text:
Talk:Indeterminate form

comment by Taku
Isn't 0 over 0 is a determinate form because if you keep multiplying very small number very few times you just get 0. -- Taku 02:25, Nov 15, 2003 (UTC)

0/0 is indeterminate since if it had some definite value, say x, so 0x=0. But any number satisfies that equation for x, so 0/0 has no definite or determinable value. Dysprosia 02:28, 15 Nov 2003 (UTC)

 * Umm, elegant! Thanks a lot, Dysprosia. -- Taku

 * But outside the realm of limits, there is very little practical use in calling 0/0 indeterminate, versus just grouping it with a/0 undefined forms (a != 0)

In the sense of this article, 0/0 is indeterminate because if f(x) and g(x) both approach 0 as x approaches something, then f(x)/g(x) can approach a number that depends on which functions f and g are. (That does not detract from the elegance of Dysprosia's observation, however.) Michael Hardy 20:18, 15 Nov 2003 (UTC)

I've just moved some material from zero divided by zero to this page performing significant changes to it to make it apropriate for this page. I am planning to move more material in a few days time but not necessarily to this page and maybe to a subpage of this page. Barnaby dawson 19:56, 20 Sep 2004 (UTC)

After an edit clash, I've added to the second paragraph. Charles Matthews 11:05, 18 Mar 2005 (UTC)

I've now done further work on that discussion. Charles Matthews 11:16, 18 Mar 2005 (UTC)

Missing Infinity ^ Infinity
The form oo^oo is missing from the list.

 * How can ∞∞ be considered an indeterminate form? If f(x) and g(x) both approach ∞ as x approaches a, then f(x)g(x) can only approach one thing, ∞, regardless of which functions f and g are.  So it's not an indeterminate form. Michael Hardy 23:34, 18 December 2005 (UTC)

It can be indeterminate in a weak sense, like 1/0 (but perhaps a bit stronger). For example, as x approaches 0 from the right, (1/x)1/x approaches infinity (from the left), while (1/x)-1/x approaches zero (from the right). But nothing of this form can converge to any non-zero finite value (because the logarithm must converge to infinity). I'm not sure if this should count as indeterminate or not; the precise definition that I've just written says that it is, but one could easily fix that so that it's not. --Toby Bartels 01:57, 23 May 2006 (UTC)

 * Remember that when one says ∞, one means "positive infinity" (i.e. increasing without bound). So your example is not the same because it raises "positive infinity" to "negative infinity". --Spoon! 11:33, 31 August 2006 (UTC)

 * What one means depends on context; compare the extended real line to the real projective line. See also this formal definition, which uses the Riemann sphere. —Toby Bartels 22:03, 20 September 2006 (UTC)

On merging with defined and undefined
User:Michael Hardy, at 15:37, 18 December 2005, wrote in this diff:

 * This merger proposal has no merit. Too many readers keep failing to see that "indeterminate form" does NOT mean something that is undefined. Maybe we should hit that point much harder.

(copied here for future reference by Oleg Alexandrov (talk) 00:16, 19 December 2005 (UTC))

 * To be honest I'm not exactly sure what MH's getting at there. An indeterminate form per se is not undefined, but its value is, and they're pretty much coextensive with the expressions treated at defined and undefined. The latter is just a bad article with a bad title, and I think the real point of the merger was to figure out some way to get rid of it, a project I'd support. --Trovatore 16:58, 13 February 2006 (UTC)

 * No -- they're NOT coextensive, since 1/0 is NOT an indeterminate form. 0/0 is an indeterminate form; 5/0 is not.  That means that if f and g both approach 0 as x approaches something, then f/g could approach anything.  That is not true if f approaches 5 and g approaches 0. Michael Hardy 22:08, 13 February 2006 (UTC)

 * OK, true, 1/0 is not an indeterminate form, but it's not enough by itself to justify a separate article from this one, particularly one with as bad a name as "defined and undefined". And all the indeterminate forms are denotationally undefined (unless you're going to take the 00=1 approach; I'd be against presenting that as the default approach, though it ought to be mentioned). --Trovatore 22:16, 13 February 2006 (UTC)

Determinate and inderminate discussion
The "Discussion" section had a problem paragraph (reading "By contrast "1/0" is not an indeterminate form because there is no range of different values that f/g could if f approaches 1 and g approaches 0."). There was at least a missing word (between could and if). I attempted to correct that and add more detail to clarify it, but then realized I still don't understand the distinction which is the whole point of the article.

Several times 1/0 is contrasted with 0/0, but the only characterization of this contrast is that there is "no range of different values" without elaboration. But f/g could have a limit of either +∞ or -∞ depending on whether the function g approaches 0 from the + or - side. So there seems to be more than one "value" possible among limits approaching 1/0. Just how is this determinate while 0/0 is not? Whatever the answer, I think the article needs improvement in explaining this. -R. S. Shaw 00:58, 13 March 2006 (UTC)

I think that the whole article is rather unclear in a few ways, first of all that this is fudnamentally about limits (although of course, that is not your problem, R. S. Shaw). With this in mind, I'm going to rewrite it a good deal right now, and I should be able to address your (Shaw's) problem at the same time. -- Toby Bartels 20:29, 22 May 2006 (UTC)

Lead sentence
The lead sentence says
 * In calculus and other branches of mathematical analysis, an indeterminate form is an algebraic expression whose limit cannot be evaluated by substituting the limits of the subexpressions.

This is clearly wrong - an indeterminate form may be obtained by substituting limits of subexpressions, but the form itself is just a few symbols and doesn't have a "limit", it has a "value" (or more precisely, it does not have a value). That is, the question "What is the limit of 0/0?" is meaningless. CMummert 02:22, 29 December 2006 (UTC)
 * Yep. How about:
 * …an indeterminate form is an algebraic expression representing a limit that cannot be evaluated by substituting the limits of the subexpressions.
 * –EdC 17:02, 29 December 2006 (UTC)

Untitled comment
I think the situation is being looked into the wrong way. Say you had (x^2-9)/(x+3). Someone that didn't know that x+3 is a conjugate of x^2-9 would tell you that -3 gives an undefined answer. Yet someone who factored would get x-3, and say that all values are defined. I take this to mean that in a situation like x^2/x, 0/0 is 0. However, in the cases where it's 0(a)/0, where a is any number except 0, the result is a as 0/0 would negate eachother and equal one. An example is above. And don't gimme a "but my calculator says no." k, I want to know what you guys think. Also, another example is x/x, every number outside of zero gives a 1 answer, so it would be expected that 0/0 gives a one answer. This even works with x^2/x. The reason (0,0) is a point, and not (0,1), is because when you expand ((0)^2)/0 you get 0*0/0. 0/0 is one, and that leaves 0*1, which gives 0. —Preceding unsigned comment added by Special:Contributions/ (talk)

 * Well, some of your terminology is nonstandard: you say "conjugate" when you mean "factor" and "negate" when you mean "cancel". But the "0"s in the expression 0/0 do not cancel each other.  When you have, e.g., 3/3, the two "3"s cancel each other because you can divide both the numerator and the denominator by 3.  But in 0/0 you cannot divide both the numerator and the denominator by 0.  But I'm not sure what your question is otherwise.  It is certainly true that x/x approaches 1 as x approaches 0, and x2/x approaches 0 as x approaches 0, and (x2 − 9)/(x + 3) approaches −6 as x approaches 3.  But what is in the article that is being "looked into the wrong way"? Michael Hardy 00:45, 4 January 2007 (UTC)
 * Conjugate is being used the right way, just outside of complex numbers. I'm using the other factor to remain with a 2 variable-number combination (or however you want to say it). The article says that 0/0 is 0 when it's X^2/x (or just call it x if you want), but x/x is not 1 when x=0. I say it's being looked at the wrong way because if you have 0^2/0, it expands to (0/1 * 0/1) / 0/1. and since a top and the bottom 0/1 cancel out, you get (0/1 * 1) / 1 which is zero. However, when you have x/x, any real number / any real number is 1. It would also make sense that if you had (0/1) / (0/1) it would equal one. And if that doesn't make sense, a perfect example is limits. If you do the differential equation, you have to cancel out the h on the bottom (if my equation's the same as yours) with the hs' on top. This gives you the formula for any point. But in any situation, if you used the original formula with h=0, You get the numerator over 0, which be definition is indeterminate or at the minimum undefined, which refutes the ability to use limits on points. Therefore this means that division comes before multiplication when using the order of operation because otherwise, with the above x/x and differential equation, the equations wouldn't be true in any situation, so the only thing can be true is 0/0 is 1 —The preceding unsigned comment was added by 66.66.92.167 (talk) 23:12, 8 January 2007 (UTC).
 * No, the article says that x^2/x has the limit 0 as x tends to 0, and x/x has the limit 1. Neither formula has a value at x=0.
 * You can't cancel out zeroes from a fraction; you can only cancel out non-zero common factors.
 * I don't understand which differential equation you're referring to.
 * I don't think refute means what you think it means.
 * I don't understand your last point. In mathematics the order of operations is explicit. –EdC 00:32, 9 January 2007 (UTC)

 * You can cancel out zeroes, which is why the limits are what they are. The differential equation I'm referring to is the basic slope formula or whatever other areas call it. When you use f(x) instead of y, the equation is dy/dx = (f(x+h) - f(x)) / h. This is the equation used to find the slope of a line between two points. When this is done with any equation with a leading x power > 1, you result has xs and hs. To determine the "instantaneous veloctiy", "slope", whatever you want to call it, 0 is substituted for every h value because you want the slope at that point. Now this part is crucial. Slopes work, and they are proven to work, otherwise we wouldn't use them because they wouldn't work. To say that you can't divide by 0 makes the whole beginning equation moot. And yes, the definition of refute is "to prove to be false or erroneous, as an opinion or charge." You would be refuting the whole ability to find the slope on a given point, which is a significant part of calculas. I know you will rant about "Slopes work, and they are proven to work, otherwise we wouldn't use them because they wouldn't work." but there is no other way to word it. It's like telling me to prove adding works. It works because it does, it doesn't need a reason. —The preceding unsigned comment was added by 66.66.92.167 (talk) 02:17, 17 January 2007 (UTC).

