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Talk:De Moivre's formula

Assert true for all real powers
HI De Moivre's formula is actually true for all complex numbers x and all real numbers n, but this requires careful extension of several functions to the complex plane. <-- Loisel I'm not so sure this makes any sense$$. $$

 * From (cosy+isiny)^2=cos2y+isin2y, one obtains cosy+isiny=±SQRT(cos2y+isin2y), or SQRT(cos2y+isin2y)=±(cosy+isiny). Now, let y=x/2. Then (cosx+isinx)^(1/2)=±[cos(x/2)+isin(x/2)]. That is what De Moivre's formula should be when a square root is involved:; the square root must have ± value. One of them is a false root. If so, the example when x=0 and x=2*pi will not be ambiguous as long as the false root is discarded. Therefore, the section that says De Moivre's formula fails for non-integer power has used the wrong form of De Moivre's formula and has drawn an incorrect conclusion from the wrong formula. Can the editor give a legitimate reference to back up his conclusion? 2602:306:CE35:2510:5008:B12F:27A9:9BF1 (talk) 19:53, 18 December 2014 (UTC)Mystified

 * Plus 1 is a value. Minus 1 is a value. However plus or minus 1 is not a value. it is two possible values. Waving your hands saying one root is false does not fix anything. How do you know which is the 'false' root except in that it doesn't correspond with what you first thought of? Have you tried reading the section De Moivre's formula? Dmcq (talk) 00:04, 19 December 2014 (UTC)

 * Yes, I have. That is why I am asking where one can find a widely referenced paper (published in a reputable journal), or a textbook, or a handbook that says De Moivre's formula fails when the power is not an integer. Then and only then do readers accept that Section in Wiki. To me, that section in Wiki is baseless. Sqrt (1) = ±1. One should keep it as such until information is available to make a judgment. That section in Wiki has no basis to pick one value over the other to support the false claim. 108.227.82.81 (talk) 03:18, 19 December 2014 (UTC) Mystified

 * Are you not making an unduly strong assertion about a false claim? The maths involved is clear enough to most of the experienced maths editors here. One can actually argue this case without reference to de Moivre's formula, only referencing material about exponentiation involving complex numbers. The Exponentiation article gives all the necessary background. —Quondum 03:43, 19 December 2014 (UTC)

 * The example given by Wiki does not support the claim. When the power of a complex number is 1/2, De Moivre's formula should be either (cosx+isinx)^(1/2)=±[cos(x/2)+isin(x/2)], or (cosx+isinx)^(1/2)=cos[k*pi+(x/2)]+isin[k*pi+(x/2)], k=0 and 1. When x=0 and x=2*pi, either form of the formula gives sqrt(1)=±1, as should be. My point is, Wiki had better use some other example to make the claim. 108.227.82.81 (talk) 13:42, 19 December 2014 (UTC)Mystified

 * Your point is simply wrong, and your insistence on repeating it over and over again is very dull. The article is correct as written, contains a reasonably clear explanation of why it is correct, and also contains a not-wrong explanation of how to handle non-integer powers.  This has been explained to you repeatedly.  So please stop wasting the time of other editors by repeating yourself over and over without trying to understand what you have been told.  To the extent that your problem is failing to understand the simple point that "±1" is not a number (but rather, a set containing two numbers) there is very little I can do for you, but perhaps you could go bother the people at the math reference desk or math stackexchange instead. --JBL (talk) 14:13, 19 December 2014 (UTC)

Can you answer my request to the point? Give me a reference in a textbook, a handbook, or a journal paper. 108.227.82.81 (talk) 23:13, 19 December 2014 (UTC)Mystified

Thanks for pointing that out; n has to be an integer. x can be complex however, since Euler's formula
 * e^(ix) = cos(x) + i sin(x)

works for all complex x. --AxelBoldt

Is the function described as a [multivalued function]? in fact a periodic function?
 * Yes it is. It is 2π periodic, because sin and cos are. --AxelBoldt

Is the function described as a multivalued function in fact a function? If so, what is the domain of this function? z,w being fixed chosen numbers, it is is some sense rather a multivalued constant, isn't it? (I mean, a constant can be seen as a nullary function, but this would be even more nonsense hair-splitting... and again, I can't see on what domain it is defined - don't say the empty set, because it is well known that there is only one universal function defined on this set, and it definetely has not multiple complex values.)  — MFH: Talk 19:46, 10 May 2005 (UTC)

Hey Loisel. I looked in a few advanced level math books that featured DeMoivre's Theorem, and n is always stated as a RATIONAL, but no proof allows for the irrational component of the reals. The high school textbook I looked at claims that n is a POSITIVE integer. Also, this is the only case where I have seen x used in the statement of DeMoivre's formula -- all the books I have referenced use $$\theta$$. Thelema418 (talk) 22:25, 18 August 2012 (UTC)

I now found a few books that state n is real. And after finer reading of the books that say RATIONAL, it appears that they claim n is real in later analysis. What is truly interesting is that several texts make no statement concerning the domain for n. The PRAXIS exam guides created for the math requirements contains the theorem, but does not give clarification. Also, a few of Larson's texts do not state it either. Thelema418 (talk) 03:58, 19 August 2012 (UTC)

Page might want tidying
I've just edited the page in large sections but am not too familiar with wiki editing, as such my edit may seem to be laid out in a somewhat messy fashion for the sections. You may also want to check the whole article to check that the tone stays constant throughout as I have not edited the entire page, only the sections that warranted it. Thank you. :) sumandark8600 (talk) 01:01, 17 January 2015 (UTC)

proof
Is the proof in the article the original proof of the formula? I mean, is it the way DeMoivre obtained or proved it?

