never seen such a distinction that was really worth making. (Once you get to infinite ordinals and cardinals, there is a distinction.)

There is most definitely an ordinal number zero, in the usual formulation, and you can definitely add, multiply, and exponentiate ordinals. You should check out the links I gave, for details.

As I say, this is not to cast aspersions on the way you look at it; if it works for you, that's fine. But it is not standard. --Trovatore (talk) 08:45, 24 April 2014 (UTC)

I am not talking about infinite ordinals and cardinals, but about elementary finite ordinals and cardinals. Now is the twentyfirst century. The first century was the first in the current era, and the century immediately before that was the first century before the current era, BCE. There was no zeroeth century. Zero is not an ordinal number. One century plus three centuries equals four centuries, but the first century plus the third century is not equal to the fourth century. The sum of two ordinal numbers has no interpretation as an ordinal number. Bo Jacoby (talk) 07:40, 25 April 2014 (UTC).

That's an artifact of human language; it has little to do with mathematics. That sort of "ordinal" is not a mathematical object, just a kind of word. --Trovatore (talk) 07:46, 25 April 2014 (UTC)

Trovatore thinks that counting has little to do with mathematics. I disagree. Bo Jacoby (talk) 18:53, 25 April 2014 (UTC).

I did not say that. The sort of ordinals you talk about have little to do with mathematics. That's why our article about them is at ordinal number (linguistics). --Trovatore (talk) 19:22, 25 April 2014 (UTC)

Ordinal numbers are not operated by addition, multiplication etc, but only logically by comes before, comes after, and equals. So they have 'have little to do with mathematics'. But they are the reason why 'the counting numbers that we know and love, (1, 2, 3, 4, 5, 6, 7, 8, 9, 10, and so on...) are the natural numbers' as the OP wrote above. Ordinal fractions (see www.statemaster.com/encyclopedia/Ordinal-fraction), however, are operated mathematically, but that is another story which cannot be told on wikipedia because Trovatore does not yet know about it. Bo Jacoby (talk) 03:14, 26 April 2014 (UTC).

You do realize that "ordinal number", in a math context, is going to almost universally refer to the set theoretic notion - which does support addition, and other such.Phoenixia1177 (talk) 03:30, 28 April 2014 (UTC)