 * No. As the article on differentiation describes, the slope of a line is found by taking the limit of the equation (f(x+h) - f(x)) / h as h tends to 0: $$\frac{dy}{dx} = \lim_{h\to 0}\frac{f(x+h) - f(x)}{h}$$.  At no point is a division by 0 performed.  Imagining that taking a derivative involves cancelling zeroes leads to the same mistakes that motivated the development of mathematical analysis on a rigorous footing. –EdC 03:11, 17 January 2007 (UTC)
 * Note that you can, if you believe in non-standard analysis, set h to be an infinitesimal. However, that's not the same as zero. –EdC 03:19, 17 January 2007 (UTC)

List given in introduction
I've changed "The indeterminate forms include [...]" to "The indeterminate forms are [...]" on the assumption that the list is exhaustive. Previous wording admitted interpretation of list as a subset.

If the given list is not exhaustive, I believe the best compromise is "The indeterminate forms (as described this article) are [...]". That is to say, whether or not the list is exhaustive, the current wording is needlessly ambiguous. 23:29, 7 March 2007 (UTC)

List given in introduction (second concern)
A separate critique is that it seems better to say "are expressed by/using/as" or "are referred to by/as" in place of "are". It may be confusing to many readers to say such-and-such is 0/0 as it might seem to imply (as we almost always intend when writing such expressions) that 0/0 represents a number rather than a form. I have not made this change; I don't know what wording best resolves this potential misreading. Please discuss here. 23:35, 7 March 2007 (UTC)

 * The real problem is that although numerous books use the term "indeterminate form", I have never seen published a definition of one. Can you think of a textbook that defines them? CMummert · talk 00:25, 8 March 2007 (UTC)

Indeterminate form, but continuous
Example: Define f as f(x)=(x+3)(x-3)/(x-3) for x other than 3, and f(x)=6 for x=3. This function is continuous at x=3, even though its limit as x goes to 3 is of the indeterminate form 0/0. FilipeS 18:06, 11 June 2007 (UTC)

 * The indeterminate form is because the division function is discontinuous at (0,0). Note that the indeterminate form is "0/0", not 'f(0)', which is not an indeterminate form. Also, the sentence you are editing is describing a binary operation, not a unary function as your example above provides. — Carl (CBM · talk) 20:49, 11 June 2007 (UTC)
 * Yeah, but the passage about continuity and indeterminate forms in general is problematic anyway. I'd go so far as to say it's original research. As far as I know, the literature dealing with indeterminate forms never gives a general definition of "indeterminate form"; it simply enumerates them. --Trovatore 21:44, 11 June 2007 (UTC)
 * It's more a question of how closely you want the sources to match the phrasing of the article. Searching on google books, I can find several sources that are very close to the statement here:   .  I don't think it's an unacceptable deduction from the statements in these references and what is in the article. I'll clarify that the term is ordinarily only used for algebraic expressions. — Carl (CBM · talk) 23:42, 11 June 2007 (UTC)

 * Carl, that is needlessly obscure. "Indetermination" is a purely formal, and not entirely rigorous concept, yet a useful one. 0/0 is an indetermination, right? Trying to compute directly the limit of f as x goes to 3 gives 0/0, right? Then it's an indetermination! And yet the function is not discontinuous, as one can easily see by solving the indetermination. The article was wrong. FilipeS 11:56, 12 June 2007 (UTC)

 * 0/0 is an indeterminate form because the binary division operation is discontinuous. Your function f is not a binary operation. Your revert also removed the clarification about what is meant by algebraic operations and the important fact that the extended real line is being considered. I added the word binary in a few more places to further clarify that this paragraph is referring to binary operations only. — Carl (CBM · talk) 13:42, 12 June 2007 (UTC)

What is your source for claiming that being an indeterminate form has anything to do with the arity of the operations involved? I've never heard of such a notion! Exponentiation is a unary operation, yet 1 to the power of infinity is still an indetermination. FilipeS 13:53, 12 June 2007 (UTC)

 * Exponentiation (a^b) is a binary operation. Every indeterminate form involves a binary algebraic operation: 0^0, 0/0, 0 * infty, etc. — Carl (CBM · talk) 14:02, 12 June 2007 (UTC)

It depends on how you define it: the exponential function is unary. And you can regard division as binary, f(x, y) = x/y, or unary, g(x) = x/b, or h(x) = a/x. Which is why bringing arity into the conversation makes no sense to me. Indeterminate forms have to do with calculus and limits, not algebra and arity. FilipeS
 * But there are no indeterminate forms of the form exp(_). And for an indeterminate form of the type $$0/0$$ to exist, both divisor and dividend must be free (to range as a function of x); if one is fixed as in your h(x), g(x), the indeterminate form does not exist. –EdC 22:41, 12 June 2007 (UTC)

 * f(x)=1^x and g(x)=0^x can also be called exponential functions.
 * And what do you suppose you get if you apply exp to 0/0?...
 * As for your other objection, the numerator and the denominator in f(x)/g(x) are both free to vary. You don't need a two-variable function to get that. FilipeS 23:28, 12 June 2007 (UTC)

 * If you apply exp to 0/0, you get to deal with the indeterminate form 0/0 and then apply exp. JumpDiscont (talk) 04:15, 5 January 2010 (UTC)

Generality, abstraction, sophistication
(This goes back to Carl's remark that "It's more a question of how closely you want the sources to match the phrasing of the article.") I don't think that's the whole issue. While no doubt someone has done it, I don't believe it has ever been standard to give a general, abstract definition of "indeterminate form". If you want to discuss discontinuous funcions, you talk about discontinuous functions; the "indeterminate form" terminology is a fifth wheel when you get to that level of abstraction.

Historically the phrase has been used to give calculus students a way of detecting situations where they have to apply extra care when computing a limit, without bringing up anything as sophisticated as the definition of a continuous function from R2 to R. It's an enumerated collection of situations; no rule is given for extending it. I think that should be the focus of this article. Something like, say, ei∞ could easily be added to the list, but historically, it has not been.

If generalizations can be sourced, they can be added, but they should be clearly marked as generalizations of the notion, not part of the standard interpretation of "indeterminate form". --Trovatore 20:13, 12 June 2007 (UTC)

 * Another example of an expression which could easily be argued to be an indetermination, but corresponds to a clearly unary operation, is arg(0), where 'arg' stands for the complex argument. Although some authors apparently define it as zero by convention, arg(z) can actually approach any real number as z goes to zero, depending on the path followed by z in the complex plane. FilipeS 20:38, 12 June 2007 (UTC)
 * But you're still arguing about the details of the presentation of the general case. My point is that, historically, there is no general case. There are exactly seven indeterminate forms. --Trovatore 20:51, 12 June 2007 (UTC)

 * I'm agreeing with you, if you haven't noticed. As for the number of indeterminate forms, it depends on whether you regard infinity and minus infinity as separate entities, or identify them with each other. FilipeS 20:53, 12 June 2007 (UTC)

I would like to clarify the article, but apparently I'm not succeeding. I agree there are only seven indeterminate forms. If the purpose of that paragraph was only to generalize them, it should be removed. But I think it can be salvaged, and has merit, as an explanation of why these particular seven expressions are indeterminate but the rest aren't. The point is that they are the only expressions involving one of the operations of addition, subtraction, multiplication, division, and exponentiation and two values from the extended real numbers such that the value of the operation on those two values is not determined by continuity, and therefore it isn't possible to commute the limit across the operation. That point ought to be conveyed by the article, but if the current paragraph isn't doing it, I'll be glad to rewrite it from scratch. — Carl (CBM · talk) 23:44, 12 June 2007 (UTC)

 * My advice: don't try to explain. It's fairly obvious why, and I see nothing to be gained from making the notion of 'indeterminate form' strict. If some mathematician somewhere decided to state that arg(0) is indeterminate, what would be wrong with that?... FilipeS 23:52, 12 June 2007 (UTC)

 * Please note I am not in any way trying to make that paragraph a definition, only an explanation. Also there is complete agreement at present about the term "indeterminate form" in every reference I have looked at, and I have looked at quite a few in the process of discussing this article in the last few months. — Carl (CBM · talk) 00:25, 13 June 2007 (UTC)

 * I don't feel that the status of the indeterminate forms as the discontinuities of the standard arithmetical operations on the extended reals is in any way obvious, especially to the lay reader. I agree that the seven indeterminate forms are pretty much fixed, and would support User:CBM's proposed rewrite. –EdC 21:58, 17 June 2007 (UTC)
 * I'm against getting overly clever at explanations. Note also that if we refer to discontinuities, we have to be careful, as there is no such thing as a function being discontinuous (or continuous) at a point not in its domain. The proper phrasing is that the function does not have (respectively, has) a continuous extension containing that point. And I think that's getting too involved for this article. --Trovatore 22:04, 17 June 2007 (UTC)