Since DM's formula is a lot easier to prove using Euler's formula, I see no other reason for this proof to be in the article (except curiosity)... -- Euyyn (March 26 2005, 2346 GMT)

unreal numbers are things like -20 centimeters and the square route of -5. —Preceding unsigned comment added by 66.189.214.66 (talk) 03:20, 5 May 2009 (UTC)

DeMoivre never explicitly stated this formula; it was named in honor of him, but I'm not certain if he even used it. I added a citation to this article becuase I think that matter is confusing. Thelema418 (talk) 22:05, 18 August 2012 (UTC)

Doing some more research I found that the following 2 page article by De Moivre is cited as his implicit proof. Note that versine was commonly used at the time. http://cerebro.xu.edu/math/Sources/Moivre/de_%20sectione_anguli.pdf Thelema418 (talk) 02:39, 23 August 2012 (UTC)

first proof
I don't think the first proof is correct, because we are assuming without proof that the exponential law is true also for imaginary numbers.

cos(x+i)
I think you guys have the parentheses in the wrong places, don't you?

cos(x+i) sin x is not the same thing as cos x + i sin x, and I'm pretty sure the latter is what you mean. When someone who knows how to edit HTML fixes this, feel free to delete my comments here on the talk page. --anon

 * You are right, there was a bug in the proof. Fixed now. Oleg Alexandrov 19:33, 3 August 2005 (UTC)

de moivre formula is valid for all n
i dunno why this article says only integers, but i can prove its true for i. all i did is the hyberbolic trig functions and i proved that cis(ix)=e^-pi or something, my point is its valid for all complex n. is there something wrong with what i did? --anon

The big problem is that for nonintegers n, the n-th power is not defined uniquely. If you try to use De Moivre's formula then, you run into trouble. See here:

 * $$-1 = \cos \pi + i \sin \pi\,$$

but also

 * $$-1 = \cos 3\pi + i \sin 3\pi\,$$

By De Moivre's formula:

 * $$(-1)^{1/2} = \left(\cos \pi + i \sin \pi\right)^{1/2} =\cos \frac {\pi}{2} + i \sin \frac{\pi}{2}= i\,$$

but also
 * $$(-1)^{1/2} = \left(\cos 3\pi + i \sin 3\pi\right)^{1/2} =\cos \frac {3\pi}{2} + i \sin \frac{3\pi}{2}= -i\,$$

so you get two different answers which is not good. Does this help? Oleg Alexandrov (talk) 05:51, 16 October 2005 (UTC)

 * -1 is square of +i or -i. So square root of -1 is naturally +i and -i. -i should be viewed as -(-1)^{1/2}. In this way, the ambiguity is avoided and De Moivre's formula remains applicable. 70.53.228.108 (talk) 02:13, 21 November 2014 (UTC) Cucaracha
 * The cubic root of -1, obtained by De Moivre's formula, is 0.5+0.866i, -1, 0.5-0.866i. I do not see a problem with the formula when n is a rational number. 70.53.228.108 (talk) 02:38, 21 November 2014 (UTC)Cucaracha
 * The cube root of −1 is also −1 using your logic and De Moivre's formula so all three are the same by your reasoning. There is a section [Failure for non-integer powers which explains why it doesn't work for non-integer powers. Dmcq (talk) 09:39, 21 November 2014 (UTC)
 * De Moivre's formula is valid for all rational n. I said the three cube roots of -1 are 0.5+0.866i, -1, 0.5-0.866i, NOT ALL ARE -1. The section on [Failure for non-integer powers] should be revised. 70.53.228.108 (talk) 16:01, 21 November 2014 (UTC) Cucaracha
 * I did visit that Section. I found the example of using i=(-1)^(1/2)=(1/-1)^(1/2)=1/{(-1)^(1/2)}=-i is wrong. The correct expression is i=(-1)^(1/2)=(1/-1)^(1/2)=±1/{(-1)^(1/2)}=±i. -i is a fictitious root from taking the square root and should not be reported. The correct result is i=(-1)^(1/2)=(1/-1)^(1/2)=-1/{(-1)^(1/2)}=i. This section should be revised.70.53.228.108 (talk) 16:46, 21 November 2014 (UTC)Cucaracha
 * Please refer to the section. It doesn't talk about or use i in the example. Dmcq (talk) 17:51, 21 November 2014 (UTC)
 * PLEASE go to the link "Failure of Power and Logarithm Identities" in that Section to see the example I quoted. I hope someone in charge of editing here would listen to me and would revise this Section. One of the examples this Section based on is flawed. 70.53.228.108 (talk) 21:04, 21 November 2014 (UTC)Cucaracha

I can't decipher what Cucaracha is saying, but it is certainly true that the section in question is not written clearly and could use some serious editing. --JBL (talk) 23:26, 21 November 2014 (UTC)
 * The example in "Failure of Power and Logarithm Identities" that shows i=-i is wrong. It forgets (1)^(1/2) can be ±1. If (1)^(1/2)=-1 instead of 1 is used, i=i will follow. In a math article, how can this kind of mistake not be corrected? 70.53.228.108 (talk) 02:46, 22 November 2014 (UTC) Cucaracha
 * (1) This is a comment about a totally different article, why are you making it here? (2) You don't understand the example you are objecting; the precise point is that it is not possible to define these functions as honest functions in a consistent way.  Saying "but it can take multiple values" is to completely miss the point of the exercise. --JBL (talk) 15:06, 22 November 2014 (UTC)
 * This article makes reference to the link. The link uses an invalid result to make a point. How can you say the link is irrelevant and not see the need to correct a glaring error? Please do not reply to my comment unless you are an editor. It is the editor, not you, who bears the responsibility of making Wikipedia correct and accurate. 70.53.228.108 (talk) 22:14, 22 November 2014 (UTC) Cucaracha
 * Everyone can be an editor on Wikipedia. What is said at the linked place is correct. It seems you disagree with the people there and here so what you need to do to get any further is a WP:Reliable source that says what you say. Without that no 'correction' to conform to your ideas will take place. Dmcq (talk) 23:36, 22 November 2014 (UTC)
 * Come on. n can even be a complex number, let alone a real number. Let us raise e^(ix) to the power i. Then, e^(-x)=cos(ix)+isin(ix)=cosh(x)-sinh(x). So who say n has to be an integer?? Why don't you admit that the example in the link is flawed and n need not be an integer. 70.53.228.108 (talk) 19:56, 23 November 2014 (UTC)Cucaracha
 * Ok, here's what someone said earlier in TALK: ".... I looked in a few advanced level math books that featured DeMoivre's Theorem, and n is always stated as a RATIONAL...." 70.53.228.108 (talk) 20:10, 23 November 2014 (UTC)Cucaracha
 * A citation is where you give the name and author of a book and the page number. And preferably for the discussion here say what it says. I'll copy the demonstration of the problem in the section 'Failure for non-integer powers'
 * For x = 0 the formula gives 1½ = 1
 * F:or x = 2$\pi$ the formula gives 1½ = −1.
 * As that section shows there is no corresponding problem with the complex exponential function equivalent to de Moivre's formula. Basically there are two possible values for 1½ according to de Moivre's formula but a function can only have one value for each input. What you are talking about is in the article as a generalization which can be used to find the various roots of a complex number. Dmcq (talk) 23:02, 23 November 2014 (UTC)
 * No. The example should be understood as cos(kπ+x/2)+isin(kπ+x/2)=±{sqrt[e^(ix)]}, where k=0 and 1. When x=0, the relation becomes cos(kπ)+isin(kπ)=±1. For +1, k=0. For -1, k=1. The result is one-to-one corresponding, The ± from taking a square root cannot be dropped. When you take the square root of 1, do you not get ±1? Don't you see why I keep telling you the example is flawed? 70.53.228.108 (talk) 00:10, 24 November 2014 (UTC)Cucaracha
 * Formulae don't understand anything. It is you who are keeping in mind where the 1 comes from. The 1 does not remember anything. Dmcq (talk) 01:37, 24 November 2014 (UTC)
 * (1) If you stick to i^2=-1, sqrt(1)=±1, and sqrt(-1)=±i, you can never conclude from the example that i=sqrt(-1)=sqrt(1/-1)=-i. (2) Let us assign λ=sqrt(2) in De Moivre's formula [cosh(z)+sinh(z)]^λ=cosh(λz)+sinh(λz). For, say, z=1, we obtain LHS=(1.543081+1.175201)^sqrt(2)=4.113251, and the RHS=cosh[sqrt(2)]+sinh[sqrt(2)]=2.178184+1.935067=4.113251. Next, for, say, z=i, we obtain LHS=[cosh(i)+sinh(i)]^sqrt(2)=[cos(1)+isin(1)]^sqrt(2)=e^[isqrt(2)]=cos[sqrt(2)]+isin[sqrt(2)], and the RHS=cosh[isqrt(2)]+sinh[isqrt(2)]=cos[sqrt(2)]+isin[sqrt(2)]. It is seen, the formula also works for an irrational power. My points are: (A) The article should not restrict the power to integers only. (B) The example used to support the restriction is flawed. 70.53.228.108 (talk) 16:43, 24 November 2014 (UTC)Cucaracha

And you are wrong on all points. Specifically, there is no "±" in de Moivre's formula. At n = ½, de Moivre's formula unambiguously assigns the value 1 to 1½ (when we take x = 0) and also unambiguously assigns the value −1 to 1½ (when we take x = 2π). This inconsistency is precisely what is meant by "the formula doesn't work for non-integer values of the exponent." This is explained clearly in the article (well, at least I feel that it is clear after I have rewritten it). --JBL (talk) 17:28, 24 November 2014 (UTC)
 * So you are the author. Please go to Eq. (1.3-9) in Korn and Korn to see it for yourself. I have decided not to write anymore. Is it not that the title of this Section says de Moivre formula is valid for all n? I happen to agree with it. And you do not. Bye now. 70.53.228.108 (talk) 18:53, 24 November 2014 (UTC)Cucaracha
 * Wikipedia is a collaborative enterprise, there is no one author.
 * I just had a look on the web and it seems there are a number of people saying they have proof for all n or all rational n. Those proofs are not correct because they assume that exponentiation works cleanly and is unique for complex numbers. Dmcq (talk) 23:05, 24 November 2014 (UTC)
 * FWIW, Cucaracha appears to be referencing this book: which I do not have access to from home. --JBL (talk) 23:21, 24 November 2014 (UTC)
 * Also, a brief run around the internet suggests that at least some sources use the name "de Moivre's formula" for the (correct, but different) formula that appears in the Applications subsection. Possibly this is related to (or is the cause of) the underlying misunderstanding. --JBL (talk)
 * I don't have that book either, seemingly there's an old Schaum book for schools which also says something like that but doesn't try and give a proof. Amalgamating those sections makes it clearer what is being said I think, thanks. It is worrying when mathematics becomes a question of faith for people. Dmcq (talk) 09:36, 25 November 2014 (UTC)
 * I apologize for going back on words by writing again. But for the sake of math, I accept to be scolded. I believe the confusion will go away if the example in WP is viewed as follows. cosx+isinx=cos(2kπ+x)+isin(2kπ+x). Then, sqrt(cosx+isinx)=±[cos(kπ+x/2)+isin(kπ+x/2)], k=0, 1. Now, it does not matter if x=0 or x=2π, the above expression will yield sqrt(1)=±1, which is the way it should be. The difference between WP's view and my view is ± after a square root is taken. If my view is correct, then the subject example cannot be used to cast doubt on De Moivre's formula when n is not an integer. An Indian proverb says: "When you lie in bed, it matters not if your head or your feet are at the headboard, your bum is always in the middle of the bed." The moral is the fact cannot change.70.53.231.244 (talk) 17:17, 26 November 2014 (UTC)Cucaracha
 * Well whatever makes you happy but it would be better if you gave up this idea that one thing can be equal to two different things. It is possible to deal with multivalued functions as giving a set containing the results but that's very messy and people try to avoid anything like that nowadays. They started going out of fashion with Riemann surfaces I believe. Dmcq (talk) 17:42, 26 November 2014 (UTC)
 * I commend you on your efforts here User:Dmcq. I feel that part of the problem here stems in part from High School math and in part from the English language.  Case in point, in my High School I was taught the quadratic formula $$ ax^2 + bx + c = 0 \Rightarrow x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2}$$  Now, what could (in general) $$ x = a \pm b $$ mean?  To make sense of this, a naive interpretation would be $$ (x = a + b) \lor (x = a - b) $$  This comes from the English "x is equal to a plus or minus b," transpiled into "x is equal to a plus b or a minus b."  The correct mathematical statement is $$ x \in \{a+b,a-b\}$$, but I don't recall anything that subtle being discussed in any math classes I had in high school, and such a discussion never arose in my University as more precise language is demanded from the outset.Nathan.s.chappell (talk) 11:00, 28 January 2020 (UTC)