 * I agree. Plus, f(x, y)=x/y does not have a continuous extension for y=0, x nonzero either, yet the article claims that x/0 is not an indeterminate form. FilipeS 17:38, 19 June 2007 (UTC)

I'm confident that the four people in this discussion all understand perfectly what's going on with indeterminate forms. The question is: if someone in 11th grade asked you for a one-sentence explanation of why these seven forms are called indeterminate but the others are not, what would you say? — Carl (CBM · talk) 18:58, 19 June 2007 (UTC)

 * I would say that indeterminate forms are mathematical dead ends. When one shows up in a limit you're trying to compute, that's a sign that you must turn back and take a different route to the calculation. FilipeS 01:03, 22 June 2007 (UTC)

 * Surely her next question would be why they are dead ends? — Carl (CBM · talk) 01:13, 22 June 2007 (UTC)

 * To convince her, I might give a few examples of different functions whose limit at a given point leads to some indeterminate form (say, 0/0), but that when computed in a different way turn out to be different numbers. FilipeS 01:18, 22 June 2007 (UTC)

 * You're beating around the bush. It's fully reasonable, and expected, for an encyclopedia article on indeterminate forms to do more than give a list, an example of two limits, and no insight whatsoever. Doing so is really a disservice.  I'm planning to remove the paragraph in question - in fact, I'll do it right now - but it should be repalced by something that conveys some intuition about why these forms are indeterminate but others are not. — Carl (CBM · talk) 01:31, 22 June 2007 (UTC)

 * It's not very civil of you to ask a question and then say you don't like the answer. This is your pet article, not mine. You sort out the OR mess you've made of it. Don't expect any help from me. FilipeS 01:33, 22 June 2007 (UTC)

 * I don't agree there was OR in the previous version - I've read a lot of published sources and it was a relatively clear statement of the general principle. If you don't wish to help, I encourage you not to respond to my requests for comment. — Carl (CBM · talk) 01:39, 22 June 2007 (UTC)

 * Funny, then, how you can't seem to be able to come up with any references, or even a coherent definition. If you'd taken any longer to delete that paragraph, I would have done it myself. FilipeS 01:51, 22 June 2007 (UTC)

false statment that should be deleted/fixed (by BrainScratcher)
in the 0^0 form it sayed that if f & g both analytic and g in not fixed zero in the nighberhood of the limit than f(x)^g(x) is always 1 in that limit which is just false it is often the case but consider the example: f(x)=3^x which is positive for all x & g(x)=1/x both zeroing at negative infinity but f(x)^g(x) is clearly 3 whats up with that ?
 * Your function f is not analytic in a neighborhood of infinity (in fact, it's not even continuous at infinity), and in any case the statement talks about neighborhoods of a point on the complex plane. Infinity is not a point on the complex plane (although it is a point on the Riemann sphere). --Trovatore (talk) 00:50, 25 September 2009 (UTC)

negative infinity over negative infinity
I didn't see any mention of negative infinity in the article. I only mention this because my Calculus book (Calculus - Early Transcendentals 6th Edition by James Stewart) says that -∞/-∞ is also indeterminate. Also, what about the forms of -∞/∞ and ∞/-∞? Are they all indeterminate forms? Thanks. BuddhaBubba (talk) 00:03, 20 July 2010 (UTC)

 * Yes they're all indeterminate,they're all just variants of ∞/∞, the signs make no difference in a multiply or divide except to determine the final sign. Dmcq (talk) 09:26, 20 July 2010 (UTC)

Tan(90 degrees)
Would Tan(90 degrees) be an indeterminate form. The tan wave goes to infinity at 90. Also if you imagine a right angled triangle and imagine the adjacent side getting smaller and smaller theta gets closer and closer to 90 making tan(theta) closer and closer to tan(90) and therefore undefined. To have a right angled triangle with Tan(90) it's adjacent side would have a length of zero and the hypotenuse and opposite sides would be the same length. 90 would be theta, the angle between the adjacent and opposite sides would be 90 and the third angle would be 0. In effect it would be a straight line. However basic trigonometry still works.

For example, if the hypotenuse and opposite had both got a length of 1 then;

Sin(90)=1/1 which is correct. Cos(90)=0/1 which is correct. Sin(0)=0/1 which is correct. Cos(0)=1/1 which is correct. Tan(0)=0/1 which is correct.

However, Tan(90)=1/0. Which is an indeterminate form. This means that Tan(90) can be converted to 1/0 (and other determinate forms using l'hopitals rule).

If we were trying to work out the length of the hypotenuse and opposite it wouldn't work because we'd get Tan(90)=Hypotenuse/0. This happens because the hypotenuse and opposite could be any length as long as they are the same, the angles would be unchanged.

As well as this Tan(theta)=sin(theta)/cos(theta) so Tan(90)=Sin(90)/Cos(90). Sin(90) is 1 and Cos(90) is 0. Therefore Tan(90) again equals 1/0.

So is Tan(90) an indeterminate form? It tends towards infinity on the Tan graph. The limit of Tan theta as theta approaches 90 is ±∞. Also are all of the above valid arguments? — Preceding unsigned comment added by 86.140.32.172 (talk) 14:40, 20 July 2012 (UTC)
 * 1/0 is not regarded as an indeterminate form, as the limits of quotients that lead to it, if they exist, can only be positive or negative infinity, which isn't an infinite number of possible values. Double sharp (talk) 10:06, 26 October 2012 (UTC)

1^∞ and 0 × ∞ Indeterminate?
These don't seem to be indeterminate based ont he way the article explains the term.

1^∞

A power is the number times itself that many times. 1 times itself a unlimited number of times would be 1.

0 × ∞

Standard rule is any number times zero is zero ie 0 x X = 0 so it doesn't matter what X is.

If these are indeterminate the article should explain why they are.216.31.124.44 (talk) 05:31, 18 January 2013 (UTC)

 * It doesn't look to me like you've read the article all the way through. FilipeS (talk) 12:37, 24 February 2013 (UTC)

 * to Dr. William Heinz

 * $$\quad \lim_{x \to 0^+} \left(e^{x}\right)^{{a}\over x} = e^{a}$$

 * So, We need 1^∞ . — Preceding unsigned comment added by 106.132.205.157 (talk) 04:30, 13 August 2015 (UTC)

What is the definition of "indeterminate form" (is there a widely accepted definition?)
Some sources say that 0/0 is an indeterminate form, but not all. For instance, in "Mathematical Analysis I, Volume 1 By Claudio Canuto, Anita Tabacco", when f(x),g(x) converge to 0, the expression f(x)/g(x) is an indeterminate form when x -> 0, while 0/0 denotes an indeterminate form.

In the subsection "Evaluating indeterminate forms", the word "indeterminate form" refers to f(x)/g(x) (with x -> 0), just like in the definition in the above mentioned book.

In many books, the indeterminate forms are frequently simply listed, without giving any definition. If the main page gives a definition, it should probably cite something for its definition. Moreover, when writing the definition, keep in mind that in some definitions, floor(0) is an indeterminate form, whereas in other definitions, it is not  (I have no preference between those options; this is not a form that a calculus student is likely to encounter. But it would be good if the definition is clarified one way or the other). MvH (talk) 16:06, 12 February 2014 (UTC)MvH
 * My view is that there is no abstract definition, just a list. No, floor(0) is not an indeterminate form; at least, I have never seen it included. --Trovatore (talk) 18:56, 12 February 2014 (UTC)
 * floor(0) satisfies the current definition on the main page: "if the expression obtained after this substitution does not give enough information to determine the original limit, it is known as an indeterminate form".  Many books give no clear definition of "indeterminate form";  in that case, the phrase "intermediate form" probably means the same as "member of the list: 0/0, inf/inf, ...".   The current wording on the main page (the list "includes"..) leaves both possibilities open.    MvH (talk) 19:16, 12 February 2014 (UTC)MvH
 * Well, the "does not give enough information" bit refers to "limits involving algebraic operations", and floor is not an algebraic operation. Of course, strictly speaking, neither is exponentiation.
 * There is a strong temptation in articles like this to go beyond the literature and abstract a definition where in fact none has been given, and this temptation needs to be strenuously resisted. (Oh, I'm not saying a source can't be found somewhere that gives an abstract definition; it probably can, but it's not standard.)
 * That said, I don't object too strongly to the current wording in the first sentence. It doesn't come across to me as a once-and-for-all "definition", to which I would be sharply opposed, but more as an explanation.  Maybe there's some way to reword it to make it even more clear that it's not intending to be a definition.  I would support such a clarification as long as it doesn't make it harder to read. --Trovatore (talk) 19:27, 12 February 2014 (UTC)

Revisiting 00 over complex numbers
The article claims
 * If the functions f and g are analytic and f is not identically zero in a neighbourhood of c on the complex plane, then the limit of f(z)g(z) will always be 1.