TeX formatting/Expert plea
I did a lot of formatting with the TeX and the general organization. Since I'm not an expert in the topic, I really hope I didn't mess anything up. So, if anyone who is an expert can give it a check, I'd be very grateful. Foxjwill 19:52, 5 March 2007 (UTC)

Picture
The picture has the cube roots of unity as (-1/2 +/- i*sqrt(3/2)), where they should be (-1/2 +/- i*sqrt(3)/2).

Complex Numbers and Trigonometry
"The formula is important because it connects complex numbers (i stands for the imaginary unit) and trigonometry"

I think the reason for this article being so quality impaired is that it takes De Moivre's formula at face value and tries to bring together trigonometric functions and the exponential functions as if they were totally unrelated to begin with. This article reads like a formula sheet and nothing else. The real subject here is the complex exponential function, defined by the power series expansion. Sine and Cosine functions are nothing more than notational conveniences for the imaginary and real parts of said expansion, respectively.

If it weren't for De Moivre stumbling upon this before Euler's contributions, this formula would be nothing more than a novelty of Euler's formula. It would be in the best interest mathematically to merge this article into Euler's formula (because it is just that with integer-multipled arguments) or an article on complex variables but the encyclopedic standards say that it should not. I would try to improve the article but I am not versed in the art of editing. --Fusionshrimp (talk) 05:16, 22 January 2009 (UTC)

It says that the formula doesn't hold in it's simplest form for for fractional exponents. But is there any work on how the sine/cosine representation might "mesh" with Newton's binomial expansion applied to complex numbers? That question may not make any sense - number theory isn't my forte at all. —Preceding unsigned comment added by 216.9.142.188 (talk) 02:55, 3 February 2011 (UTC)

de moivre's formula
I will always recall my maths teacher explaining the wonder of de moivre who derived the following:

e to the power of pi i is equal to -1 ie: an indeterminant number to the power of another indeterminant number multiplied by an unreal number can exactly equal a negative integer. —Preceding unsigned comment added by 58.160.121.28 (talk) 22:24, 12 February 2009 (UTC)

Ah the joys of math teachers who only confuse themselves and their students. That's the biggest load of garbage I've ever heard. Could you ask your maths teacher to make precise the notion of an 'indeterminate' number for us? Same goes for 'unreal'. --Fusionshrimp 128.194.39.250 (talk) 07:43, 1 March 2009 (UTC)

Proof for n belonging to the set of Rational numbers!
Hello guys.

I m a mathematician and I want to present before you all a proof on De Moivre's theorem.Well, here is it..

Suppose n= p/q where p,q are integers and q is not equal to 0

we can write cos x + i sin x = cos (q/q)x + i sin (q/q)x

put x/q = a => cos x + i sin x = cos qa + i sin qa

=> cos x + i sin x = (cos a + i sin a)^q [using De Moivre's theorem for integers for q]

Decreasing the power by q on both sides, we have

 * >>>Sorry, if you are a mathematician (as you claim), how can you "decrease the power by q"? Would you also decrease the power by 2 on both sides of the equality 2^2 = (-2)^2 ? Boris Tsirelson (talk) 19:28, 2 January 2010 (UTC)

 * >>>He means take the 'q'th root of both sides. if q=2, then the square root, for example. Just like if X^2=9, then X=sqrt(9), +-3. decrease probably isn't the right way of saying it, but his maths certainly looks fine to me. 118.208.134.29 (talk) 21:47, 31 March 2010 (UTC)

(cos x + i sin x)^1/q = cos a + i sin a

Increasing the power by p on both sides,

(cos x + i sin x)^p/q = (cos a + i sin a)^p

Now we know p/q is n, so

(cos x + i sin x)^n = cos pa + i sin pa [Again using De Moivre's theorem for Integers for p]

Now we can write a= x/q,so

(cos x + i sin x)^n = cos (p/q)x + i sin (p/q) x

Since p/q is n,

therefore,

(cos x + i sin x)^n = cos nx + i sin nx

For n belonging to rational numbers... ____________________________________________________________________________________________________

By this proof, I wish to disprove the so called 'MYTH' that n can be an integer only.

I want all the support I can get from you guys.

I am 100% confident about my proof, but if you sill see a flaw, feel free to discuss it at my wikipedia page!

A publication Backup can be here given

Page 294, Chapter 43

Schaum's Outline series Theory and problems of college Maths 2/ed By:- Frank Ayres,Jr.

Philip.A.Schmidt

 * Schaum's outline 3rd edition says the proof is outside the range of the book. The book has no proof. It states something which is wrong. The proof above is your own Original research and not based on any book. You have not raised it at WP:RD/MA for others to discuss with you like I asked nor found a published proof. Dmcq (talk) 18:55, 2 January 2010 (UTC)

But it is stated there and I have supplied the proofs...so we 2 just shut up for a while and see what others have to say about it! Moreover..that book is for Degree college. That's they deleted the proof. Or if we go by you, a first grader would have all the periodicity of sin and cos and the log of e and all that(the thought makes me laugh) —Preceding unsigned comment added by Mathsislife (talk • contribs) 19:02, 2 January 2010 (UTC)

 * Have a look here for why this "proof" is wrong: http://en.wikipedia.org/wiki/De_Moivre%27s_formula#Failure_for_non-integer_powers Ulner (talk) 19:11, 2 January 2010 (UTC)

 * Gosh you're quick, thanks, I was just putting in the notice that I'd raised this problem at Wikipedia talk:WikiProject Mathematics. Dmcq (talk) 19:16, 2 January 2010 (UTC)

 * I've put in a couple of references to the generalization section into the leader and the bit about the failure, that deals with using fractional powers to find roots. Dmcq (talk) 09:05, 3 January 2010 (UTC)

In the section 'Failure for non Integer power' you stated e^{iπ} = -1 But for any value, The value of e can never fall below 0. what have you to say for it? —Preceding unsigned comment added by Mathsislife (talk • contribs) 09:18, 3 January 2010 (UTC)

 * What you are saying about e is only true for real powers. For complex powers e can assume any complex number value except 0. By the way greater than 0 isn't a very meaningful idea for complex numnbers. See the section Exponential function. See also Euler's identity which is possibly the most famous identity in mathematics. Dmcq (talk) 09:32, 3 January 2010 (UTC)

but the thing is..

In the failure for non integer power section..u took Pi as the argument...but I m giving Proof of Rational numbers. But Pi is irrational!! —Preceding unsigned comment added by Mathsislife (talk • contribs) 08:27, 4 January 2010 (UTC)
 * That was for x. No ones disputing x can be anything. The discussion is about n being a non integer value and n was taken as a half which is a rational non-integer. Dmcq (talk) 15:17, 4 January 2010 (UTC)

That's what I m talking about! In the failure section... When I saw it closely..It revealed some facts

1)When disproving N to b any number, the value of x is not kept constant, which should not be happening because after every period of 2 pi, the values differ...

2)The principal argument of any complex number has to be either -pi to pi or 0 to 2 pi.

Both of them cant happen simultaneously, but in the failure section, u have first taken principle value -pi to pi then in the second example 0 to 2 pi.

The answers are bound to differ.

Please DMCQ read the taylor series of sine and cos functions. They are periodic. —Preceding unsigned comment added by Mathsislife (talk • contribs) 09:59, 7 January 2010 (UTC)

 * What you are saying sounds a bit similar to what I said in Talk:Exponentiation about the principal argument and rational powers if x is restricted to (-π,π]. However a person is quite entitled to specify any value they like at all for x. The business of principal argument only comes into it when converting back to angles from a sine or cosine or whatever, not when handing an argument to sine or cosine. Dmcq (talk) 10:48, 7 January 2010 (UTC)

 * Well If I would have not known that the values of sine and cosine repeat itself after a period of 2π, I would have blindly agreed with you.

But,but,but I agree a person is free to choose any thing. But if you plot a graph of the x u have, firstly under -π,π then to 0, 2π, you would see the difference! —Preceding unsigned comment added by Mathsislife (talk • contribs) 09:49, 8 January 2010 (UTC)

 * How about trying it with π/2 and power 5/2. If you take the square root first then you get π/4 and then the fifth power gives 5π/4. However if you take the 5th power first you get 5π/2 which is the same angle as π/2, then taking the square root gives π/4. Dmcq (talk) 20:43, 8 January 2010 (UTC)

 * Dmcq, I'd like to know where you learnt your "maths" you speak of. First off, √(1/2) = √(2/4) = √2/2; (√2/2)^5 = (√2)^5/2^5 = 4√2/32 = √2/8. Ergo: (√(π/2))^5 = (√(2π))^5/2^5 = (2π)^2√(2π)/32 = (π^2)√2π/8. (Sorry it's so long winded, I wouldn't want you to get lost somewhere, like you did in any lesson with powers). Now, in reverse: √((π/2)^5) = √(π^5/32) = √(π^5)/√32 = (π^2)√π/4√2 = (π^2)√(2π)/8 —Preceding unsigned comment added by 82.8.237.217 (talk) 18:32, 15 January 2010 (UTC)
 * Furthermore, since I forgot partway the first error I noticed, just because cos and sin have periodicity of 2π, that doesn't mean that 5π=π in EVERY branch of mathematics. —Preceding unsigned comment added by 82.8.237.217 (talk) 18:38, 15 January 2010 (UTC)

 * I'm sorry but I really can't be bothered discussing this with you any longer, you need to discuss the problems you have with complex numbers with someone who's paid for the job or likes doing it. As I said before you need to cite a book giving your argument for your argument to be taken seriously on Wikipedia.. Square roots of complex numbers have two possible values and cube roots have three etc, they do not have unique values whereas de Moivre's formula gives a single unique value. Thus it can't be true. There simply is no more to it. The generalization gives the bit about that that the value it gives is one possible value. Dmcq (talk) 18:59, 15 January 2010 (UTC)

are you paid for Defending this Article? —Preceding unsigned comment added by Mathsislife (talk • contribs) 15:47, 21 January 2010 (UTC)

 * No, are you paid to make it wrong? Please go and learn a bit more maths or do some cleanup where maths expertise isn't required. Peoples own work is not suitable for inclusion in Wikipedia even if it is right if it isn't cited somewhere - never mind about if it is wrong. See WP:OWN about complaining about people who defend article if you wish to complain. Dmcq (talk) 17:28, 21 January 2010 (UTC)

I don't wish to complain to anyone, but What I want that the person wrong is you,not me! Please consult an expert, not consider yourself 1! —Preceding unsigned comment added by Mathsislife (talk • contribs) 06:39, 5 February 2010 (UTC)
 * If some editors disagree about an article and there seems no way for them to resolve the dispute on their own then they should follow the WP:DISPUTE process rather than continuing. I asked at the maths wikiproject for other people to look at it and they agreed with me. You still disagree and I see little prospect of agreement. I was pointing out to you the route to complain about me if you felt I was guarding the article from people correcting it. Please read the dispute policy. You can either ignore this article, raise an action yourself, or change the article and have me raise an action. Dmcq (talk) 09:05, 5 February 2010 (UTC)