This strikes me as patently false: it depends on the "operation" of exponentiation that is ill-defined on the complex numbers and cannot be defined as a continuous function in a punctured neighbourhood of (0,0). Any sensible interpretation will rely on the multivalued complex logarithm, which yields no limit even if one allows limits of multifunctions. Reliance on a single-valued definition requires a "sensible" branch of the exponentiation to be taken, as it seems to me that spiral branches exist that yield any limit that one chooses. A complex function f(z)g(z) in a bland statement like this without defining the exponentiation or mentioning the mathematically esoteric aspects seems out of place here. I notice that this has survived edits by not-so-clueless editors, so perhaps someone could explain why this claim should remain in the article, or where I've gone wrong? —Quondum 17:04, 22 February 2014 (UTC)

 * There are two quick solutions: (1) The 0^0 section in exponentiation states it better (i.e. no "complex plane"), so you can copy what it says.  Or (2) delete this line alltogether.  I'm OK with either option.  MvH (talk) 16:02, 26 February 2014 (UTC)MvH

 * I corrected it, and also shortened it a bit. I also fixed one other issue, it said that the indeterminate form 00 has been discussed for a long time, but that's misleading (the debate was whether 0^0 has a value or not; that's not the same as debating if it is an indeterminate form or not, because there is no standard definition of "indeterminate form").  There were two links on this issue in that section, I moved the second one, in order to combine it with the first.  MvH (talk) 19:56, 27 February 2014 (UTC)MvH

 * Much better, thanks. Since this is about indeterminate forms, the debate about the value of the expression is IMO irrelevant here.  I might strip that bit out.  —Quondum 23:37, 27 February 2014 (UTC)

 * That's fine (I merged two mentions of the debate into one, but I overlooked that there was a third spot where this was mentioned). MvH (talk) 14:35, 28 February 2014 (UTC)MvH

A new idea
Not sure where to place this.

http://dubai-computer-services.com/articles/infinities_by_khawar_nehal_19_mar_2014-1.pdf

If you can guide me as to where this can go, then I can place it there.

Regards,

Khawar Nehal — Preceding unsigned comment added by 94.203.214.192 (talk) 10:03, 23 March 2014 (UTC)

 * You may wish to read Wikipedia's policies on original research and verifiability, which limit what may be included in articles. Talk pages such as this are for discussing the content of these articles.  Many people have been fascinated by questions such as these and have spent a lot of time thinking about infinities, and while particular choices may seem to simplify certain problems, this does not help if this creates a logical inconsistency, which will result in paradoxes and fallacies, such as "proofs" that 1 = 2.  The result has essentially been a discovery of when infinity can be treated as a number, and when operations involving zero and infinity are reasonable (see, for example, Real projective line and Riemann sphere; also Transfinite number).  I'd suggest, if you wish to ask about aspects of consistency, the only place on Wikipedia where this might make sense is at Reference_desk/Mathematics.  —Quondum 16:01, 23 March 2014 (UTC)

Undefined forms that are not indeterminate
This article section used to read:

 * 0∞ also is sometimes incorrectly thought to be indeterminate: 0+∞=0, and 0-∞ is equivalent to 1/0.

Now it reads:

 * The limiting form 0∞ is not an indeterminate form. The form 0+∞ has the limiting value 0 for the given individual limits, and the form 0−∞ is equivalent to 1/0.

The changes made were not for the better. A simple and correct sentence was changed into one that is more convoluted, more obscure, and actually somewhat less accurate: algebraic expressions such as 0+∞ or 02 or 01/2 do not have "limiting forms"; functions do. At most, one might say that 0+∞ is a limiting form. And everyone agrees that 0+∞ can be defined as zero, so there's no mathematical sin in plainly writing "0+∞=0". FilipeS (talk) 16:27, 6 August 2014 (UTC)

 * I disagree. The statement is about limiting forms, not about algebraic expressions. As algebraic expressions they are simply nonsense. —Quondum 06:33, 7 August 2014 (UTC)

is there such a thing as a "form"?
The expression "indeterminate form" is as unanalyzable as the derivative dy/dx in the standard approach. Namely, an indeterminate form is NOT a form that's indeterminate rather than being determinate. Therefore the page should avoid using the term "form", as in the phrase "such-and-such a form is not indeterminate". Tkuvho (talk) 09:10, 17 December 2014 (UTC)
 * Perhaps in this context it can be taken to mean "formula"? —Quondum 14:33, 17 December 2014 (UTC)
 * What is an "indeterminate formula"? What is a "determinate formula"? Tkuvho (talk) 18:26, 17 December 2014 (UTC)
 * Wolfram MathWorld seems to use the two terms with separable meanings. The term "form" appears from the context to mean something along the lines of a notation representing an expression of a particular form, or perhaps even simply an expression of a particular form. Combine this with their statement "A mathematical expression can also be said to be indeterminate if it is not definitively or precisely determined", and it seems to make sense.  So one might say that the inverse of a quadratic function is indeterminate except at one point of the codomain of the original function, and since the inverse of a function is only defined where it is determinate, it is only defined at one point. —Quondum 18:44, 17 December 2014 (UTC)
 * OK, please don't rely on MathWorld for nomenclature or terminology.
 * I take Tkuvho's complaint to be referring to the stand-alone word "form". You're not objecting to "indeterminate form" as a fixed phrase, correct, Tkuvho?  In that case, I think the point is well-taken. --Trovatore (talk) 21:47, 17 December 2014 (UTC)
 * I take it you are saying that we should source any interpretation of the separated words "indeterminate" and "form" reliably (I at least made an attempt). Fair enough, but then the question should be revised to say "Can we find a reliably sourced definition of this use of these words? If not, the page should avoid ...". As it stands, the objection is hardly more valid than my (poorly sourced) attempt at an interpretation. —Quondum 23:39, 17 December 2014 (UTC)
 * Let's not get too hung up on the legalisms. My point is really that I don't recall anyone discussing "indeterminate forms" in a more general context of "forms".  Even if there is some RS that someone has done so, I don't think it's the standard approach.
 * My general sense of the question is that "indeterminate form" is a sort of historical artifact, a closed-ended list of seven fixed expressions, worth explaining in the terms it was used, but not worth belaboring or generalizing. --Trovatore (talk) 04:29, 18 December 2014 (UTC)
 * Browsing some books suggests that the term "indeterminate" does not stand on its own in this case: that the term "indeterminate form" is in a sense atomic, and the term "indeterminate" on its own should therefore be avoided. This is not entirely true of "form": it is used on its own occasionally. For the purposes of the article, it may make sense to define only the phrase "indeterminate form" and save the trouble of defining "form" at all. —Quondum 07:37, 18 December 2014 (UTC)
 * To summarize what seems to be a consensus: (1) the expression "indeterminate form" is so to speak an atomic expression and should not be analyzed into its component parts. (2) there is no such thing as a "determinate form" and therefore the expression should be avoided. (3) the expression "term" can be used as shorthand for "indeterminate form" when the context allows it. Tkuvho (talk) 08:44, 18 December 2014 (UTC)

a unique reference?
The unique reference provided here is to an obscure article in an obscure journal by Louis M. Rotando and Henry Korn (January 1977). Is this really the ultimate reference on indeterminate forms? Tkuvho (talk) 09:50, 11 January 2015 (UTC)

 * Most calculus books simply give the list without a definition. Membership in the list is determined by historical reasons, not mathematical reasons.  The reason we can only find obscure references for a definition is because as soon as you write down a definition, you'll find that there are other expressions like floor(0) that meet the definition, and that's a situation that most textbooks want to avoid because they don't want to include expressions weren't in the list before. Whether or not floor(0) is an "indeterminate form" depends on who you ask, it's not a concept with an agreed upon definition.   MvH (talk) 15:53, 19 April 2015 (UTC)MvH

 * I think that the added example is not going to serve to do much but confuse, so I don't see that we win any explanatory power by including it. It does not seem to fit the definition in the lead, since no substitution of subexpressions seems applicable at a first glance (it juts looks like an unambiguously nonexistent limit, sucg). But if one retains the example, we must make the fit clearer. My attempt to do so might be to consider the case $$\lim_{x \to 0} \lfloor -x^2 \rfloor = -1$$, but $$\lfloor \lim_{x \to 0} -x^2 \rfloor = 0$$. In essence, unless we present an unambiguous definition of an indeterminate form, we cannot produce examples or counterexamples without tripping over our own feet. Something about the "feel" of the thing is that it should be something of the form $$\lim_{t \to c}f(x(t),y(t),\ldots)$$, where the functions x, y, ... can be perturbed continuously within the domain of f to change the (well-defined) limiting value. As you suggest, a good definition is a thorny matter. —Quondum 17:25, 19 April 2015 (UTC)

 * I would like to se a citation for the flor function being considered an indeterminate form. Indeterminate forms refers to arithmetic operators rather than to functions. I know they are functions but the important point is that they are manipulated as operations without checking rules each time. If everything was considered as a function none of this would be necessary - but then we wouldn't be able to do arithmetic simplification without checking the rules for moving things from outside a function call into it.. Dmcq (talk) 17:43, 19 April 2015 (UTC)

 * "Arithmetic function" does not seem to be well-defined. I will discount it being only the elementary operations (+ – × ÷) as then exponentiation would be excluded, but we can consider elementary functions, which would include exponentiation (in its various forms, but only as $exp(y log x)$ if a limit of the exponent is to be involved). On the face of it, this might be a reasonable criterion for classifying something as an indeterminate form, but is this view expressed in any references? I do not see it as being referred to in the article, and cannot only be used for "editorial reasoning". —Quondum 18:46, 19 April 2015 (UTC)

 * Quondum, I searched references for "indeterminate form" some time ago. Most calculus books don't give a definition, and more advanced books generally don't use this term. The books that do give a definition, there were differences, there wasn't a good way to select one or another. For our definition "if the expression obtained after this substitution does not give enough information to determine the original limit" then all I can say for sure is that it is simply a reformulation of one of the characterizations of a discontinuous function (that's why I added a link to that). Whether or not there are indeterminate forms other than the ones listed, our article is vague on that, could be yes, could be no.  Is this vagueness due to a lack of consensus? The floor(0) example falls in the category "could be yes, could be no".  It might satisfy "the" definition, depending on which one you pick. I placed it after section "Expressions that are not indeterminate forms" because I didn't want to call floor(0) an indeterminate form, since that would seem contrary to historical practice.     PS. Is there a reference for the definition as currently stated? MvH (talk) 19:21, 19 April 2015 (UTC)MvH

 * Regarding "arithmetic function", I think the problem arises from the fact that when this list was made, the word "function" didn't mean the same thing as it does now. If any definition of "indeterminate form" was made at the time, it would probably be untranslatable to modern math.  Also, even if "arithmetic function" meant precisely what you want it to mean, it would still defeat the educational purpose of this list, because when you deal with limits, you have to be careful with every discontinuous function, not just the ones that appear in a historical list.  Basically, "indeterminate form" is a incomplete list of red flags for limits (the complete list being: discontinuous functions). MvH (talk) 19:35, 19 April 2015 (UTC)MvH

 * Dmcq's argument has some merit: that if we extend a definition to arbitrary discontinuous functions, the whole subject becomes moot (essentially the point you're making), yet we (for pedagogical purposes, not as WP) do want primary examples. I think that "form" might be synonymous with "familiar operation": "elementary function" is too broad. As to "when this list was made", I'd be surprised if the definition of a function was not pretty mature in Cauchy's time. This might suggest changing the focus of the article from assuming that there is a definition and the maths of it to more of a historical commentary on a vague concept, making it clear that it is vague. —Quondum 20:11, 19 April 2015 (UTC)
 * Yes arguments saying that divide is a function and floor is a function therefore why is floor not in the list are treating the subject too abstractly. Divide is not normally considered as a function but as an operator. One would not apply the term 'form' to functions. Overall if the sources are not definite then the article can't be definite, Wikipedia isn't a textbook but an encyclopaedia with some fairly strict content policies. Dmcq (talk) 21:00, 19 April 2015 (UTC)
 * I do not think that $$\lfloor 0 \rfloor$$ should be considered an indeterminate form unless some reference does so. In the absence of a claim that such a reference exists, I'm removing the section.  — Arthur Rubin  (talk) 21:38, 19 April 2015 (UTC)
 * I'm happy with removal. But the bigger issue remains: the article pretends to define an indeterminate form, whereas it can strictly only list examples of expressions considered so, and the current definition should be changed into a statement of utility. —Quondum 21:44, 19 April 2015 (UTC)
 * Before you delete the floor(0) example, please consider the issue that it is trying to address. The issue is that the article is vague on what indeterminate forms are:  Are they just the ones that are listed, or are there more?  If there are just the ones that are listed then there is something strange about trying to find a definition.  MvH (talk) 22:30, 19 April 2015 (UTC)MvH
 * In my experience, the notion of "indeterminate form" is typically presented without any abstract definition at all, just a list, with perhaps a few words about what they have in common, but with no attempt whatsoever to say what a general indeterminate form not on the list might be. I think that's what we should do as well.  And I don't think we should say anything about floor(0). --Trovatore (talk) 17:19, 20 April 2015 (UTC)

Algebraic operations?
Now that I've linked Algebraic operation, I notice that three of the listed indeterminate forms (namely those that involve exponentiation) relate to a function that is not an algebraic operation as defined by that article. It think that it is reasonable to deduce that the definition is flawed, since the cases of exponentiation are clearly historically notable examples. —Quondum 23:46, 19 April 2015 (UTC)
 * Books normally just list the various ones rather than saying anything too explicit. The very first example I get from Google books when I stick in 'indeterminate form' is which is practically identical to what is said here except they say function instead of expression. And a couple of paragraphs on it just lists the various indeterminate forms. I don't think I've seen any definition that talks about algebraic operators in the modern sense without continuous powers, you're talking about fields which never came into the thinking about calculus and continuous functions. Dmcq (talk) 00:23, 20 April 2015 (UTC)
 * I was talking about what the lead says, and it sounds like you agree that what it says should be changed. The book you link to seems a little inconsistent, in the sense that it defines an indeterminate form as a function that is undefined at a point of its domain (but, implicitly for which the limit may exist). So, we can look at rewording the lead to eliminate the reference to an algebraic operation. —Quondum 02:42, 20 April 2015 (UTC)
 * Okay, done. I hope that I have adequately addressed this in the lead. —Quondum 03:03, 20 April 2015 (UTC)
 * Seems fine to me. Dmcq (talk) 07:45, 20 April 2015 (UTC)

Russian translation equivalent sense for words undefined and indeterminate
0/0 is indeterminate, russian word equivalent sense will be "неоднозначно". 10/0 is undefined, russian word equivalent sense will be "неопределено". 188.208.126.82 (talk) 14:11, 9 June 2023 (UTC)

More?
Wolfram alpha uses this list of indeterminate forms,

should we add the missing ones to the list on the page? 2007GabrielT (talk) 12:59, 13 November 2023 (UTC)
 * 0/0
 * 0*∞
 * 0^0
 * 1^∞
 * ∞^0
 * (-1)^∞
 * 0^i*
 * z^∞ for abs(z) = 1
 * x^(1/0) (for any x)
 * z^∞ for abs(z) = 1
 * x^(1/0) (for any x)
 * No. In ordinary usage, there are exactly seven indeterminate forms.  It's a closed-end list.
 * Now, some of these expressions can be argued to bring up similar issues, and you can say, why shouldn't they be indeterminate forms? The real answer is, because indeterminate forms are a historical artifact that doesn't need to be extended, a way to talk about partial functions that are continuous where they're defined, but have no continuous extension to points where they aren't.  Once you have that language, you don't need indeterminate forms anymore.
 * In general, don't take anything nomenclature-related from MathWorld or from Wolfram. They're notorious for making up terminology. --Trovatore (talk) 17:36, 13 November 2023 (UTC)

title: Talk:Indeterminate form


text:
Talk:Government and binding theory

Old talk
I've made some minor changes. I'm not sure my new definition of R-expressions is entirely rigorous, but I think it may be a little clearer.--Nid Flocken 05:59, 11 November 2005 (UTC)

It was difficult to find a "government" deffinition. I suggest a link from this page  to that page  dont know how to do it... 84.106.157.114 (talk) 11:05, 21 October 2009 (UTC) This last link was more helpfull to me than the original article section on Government ! 84.106.157.114 (talk) 11:05, 21 October 2009 (UTC)

—The preceding unsigned comment was added by Special:Contributions/ (talk)

Why is the "b" in the title capital? If it's the title of Chomsky's book, it should be capitalized, but if it's a common noun, it should not. Wikipedia article titles do not usually capitalize common nouns. Michael Hardy 02:25, 15 Nov 2003 (UTC)

--- The above comment (re: capitalization in the title of the article) is, as I understand it, both misguided and mistaken. First, I think that the standards for capitalization are certainly satisfied by the properties of the usage of "binding" used here; i.e. i think it is perfectly acceptable to capitalize it, e.g. "Binding." But moreover, I think it atraditional and antiquated to say anything about whether a word "should not" be capitalized; perhaps Wikipedia article titles are usually not capitalized, but I think it strikingly presriptivist (cf. prescriptive) to assume that fact to entail that this one should not be. --snwright

---

I should really learn not to worry about these things. I had a read of the Naming conventions (capitalization), and had a look at usage on the internet. It appears anything goes. So, if you want to change the article back, I promise to leave the title alone. I'm still hoping to add to the content - it's on my mental to-do list. Regards Dduck 12:31, 15 Nov 2003 (UTC)

Making this article coherent
This page needs to show the various parse trees it discusses so readers can follow along. Instead of giving the four examples up front and then running through the binding principles, it should devote a subsection to each principle, and illustrate each with one or more examples.

Even while writing it, I felt the explantion of the c-command in "John saw his mother" was hand-wavy, so I'm not sure it's entirely correct. -- Beland 02:13, 19 Apr 2005 (UTC)

Removal clean up tag
Have got rid of the above as I can not see the reason for it. The article may not be the clearest explanation ever of GB theory but its not bad enough to deserve the tag. Marcus22 17:49, 8 April 2006 (UTC)

Tree Diagram
The tree is kind of weird. The "ming vase" DP should be a sister to V, and the PP an adjunct to V. Also, as it stands, with PP as a complement to DP, the reading must be something like, "He smashed the ming vase, and this ming vase owned a hammer." This looks like a botched double VP shell or something. —Preceding unsigned comment added by 24.131.170.183 (talk) 02:22, 20 November 2008 (UTC)

I agree, the diagram for The mother of John likes himself is incorrect as well, there ought not to be any ternary branching. Here is the corrected version in bracketed notation [TP [DP [D' [D The] [N mother]] [PP [P of] [John1]]] [T' [T ~s] [VP [V like] [NP [N himself1]]]]]. Someone ambitious could draw it using phpSyntaxTree. Thiuda (talk) 14:49, 19 November 2010 (UTC)

I agree too. After 7 years, the constituency in the tree diagram for the "Ming vase" sentence still seems to be wrong. Mmahdavim (talk) 05:14, 13 November 2015 (UTC)

Incomplete
I got my graduate degree in Linguistics in the late '60s, as the Transformational model was vanquishing the Taxonomists. There were any number of theories of Deep Structure floating around at the time. (Since then I've been a computer guru.) This account does not explain to me what has happened since then; if we are to coherently account for third-person pronouns, then the semantic domain has to be the discourse, which was something many of us suspected at the time. But the article as it stands gives no hint of this. -- Craig Goodrich 24.14.168.244 (talk) 02:24, 28 April 2009 (UTC)

If someone could define what a governor is it would be incredibly helpful —Preceding unsigned comment added by 83.179.25.167 (talk) 18:28, 8 May 2010 (UTC)

I second the anonymous commenter above. If the author of the article could elaborate on what he or she wrote, that would be helpful. Things that need explaining: the word "dominates" - this can mean so many different things... and it is not at all clear what it means here. How does government relation make case assignment unambiguous? At least a one line explanation? At the same time, can the first tree have a concrete example? What is A prime in it? Term "maximal projection" needs to be explained. Even a search through wiki doesn't help me to understand what is meant by it.