Unwarranted assumptions about whether "n" is odd or even
The section titled "De_Moivre's_formula" gives a formula for cos(nx) that assumes "n" is even. Likewise, it gives a formula for sin(nx) that assumes "n" is odd. The correct formulae that don't make these unwarranted assumptions are here. I will fix.Anythingyouwant (talk) 05:10, 27 August 2011 (UTC)
 * As far as I can see the old formulae were correct and the new ones are also correct. I can't say I'm particularly happy with either. Neither are very obvious from the original formula. However the new one seems a bit worse to me as having a funny construction for generating just one or minus one and removing half the entries. I've put a new version at here which should make the correspondence more obvious. Dmcq (talk) 11:37, 27 August 2011 (UTC)
 * I've commented over there at List of trigononetric identities. I think it would be better to just add a sentence about the effect of the sine function.  I'll try that in this article.Anythingyouwant (talk) 16:41, 27 August 2011 (UTC)

Refernces, etc.
Allegedly, the Abramowitz and Stegun Handbook of Mathematical Functions was used to source this article. Yet, the source only contains 3 lines about De Moivre's formula. This material says that if x is restricted to a 360° range, then n can be any number; if the restriction is not made, then n must be taken as an integer. This does not agree with the Wikipedia article's presentation of the subject. For that reason I am encouraging editors to find more citations for this article. Thelema418 (talk) 17:16, 19 August 2012 (UTC)

Abraham de Moivre's formula
This formula is named after Abraham de Moivre in which I assert that it should be properly titled de Moivre's formula (lowercase title) as opposed to De Moivre's formula as the formula is not named after Abraham De Moivre. Technical 13 (talk) 18:10, 19 May 2013 (UTC)

 * Here is a correctly capitalized sentence: "De Moivre's formula does not, in general, hold for non-integer powers." The title "De Moivre's formula" is correctly capitalized for the same reasons the sentence is correctly capitalized.  On the other hand, the two capital "D"s bolded in the first sentence should be lower-cased.  --JBL (talk) 18:36, 19 May 2013 (UTC)

 * The exception here is that fact that his name is "de Moivre" and it does not matter where it comes in the sentence since it is a noun. You wouldn't say "EBay's auctions are final." or "IPod's sales are in the multi-millions." It would still be "eBay's auctions are final." or "iPod's sales are in the multi-millions." respectively. So, considering it is a proper noun no different than the other two examples, I maintain that the article should be Lowercase title and that any usage of it would be properly represented as "de Moivre's formula does not, in general, hold for non-integer powers." Technical 13 (talk) 18:52, 19 May 2013 (UTC)

 * WP:NAMECAPS supersedes personal names and specifies: "Proper names of specific places, persons, terms, etc. are capitalized in accordance with standard usage: Winston Churchill, John de Balliol, Wales, Tel Aviv, Three Great Gardens of Japan, etc." Based on that, I maintain that the article should use the Lowercase title template regardless of how it is linked to from other articles. Technical 13 (talk) 21:21, 19 May 2013 (UTC)
 * Okay, had to save to go find the diff... when  added Lowercase title here. Technical 13 (talk) 21:25, 19 May 2013 (UTC)
 * I would maintain that "standard usage" in this case is to use the capital letter when the name appears at the beginning of a sentence (although not, of course, elsewhere), and see no statement that NAMECAPS "supersedes" the rest of MOSCAPS. Indeed, NAMECAPS doesn't mention the specific circumstance questioned here, while, elsewhere in the same article in which it appears, there is a specific statement to the effect that it doesn't apply in this situation. Although I obviously don't have the final word on this, IMO, the specific guideline would supersede the more general one. Otherwise, why have it? Anaxial (talk) 22:00, 19 May 2013 (UTC)
 * That is exactly my argument, the policy supersedes the general usage page. The only question here is whether or not the article should have the Lowercase title template exactly the same as this talk page has as was instituted by an administrator/oversighter/arbitrator. Technical 13 (talk) 22:29, 19 May 2013 (UTC)

Technical13: can you find any evidence anywhere of anyone treating the "de" in a French name the way you suggest at the beginning of a sentence? JBL (talk) 23:54, 19 May 2013 (UTC)
 * I'm not sure that we on the same page. It doesn't matter where it comes in the sentence. The article should be de Moivre's formula with the Lowercase title just like an administrator/oversighter/arbitrator  almost three years ago.  Technical 13 (talk) 23:59, 19 May 2013 (UTC)
 * That would be true if and only if it is standard to treat capitalization in French names in this way. Do you have any evidence at all to support the idea that this sort of nonstandard capitalization is used anywhere, by anyone?  JBL (talk) 00:55, 20 May 2013 (UTC)
 * So, you want an WP:OTHERSTUFFEXISTS list?

Hey, there we go... Another mathematical thing exactly as in this instance that uses Lowercase title just like I think this article should... There is your standard to treat capitalization for French names that are used as proper names for mathematical formulas and theorems in this way. I hope this satisfies your request. Technical 13 (talk) 01:18, 20 May 2013 (UTC)
 * de Bie
 * de Blob
 * de Blob 2
 * de Buddy's
 * de Bourdeille
 * de Branges's theorem