I find this sentence most frustrating as despite my best effort I can't understand it / it looks broken to me:

Governors are heads of the lexical categories (V, N, A, P) and tensed I (T).

Who or what is tensed? What do the letters after the word "tensed" mean? What do the parenthesis mean here? Are they punctuation or are they a mathematical notation, tuples, intervals... maybe something else? I.e. I can read this sentence in two ways:
 * 1) Governors are parts of speech, which are tensed to be "I (T)", whatever "I (T)" means, with further interpretations of I to be either the Roman numeral, or the first letter of some word, and then it must read as "tensed by I to be [of] T[ense]"?
 * 2) Governors are parts of speech, and, in addition to it, there are such governors, which are the tensed "I (T)", whatever "I (T)" means.79.177.202.142 (talk) 12:58, 24 May 2013 (UTC)

 * I have only contributed one sentence to this article, but I will nevertheless try to answer a couple of your questions. First, however, note that there has been very little action on this article for a long time. Someone who knows GB well needs to redo the article entirely.

 * The I stands for inflection. In GB, the root node (highest node) in the tree for a declarative sentence for a long time was IP, indicating that the sentence was an inflection phrase. The category tense was introduced into the framework around 1990. One saw the need to split IP into two categories, tense (T) and agreement (Agr), which led to tense phrases (TPs) and agreement phrases (AgrPs). The term tensed is applied to verbs and clauses. A finite verb is always tensed, and a clause that contains a finite verb is a finite or tensed clause. When the author(s) of this article wrote "I (T)", he/she was trying to take these changes into account, i.e. trying to indicate that the relevant functional category that heads the clause is I or T, depending on which version of GB you want.

 * The parenthesis do not indicate tuples or intervals. GB is not computational linguistics. Most grammarians who adopt and work with the GB framework are theoreticians, not computational people. The parenthesis have no special meaning.

 * There were various definitions of government (and governor) produced in the GB framework, none of which was really clear in my view, and all of which seemed to be subject to change. See the article on government (linguistics). If one states that a governor is "tensed", one is probably referring to the finite verb or to a projection of the finite verb, since the finite verb is the only word in a clause that can indisputably bear tense. The concept of government is clearer in dependency grammar (my opinion), where specific words govern other words, e.g a preposition governs its object, a finite verb governs a non-finite verb, a noun governs a determiner, etc. --Tjo3ya (talk) 15:37, 24 May 2013 (UTC)

 * Thank your for the explanation, Tjo3ya! I really wish there were more links on the page itself linking to the terms used here. While I still have some struggle understanding the article, I did some more research and am trying to read the relevant literature (I'm a math/CS student collecting information for a course in natural language processing). Here's one more thing that makes absolutely no sense to me: the "maximal projection". I've found this article: Projection Principle, which may be related. But the article is really very vague all by itself and never mentions that projections can be maximal and... what other kinds may there be? I've found a single other mention on Wiki, here: Stripping (linguistics), which also mentions a "minimal projection". There are three other mentions not qualified with size/strength, all point to X-bar theory, which, unfortunately, doesn't mention projections at all (there is only one instance of the verb "project" there, which may or may not be related). 79.182.106.53 (talk) 14:14, 1 June 2013 (UTC)

 * A maximal projection is the highest projection of a given category (word). One distinguishes between minimal projections, intermediate projections (= bar-level projections), and maximal projections. These distinctions are the most pronounced in X-bar theory. X-bar theory assumes that every lexical or functional category has all three of these projection levels. For the most part, the maximal projection of a noun is NP, the maximal projection of a verb is VP, the maximal projection of a preposition is PP, etc.

 * For an example, take a look at the big tree in the article on X-bar Theory. Every lexical and functional category has at least three projections according to the X-bar schema. The subject pronoun he, for instance, has the minimal projection N, the bar-level projection N', and the maximal projection NP. The finite verb studies has four projections, the minimal projection V, the first bar-level projection V', the second bar-level projection V', and the maximal projection VP.

 * By distinguishing between the projections of a given category, one can acknowledge lesser and greater groupings of words. Minimal projections always correspond to single elements (usually words). Intermediate projections often correspond to two or more words, and maximal projections will correspond to even more words. Various phenomena of syntax are sensitive to these distinctions.

 * Interestingly however, the distinction between projection levels is mostly absent from dependency grammar (DG) structures. A word in a DG tree has a single projection only, and that projection corresponds to all three levels of X-bar theory.

 * Your question suggests that an article on syntactic projections is needed.

 * Sorry for long time between responses. Yes, an article, or even just a note on what is meant when the word "projection" is used would be helpful. After all it's a word that has many other meanings in other contexts. There was also a discussion on the page dedicated to c-command, which argued for a separate page dedicated to government relation. Which made me think that if there was a glossary of terms, on, say the page dedicated to the grammar or linguistic (although these two seem too unspecific) similar to how this page: http://en.wiktionary.org/wiki/Appendix:Glossary_of_set_theory is the glossary of all terminology used in set theory. It is otherwise difficult to infer a hierarchy of terms / other relations established between the terms. 109.67.113.85 (talk) 08:42, 8 June 2013 (UTC)

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Talk:Government and binding theory


text:
Gayo language

Gayo (alternatively rendered as Gajo) is an endangered Austronesian language spoken by some 275,000 people in the mountainous region of the Indonesian province Aceh on the Northern tip of the island of Sumatra, specifically around the Central Aceh, Bener Meriah and Gayo Lues regencies. It is classified as belonging to the Western Malayo-Polynesian branch of the Austronesian languages, but is not closely related to other languages. Ethnologue lists Bukit, Dëret, Lues, Lut, and Serbejadi-Lukup as dialects.

Gayo is distinct from other languages in Aceh. The art and culture of the Gayo people is also significantly different compared with other ethnic groups in Aceh.

In 1907, G.A.J. Hazeu wrote a first Gayo–Dutch dictionary for the colonial authorities of the Dutch East Indies.

title: Gayo language


text:
Small matter of programming

In software development, small matter of programming (SMOP) or simple matter of programming is a phrase used to ironically indicate that a suggested feature or design change would in fact require a great deal of effort.

It points out that although the change is clearly possible, it would be very laborious to actually perform. It often implies that the person proposing the feature underestimates its cost.

Definitions
The 1983 Jargon File describes an SMOP as follows:

"SMOP (ess'em'oh'pee') noun.

An acronym for "a Small Matter Of Programming". A piece of program code, not yet written, whose anticipated length is significantly greater than its intellectual complexity.

This term is used to refer to a program that could obviously be written but is not worth the trouble. It is also used ironically to imply that a difficult problem can be easily solved because a program can be written to do it. The irony is that it is very clear that writing such a program will be a great deal of work.

Example: "It's easy to change a FORTRAN compiler to compile COBOL as well; it's just a small matter of programming.""

The IBM Jargon Dictionary defines SMOP as:

"SMOP (smop) n. Something quite possible, but requiring unavailable resources to achieve. "Why isn't that function available in the program?" − "It's just a Simple Matter Of Programming". (The implication being that, given a few person-centuries, all things are possible.) Also SMOUP (smoop), a Simple Matter Of Micro-Programming (if handwritten, using a Greek mu). See also how hard would it be."

Usage
SMOP was among the "games" described in an article as paralleling the Games People Play identified by Dr. Eric Berne in the field of self-help psychology. The game essentially consists of proposing seemingly simple adjustments to a design, leading to unexpected consequences and delays.

Alternative phrases such as simple matter of software or small matter of software are occasionally used in the same manner. However, the phrase is also used without irony to indicate that straightforward software development is all that is required to resolve some issue. This usage is often invoked when the speaker wants to contrast the implied ease of software changes with the suggested greater difficulty of making a hardware change or a change to an industry standard. This non-ironic usage is more often invoked by senior management and hardware engineers, than it is by software engineers.

The term was also explored and expanded upon by computer scientist Bonnie Nardi in her 1993 book A Small Matter of Programming: Perspectives on End User Computing.

title: Small matter of programming


text:
Iban language

The Iban language (jaku Iban) is spoken by the Iban, one of the Dayak ethnic groups, who live in Brunei, the Indonesian province of West Kalimantan and in the Malaysian state of Sarawak. It belongs to the Malayic subgroup, a Malayo-Polynesian branch of the Austronesian language family.

Iban has reached a stage of becoming a koiné language in Sarawak due to contact with groups speaking other related Ibanic languages within the state. It is ranked as Level 5 (i.e. "safe") in term of endangerment on Expanded Graded Intergenerational Disruption Scale (EGIDS). In 2024, the Iban language was included in Google Translate and became a historic moment as the first Borneo language to be registered into Google Translate and as the first Malaysian language to be registered into it other than Malay.

Classification
Iban comes from the Ibanic language group spoken in Sarawak, West Kalimantan, and Brunei within Borneo island. part of the Malayic subshoot of the Malayo-Polynesian branch in the Austronesian language family. The Malayic languages originate from western Borneo, thus Iban is closely related to Malay, especially the Sarawakian dialect. Other isolects in the Ibanic group of languages are Sebuyau, Mualang, Kantu, and Seberuang. These groups of languages can be identified by the word-final position in certain lexical forms of /-ai/. These lexical forms are similar to other Malayic languages with lexical forms of /-an/, /-ang/, or, less frequently, /-ar/.

The Iban language is also related to other dialects such as Sebuyau, Kendayan, Balau and Selaku.

History
According to the oral history of the Iban people, Benedict Sandin, in 1968, plotted the ancestry of the Iban people as descendants from the Upper Kapuas, West Kalimantan, Indonesia. The Iban people arrived in Sarawak in the 16th century, and settled in the regions of Batang Lupar drainage basin and Undop river in southern Sarawak. From there, they migrated north, east, and west, and expanded into Saribas, Batang Sadong, Batang Layar, and Batang Lupar rivers. In the 1800s, they moved into the Rajang basin (middle region of Sarawak) from Batang Lupar river, Katibas river, and Saribas river (Saribas is a tributary of the Rajang River). By 1870s, they had reached Mukah and Oya rivers. In the early 1900s, they reached Balingian, Tatau, and Kemena rivers (near Bintulu). They also reached the Baram area and Limbang rivers around the same time in northern Sarawak and would become the largest ethnic group in Sarawak.

Brooke administration
Fearing that the Iban tribes outnumbered the pre-existing local tribes with detrimental environmental effects on lands intended for shifting cultivation, the Brooke government restricted the Iban people from further migration to other river systems such as the Baleh river. However, the Brooke government allowed the Ibans to settle in other areas such as Lundu, Balingian, Bintulu, Limbang and Baram to consolidate the government's authority there. As a result of this policy, several minority ethnic groups such as Bukitans living along the Batang Lupar River were assimilated into the Iban people, thus contributing to the growth of Iban tribe and the expansion of the Iban language in the state. The Iban language was taught in schools in the 1940s during the Brooke era.

1958-1977: Borneo Literature Bureau (BLB) foundation
During the period of Crown Colony of Sarawak, the Iban language was used in government official letters, courts, announcements, and notices. Radio Sarawak, started by the British, offered Iban language programmes. The Iban language, known under the name of "Asian language", was offered as an examination subject in the Sarawak Junior Certificate. The "Asian language" was renamed to "Iban language" in 1963. Borneo Literature Bureau (BLB) was founded by the British in 1958 to collect and document oral Iban literature. BLB published more than 60 Iban language books during its lifetime until 1973 when it was replaced by a Malaysian federal government agency Dewan Bahasa dan Pustaka (DBP) in 1977. After that, the publication of books in Bornean languages came to a halt. The publication of the Nendak magazine, which was started by BLB in 1967 also came to a halt. Jimbun Tawai, the former vice chairman of Sarawak Dayak Iban Association, called this period under Crown Colony as "golden era" of the Iban language.

1977-2000s post BLB closure
After the closure of BLB, other smaller publishers continue in this niche such as the Kuching-based publishing company named Klasik. Examples of works include ensera (Iban epic story) and cherita kelulu (morality novellas). Christian churches such as the Catholic church publish prayer books that adopt certain aspects of Iban adat (culture). Thus, Christian texts bear greater significance as cultural repositories of the Iban language when to compared to other genres after the demise of BLB.

State-sponsored media such as Berita Rakyat was founded in 1974 and ended in the 1990s. The magazine was started by Rajang Security Command (RASCOM) in Sibu to defeat the communists' activities in the Rajang basin. The magazine stopped publication after the cessation of Communist insurgency in Sarawak in 1990. Another magazine named Pembrita was published by the Information Department of the state government and was aimed at providing developmental news to the rural Iban populace such as exemplary longhouses, lucrative cash crops, and animal husbandry. The magazine also called on the rural Ibans to modernise their ways of farming. There were no Iban newspapers in the 1990s and early 2000s. The high cost of imported paper materials and low advertising revenues contributed to the difficulties of Iban newspaper publishing.

Tun Jugah Foundation was established in 1985 after the death of Tun Jugah to record the oral history of the Iban people, producing Iban dictionaries and surveys of the rural-urban migration of the Iban people.

Radio Televisyen Malaysia (RTM) expanded their Iban radio broadcasts to 10 hours on Sundays and 9 hours on rest of the week by the 1980s as WaiFM Cats FM is the first commercial radio station to broadcast in Iban opening in 1997.

The Iban language was included in the primary school curriculum in 1968 and a few secondary schools in 1988. In 1968-1969, teachers' training colleges offered Iban as an elective subject. It was only in 1988, that Iban was formalised as part of the Malaysian national curriculum. There are no Iban-medium schools in Sarawak. In 2008, Iban was taught as an elective language subject in Malaysian Form 5 secondary schools for Sijil Pelajaran Malaysia (SPM) certificate examination. A survey done in Sarawak in 2008 showed that a total of 367 primary schools and 55 secondary schools have taught the Iban subjects since 1968. The number of primary schools offering the Iban language subject increased to 1,264 in 2015 while the number of secondary schools reduced to 52 in 2015. Most of the schools are located in Kapit, Sibu, Sri Aman, and Sarikei Divisions where there are significant Iban population. The Iban language subject is also offered in undergraduate programmes in two teachers' institutions in Sarawak. In Sultan Idris Education University, Perak, the Iban language is offered as a minor subject for Iban students majoring in Malay studies.

In 2003, Malaysian federal authorities banned the Iban Bible (Bup Kudus) as its use of the word Allah overlaps with the use of Allah (Taala) as the name of the God worshipped in Islam (Islam given official status on the Federation level) which the federal government argues can "confuse" the Muslim populace in the state. The ban was lifted by the then deputy prime minister Abdullah Ahmad Badawi after persistent protests.

2010-present: emergence of vernacular mass media
Newspapers The Borneo Post and Utusan Borneo started Iban language sections in 2010 and 2014 respectively. The Borneo Post stopped the Iban section in 2017. A monthly Iban magazine named Pegari was also published by a small company named PEGARI Iban Production from 2012 to 2018. Borneo Media Solutions, a subsidiary of PEGARI Iban Production, also published several books in Iban language.

RTM opened their first Borneo-oriented channel TVi in 2011 which later became TV Okey in 2018 which includes a 30-minute Iban news slot. TV Sarawak started the Iban language section in October 2020. Iban language support was added to Google Translate in 2024.

Extent of use
The Iban language is allowed in the Sarawak State Legislative Assembly with the special permission from the Speaker and simultaneous interpretations will be provided during the assembly sitting and when written into the Hansard.

Rentap's battle cry while fighting against James Brooke in 1860s “Agi Idup, Agi Ngelaban!” (“I will fight as long I will live!”) is adopted by Sarawak Royal Ranger Regiment as their motto. The battle cry is also used in speeches and car stickers to evoke the warrior spirit of the Iban people. The word "Oo-ha", an Iban call for celebration, was popularised by the former chief minister of Sarawak Adenan Satem as a form of "hello" before giving speeches in order to motivate a crowd. The Chinese-predominant Sarawak United Peoples' Party used the Iban word "Sa'ati" (United) as their party slogan. Another Chinese-predominant Sarawak Democratic Action Party has been using the Iban language to garner support from the Iban population. Other words include "Segulai sejalai" (going together) that was selected as the slogan for Malaysian national unity, and "Ngap Sayot" (literally means "eat vegetables") used by Sarawak FA football team battle cry to signify taking down opponents just like eating vegetables.

Iban churches in Sarawak conduct services in the Iban language.

Trades in the Sarawak bazaars are also frequently conducted in the Iban language.

Consonants
Iban has the following consonant inventory:

Vowels
Iban has a six-vowel system, with five cardinal vowels plus schwa:

Vowel sounds are nasalized when preceded by a nasal consonant.

Grammar
Lexical roots can be expanded by many affixes in Iban, as exemplified here with the verb gagai.

There are four types of affixes in Iban, namely prefixes, suffixes, circumfixes and infixes.
 * gagai 'chase'
 * begagai 'chasing/playing with each other'
 * begagaika 'chasing something/someone'
 * ngagai 'to chase'
 * digagai 'being chased by'
 * dipegagaika 'being chased by many'
 * pengagai 'chaser'
 * tegagaika 'outrun/outpace'

Other examples:

 * Sayau 'love'
 * Dikesayauka 'was loved by'
 * Penyayau 'affection'
 * Kiruh 'busy'
 * Ngiruhka 'to make someone busy'
 * Pengiruh 'preoccupied'
 * Pengiruh-ngiruh 'really preoccupied'
 * Enjuk 'give'
 * Berenjuk 'giving each other' (present)
 * ngenjuk
 * Dienjuk 'gave' (past)
 * Deka ngenjuk 'will be given' (future)
 * Pengenjuk 'giver'
 * Kangau 'call'
 * Bekangau 'calling each other' (present)
 * Ngangau 'calling' (present)
 * Dikangau 'was called' (past)
 * Deka dikangau 'will be called' (future)
 * Pengangau 'caller'

Personal pronouns
Iban has separate words for inclusive and exclusive we, and distinguishes singular, dual, and plural.