de Moivre's formula is currently grossly incorrect
My corrections were perfectly valid whereas the previous (and now current) version of the page are wrong in their majority. For proof that de Moivre's theorem is true for all ℚ at the very least (though no actual proof is given) please see pg 36 of book with ISBN number: 978-0-435519-21-6. Un-fortuitously I am unaware of any books that provide a proof beyond an integer power but I have already provided a proof and if you find that insufficient I can provide you with other internet sources that cite it. I can understand and forgive a non-mathematical page containing errors as non-mathematical phenomena are not clear cut nor are they ever entirely obvious and often it is plausible to find support for all sets of contradicting information. However this is not the case with mathematics which is the only subject that can have all of its information conclusively proved or disproved. As such maths should never be quoted incorrectly in any Wikipedia article as it is extremely black and white, i.e; things are either purely correct or incorrect and as it stands my corrections were correct whereas the current page reversion is not. I am amazed at how this page is able to pertain information of such mass, gross inaccuracy especially given the simplicity of de Moivre's theorem and the prerequisites needed to understand it. It is after all something that all students of complex numbers by extension of studying further mathematics learn at the age of 16 in England, UK. Though I appreciate that even a graduate of a masters in mathematics may be mis-informed on this subject given that they have never studied complex numbers but as such they should not consider themselves versed enough to be able to misinform others by way or publishing erroneous statements in this article. For anyone unsure of my edit to correct this page please see https://en.wikipedia.org/w/index.php?title=De_Moivre%27s_formula&oldid=642849732 so that I may not have to explain myself to too many people. Thankyou. (Sumandark8600 (talk) 21:00, 17 January 2015 (UTC))
 * To see that De Moivre's formula fails for rational exponents, consider:
 * $$(\cos (2\pi) + i\sin (2\pi))^{1/4} = 1$$
 * while
 * $$\cos ({\pi/2}) + i\sin ({\pi/2}) = i$$
 * Paul August ☎ 22:29, 17 January 2015 (UTC)

 * The error in your proof was in saying the following was true for all n.
 * $$\left( e^{ix} \right)^n = e^{inx} .$$
 * That is only in general true for integer n when applied to complex numbers. See failure of power and logarithm identities. Dmcq (talk) 23:36, 17 January 2015 (UTC)

 * @:Paul August, False; $$(\cos (2\pi) + i\sin (2\pi))^{1/4} = 1, -1, i$$ and $$-i$$ as $$(\cos (2\pi)  + i\sin (2\pi))^{1/4} =  ^4\sqrt{(\cos (2\pi)  + i\sin (2\pi))}$$ as such $$\cos ({\pi/2}) + i\sin  ({\pi/2}) = i$$ is correct for 1 value of $$(\cos (2\pi)  + i\sin (2\pi))^{1/4}$$
 * By amending de moivre from $$(\cos (nx) + i\sin (nx))$$ to $$(\cos (nx + \frac{2k\pi}{n}) + i\sin (nx + \frac{2k\pi}{n}))$$ where x ∈ ℝ, n ∈ ℂ, k ∈ ℤ such that the argument $$ (nx + \frac{2k\pi}{n})$$ falls between -π and π; i.e, -π < argZ ≤ π where Z is a complex number such that $$Z = (\cos (nx + \frac{2k\pi}{n}) + i\sin (nx + \frac{2k\pi}{n}))$$ This is rather simple mathematics so I fail to understand why it is that you find it so hard to comprehend and as such make erroneous and invalid conclusions.
 * I have provided a ISBN number above from a credible book that cites "de moivre's theorem" to be true for all powers of n where n ∈ ℚ and the subsection in the De Moivre's formula directly below 'Failure for non-integer powers, and generalization aka Roots of complex numbers correctly contradicts that de moivre's theorem fails for powers of n wheren ∉ ℤ as it is clearly true for all n; i.e, n ∈ ℂ(Sumandark8600 (talk) 23:52, 17 January 2015 (UTC))
 * @Dmcq, Again that is erroneous due to the refusal to use multi-valued functions "sigh" so much bad maths, so many great mathematicians rolling in their graves (Sumandark8600 (talk) 23:52, 17 January 2015 (UTC))

 * Here's your error.
 * $$(\cos (2\pi) + i\sin (2\pi))^{1/4} = 1, -1, i$$ and $$-i$$
 * But the left hand side is just $$1^\frac{1}{4}$$ and any any positive real raised to a real power is a real number, from e.g. the formula
 * $$a^b = e^{b \ln a}$$
 * so the left hand side is just 1. While as noted
 * $$\cos ({\pi/2}) + i\sin ({\pi/2}) = i$$
 * -- JohnBlackburne wordsdeeds 00:03, 18 January 2015 (UTC)
 * Only for 1 solution as $$1^{1/4} = ^4\sqrt{1} = 1, -1, i, -i $$ ;i.e, $$a^{1/b} = ^b\sqrt{a}$$ which has b solutions. I know my stuff, I am not an idiot (Sumandark8600 (talk) 00:10, 18 January 2015 (UTC))
 * The section on generalization is what you want if you're talking about multiple values. Even from the point of multiple values the equation
 * $$\left( e^{ix} \right)^n = e^{inx} .$$
 * is wrong when used on complex numbers with n being a half. One side only gives one value whereas the other side gives two possible values. The basic point is that a set of possible values is not a particular value. The formula gives values that are equal. Dmcq (talk) 00:20, 18 January 2015 (UTC)