Sample

 * Ke nuan 'for you'
 * Ke aku 'for me'
 * Ke kami 'for us'
 * Bup aku 'my book'
 * Bakih aku 'my friend'
 * Apai aku 'my father'
 * Gamal nuan 'your look'
 * Sulu nuan 'your beloved'
 * Sekula kami 'our school'
 * Ke pangan aku 'for my beloved'
 * Ke anak aku 'for my child'
 * Ari indai di 'from your mother'
 * Ari bakih aku 'from my friend'

Pronouns are primarily put after subjects.

Possessive pronouns
Sample phases:
 * baju tu engku 'This shirt is mine.'
 * Tu enggi nuan 'This is yours.'
 * Siti nyin enggi tua 'That one belongs to both of us.'

Demonstrative determiners
There are three demonstrative determiners in Iban. Tu 'this, these' is used for a noun which is generally near to the speaker, nya 'that, those' is used for a noun which is generally far from the speaker, and nyin, which is the furthest from the speaker.

These words can also act as demonstrative pronouns where they can stands on theirs own, replacing rather than modifying a noun.

Example:
 * Nyamai tu. 'This is good.'
 * Ok meh nya. 'That's ok.'
 * Peda di nyin dih. 'Look at that.'

Demonstrative pronouns
In Iban, demonstrative pronouns are words that show which person or thing is being referred in relation to the location of the addressee to the speaker. There are three demonstrative pronouns in Iban depending on location to the speaker. They can only be used to refer to an addressee (human) and cannot be used to refer to inanimate objects.

Examples:
 * Nama gaga iya tu baka nya? 'Why is this person acting in such a way?'
 * Kini ke iya nya tadi? 'Where is he going?' (Referring to the second closest person to the speaker)
 * Ni iya nyin tadi dih? 'Where is the other (person) one?' (referring to third person which is the furthest from the speaker)

Demonstrative adverbs
Demonstrative adverbs in Iban are closely related to the demonstrative pronouns in Iban grammar. For example, corresponding to the demonstrative pronouns are the adverbs such as kitu ('going here'), kia ('going there') and kin ('going there (farthest)') equivalent adverbs corresponding to the demonstrative pronoun this are tu, nya and nyin.

Examples:
 * Kitu nuan. 'Come here (you).'
 * Kini di kia? 'Why are you going there?' (within the sight of the speaker)
 * Aram kin tua. 'Let's go there.' (referring to location far away from speaker)

Locatives
Examples:
 * Ditu ku nganti nuan. 'I wait for you here.'
 * Dia ku nganti nuan. 'I wait for you there.' (not far from the speaker's location)
 * Din ku nganti nuan. 'I wait for you there.' (referring to a far place)

Manner
Iban also has a set of adverbs referring to manner. They are a combination of baka (ke) ('like/as') and the abbreviated determiner forms tu, nya and nyin.

Examples:
 * Aku ka iya baka tu. 'I want it to be like this.'
 * Nama di ngaga iya baka nya? 'Why did you treat him like this?'
 * Uji gaga di baka ke nyin. 'Try to do it like that.'

Interrogative words
Iban also has a few interrogative words: sapa, nama, ni, lapa, kemaya and berapa.

Sapa empu jam tu?
 * Sapa – Who

Who own watch this

Who owns this watch?

Nama gaga nuan ditu?
 * Nama – What

What doing you here

What are you doing here?

Ni ai ku tadi?
 * Ni – Where (Dini and Ba ni also used to ask for specific location)

Where water/drink my {just now}

Where is my drink?

Lapa nuan nyabak?
 * Lapa – Why (Nama kebuah also used.)

Why you crying

Why are you crying?

Kemaya tua deka betemu?
 * Kemaya – When

When we {going to} meet

When are we going to meet?

Berapa iku manuk tupi nuan?
 * Berapa – How many

{How many} CL chicken raise you

How many chicken you raise?

Bakani gaya {mua ari} saritu?
 * Bakani – How

How look weather today

How is the weather today?

Vocabulary
The first Iban-English Dictionary was published in 1900 by Rev. William Howell, an Anglican priest based at Sabu, near Simanggang (Sri Aman) and D.J.S. Bailey, a Brooke administrative officer as A Sea Dayak Dictionary. The dictionary was important in the early development of the Iban as a written language.

A Comprehensive Iban-English Dictionary, jointly published by The Dayak Cultural Foundation and The Tun Jugah Foundation in 2016, contains 31,000 entries and about 1900 pages.

The Iban-Malay dictionary was first published by Dewan Bahasa dan Pustaka (DBP), in 1989. The second edition was published in 2015. It contains 11,530 entries dan 9,710 subentries.

Writing system
According to Iban legend, an ancestor named Renggi devised a writing script on the skin of wood, but it was soaked in water and the writing vanished. Anguished with the tragedy, Renggi munched the script and swallowed it where the script became ingrained in Renggi's brain and blood and also his descendants. Since then, the Ibans became adept at memorising oral traditions, just like exactly written in books. Occasionally, the Ibans used personalised symbols as memory aids on their writing boards (papan turai).

As the Iban language had no extant writing system of its own, Christian missionaries adopted the Latin alphabet in an attempt to codify the language. During the Crown Colony era, the Borneo Literature Bureau also worked on the written form of the Iban language.

From 1947 to 1962, Dunging anak Gunggu invented an Iban syllabary known as the Dunging script. In 2010, Dr. Bromeley Philip of Universiti Teknologi MARA, who is also a grandnephew to Dunging, created digital fonts for Dunging script, named "LaserIban", available for Windows and Macintosh computers. Dr Bromeley also launched a course to promote the use of LaserIban and had transcribed several traditional folktales from Latin alphabet into Dunging script. However, the Dunging script is not widely adopted. As of 2011, only three people in the world mastered the Dunging script, namely Dr Bromeley himself, longhouse chief Tuai Rumah Bagat Nunui and teacher Ngambong Katoi.

Dialects
Iban can be subdivided into different sub-ethnic groups, each of which speak in different dialects. The most formal, intermediate, and working dialect is the Saribas dialect, and mainly Betong and Saratok. Others such as Balau, Sebuyau, Ulu Ai, and Rejang are mutually intelligible throughout the Sarawak region. The exception is the Iban Remun/Milikin dialect, which is still understood by Ibans from other districts. In West Kalimantan, dialects such as Bugau, Seberuang, Mualang, Chengkang, Sebaru, and Dau are more disparate.

Dialect comparison
Sample text (Luke 2:10-11)

Mualang
 * 10 Baroꞌ mlikat Tuhan Allah madah ke sidaꞌ: “Nang kitaꞌ takot! Ku madah brita bayek ari Tuhan Allah ke kitaꞌ, te nyuroh gaga ugaꞌ bansa.
 * 11 Malam toꞌ de kuta Daode udah adai Penyelamat kitaꞌ, Al Maseh Raja te dedanyi Tuhan Allah, nyaꞌ mah Tuhan.

Iban
 * 10 Tang ku melikat nya bejaku ngagai sida, “Anang takut! Laban aku mai ngagai kita Berita Manah ti ngasuh ati semua mensia gaga:
 * 11 sehari tu, di nengeri David, Juruselamat kita udah ada, iya nya Kristus ti Tuhan!

Sample phases in Iban Remun
 * Entai ku ngilau – Nadai aku meda. ('I did not see it.')
 * Entauk ku badak – Enda ku nemu. ('I don't know.')

Family
For extended family in Iban

Example;
 * Anak buah bini ku nya. 'That is my wife's nibling.'
 * Anak buah ke indu laki ku nya. 'That is my husband's niece.'
 * Entua laki ku nya. 'That is my husband's parent-in-law.'
 * Entua ke laki laki ku nya. 'That is my husband's father-in-law.'
 * Petunggal bini ku nya. 'That is my wife's cousin.'

Days
Example:
 * Tulat tua betemu. 'We'll meet again the third day.'

 * Ensanus ku bisi meda iya 'I saw him two days ago.'

Months
The Iban calendar is one month ahead of the Gregorian calendar as follows:

Human Rights
English: Article 1 – All human beings are born free and equal in rights. They are endowed with reason and conscience and should act towards one another in a spirit of brotherhood.

Malay: Perkara 1 – Semua manusia dilahirkan bebas dan sama dalam hak. Mereka dikurniakan akal dan hati nurani dan harus bertindak antara satu sama lain dalam semangat persaudaraan.

Standard Iban: Pekara 1 : Semua mensia ada meratai enggau hak ke sebaka. Sida diberi penau runding enggau ati tuchi lalu enda tau enda begulai enggau pangan diri dalam serakup entara bala menyadi.

Balau Iban Dialect: Pekaha 1 : Semua mensia ada bebas enggau hak ti sebaka. Sida dibehi penau runding enggau ati behesi alu enda tau enda begulai enggau dihi sama dihi dalam gehempung entaha bala menyadi.

Other Iban Dialect: Pekaro 1 : Semuo mensio ado bebas enggau hak ti sebako. Sida diberi penau runding enggau ati tuchi lalu enda tau enda begulai enggau diri samo diri dalam serakup entaro balo menyadi.

Pikaro 1 : Simuo minsio ado bibas nggau hak ti sibako. Sida dibiri penau runding enggau ati tuchi lalu endo tau endo bigulai enggau diri samo diri dalam sirakup intaro balo minyadi.

title: Iban language