 * Following my earlier point that de moivre's theorem can be amended such that the argument $$ (nx + \frac{2k\pi}{n})$$ falls between -π and π; i.e, -π < argZ ≤ π where Z is a complex number such that $$Z = (\cos (nx + \frac{2k\pi}{n}) + i\sin (nx + \frac{2k\pi}{n}))$$ where x ∈ ℝ, n ∈ ℂ, k ∈ ℤ, this can be extended to exponents such that $$ Z = e^{i(nx+\frac{2k\pi}{n})}$$ I thought that was an intuitive step that needn't be explained to you but obviously I was wrong in assuming that. This is and of itself one of the famous proofs that ii has infinite solutions all of which are real when given no restriction for the argument; i.e, not being limited to the principal argument. The fact that a set of values is not a particular value is beside the point, a multivalued function can be equal to multiple things without them being equal to each other by the nature of it being a function.(Sumandark8600 (talk) 00:36, 18 January 2015 (UTC))
 * Your amendment is a mess multiplying and dividing by n in the same expression. I think I know what you intended but it doesn't work the way you want. The fact that a set of values is not a particular value is the whole point. An expression with an equals sign means one is asserting the two sides are exactly equal, not that one can choose one side so they are equal. By your reasoning 5=0/0 is true. It isn't. Five is just one possible value of 0/0. One should be able to put the expression on each side into a computer, stick in the parameter, and it produces a value which is exactly the same for both sides. And by the way a multiple valued function is not considered a function in modern mathematics. Dmcq (talk) 00:59, 18 January 2015 (UTC)
 * 5 is not a possible value of 0/0, the only possible values of 0/0 are ±∞. My amendment uses correct formatting and is not a mess, different parts of the expression are being multiplied or divided by n, not the entire thing (which would make it a mess). Just because f(x) = a, b, c, d that does not imply that a = b = c = d but only implies that f(x) can take multiple values. Your reasoning is illogical, have you even checked my reference yet? Anyway I'm in the UK where it is 01:15 so I'm off to bed for now, night. (Sumandark8600 (talk) 01:18, 18 January 2015 (UTC))
 * See Indeterminate form about 0/0. I don't have access to that Edexcel school textbook but the fact that you had to make up your own proof rather than it giving one might indicate something to you. If it says it is true for all n then I'm afraid it is wrong and you're getting some strange ideas about equality and functions in your effort to try and make sense of it. Dmcq (talk) 02:04, 18 January 2015 (UTC)

This is not true
The "x" is NOT a complex number. It is the angle of the vector representing the actual complex number, which is "r*(cos(x)+i*sin(x)". An important part of De Moivres formula is "r" which is the length of the vector, because without "r" only complex numbers with numerical value of 1 are represented, and the formula holds for ANY complex number. 87.52.19.19 (talk) 19:07, 17 May 2018 (UTC)

 * This is a difficult one. The statement in the lead is actually true, sine and cosine have been extended so they are defined for complex numbers. However using complex numbers instead of reals in this context is unusual, and especially in elementary maths. De Moive certainly wouldn't have meant complex numbers. I think the mention of x possibly being a complex number should be removed from the lead and x just called an angle, and x possibley being complex should just be mentioned as an extension in the article. Dmcq (talk) 22:38, 17 May 2018 (UTC)

 * This would be ok with me. This only addresses one of the two issues IP raised; the other one (about adding r) is less interesting. --JBL (talk) 23:29, 17 May 2018 (UTC)

 * Good to hear both a clarification and an agreement. Must admit I didn't know of, nor consider the possibility of complex angles. I would still insist on adding "r", because it is both an essential part of De Moivres formula and an absolute necessity for the notifcation of complex numbers in polar form. 87.52.19.19 (talk) 17:28, 18 May 2018 (UTC)

Use in the proof of Euler's formula
Can this equality be used in an elementary proof of Euler's formula, at least in the logarithmic form obtained by Roger Cotes?--109.166.134.178 (talk) 18:19, 22 April 2019 (UTC)
 * You'd need to approximate and go to limits, I don't think that would be elementary. Dmcq (talk) 18:34, 22 April 2019 (UTC)

Additional simple example needed
So as to make this article more accessible to people learning about this concept (like I am right now) I would suggest adding an additional simple example that uses numbers only. Something like evaluating the expression (1 + i)^4. At the moment I am finding the page to be not as helpful a reference source as it could be.--Discott (talk) 11:59, 14 August 2019 (UTC)
 * It isn't directly relevant to that it is more to do with angles like 30° or in radians π/6 where it says
 * ( cos(π/6) + i sin(π/6) )2 = cos(π/3) + i sin(π/3) = cos(60°) + i sin(60°)
 * Perhaps you want to look at complex numbers or exponentiation Dmcq (talk) 15:56, 14 August 2019 (UTC)
 * Perhaps, but then again my example was largely arbitrary. What I am really advocating for is a simple example with integers instead of variables to illustrate the theorem in action.--Discott (talk) 21:27, 14 August 2019 (UTC)
 * 1+i is not a value that can be got by plugging in some angle it would have to be divided by the square root of 2 to be some value that could be got and that value would then correspond to an angle of 45°. You are looking at the wrong place if you are thinking of values like 1+i. Dmcq (talk) 22:02, 14 August 2019 (UTC)
 * I have added a concrete example; I think that Discott is right that this should increase accessibility of the page. --JBL (talk) 23:02, 14 August 2019 (UTC)
 * The article also really needs a section called De Moivre's formula, in part to avoid the awkward jump from introduction to proof (or now, to example). I will add a bit of material to the section about Euler's formula now, but it won't be sufficient. --JBL (talk) 23:06, 14 August 2019 (UTC)
 * Thank you JBL, I appreciate that. please do not get caught up in my suggested example, it is largely irrelevant to my request.--Discott (talk) 13:05, 16 August 2019 (UTC)

title: Talk:De Moivre's formula


text:
Dominoes

Dominoes is a family of tile-based games played with gaming pieces. Each domino is a rectangular tile, usually with a line dividing its face into two square ends. Each end is marked with a number of spots (also called pips or dots) or is blank. The backs of the tiles in a set are indistinguishable, either blank or having some common design. The gaming pieces make up a domino set, sometimes called a deck or pack. The traditional European domino set consists of 28 tiles, also known as pieces, bones, rocks, stones, men, cards or just dominoes, featuring all combinations of spot counts between zero and six. A domino set is a generic gaming device, similar to playing cards or dice, in that a variety of games can be played with a set. Another form of entertainment using domino pieces is the practice of domino toppling.

The earliest mention of dominoes is from Song dynasty China found in the text Former Events in Wulin by Zhou Mi (1232–1298). Modern dominoes first appeared in Italy during the 18th century, but they differ from Chinese dominoes in a number of respects, and there is no confirmed link between the two. European dominoes may have developed independently, or Italian missionaries in China may have brought the game to Europe.

The name "domino" is probably derived from the resemblance to a kind of carnival costume worn during the Venetian Carnival, often consisting of a black-hooded robe and a white mask. Despite the coinage of the word "polyomino" as a generalization, there is no connection between the word "domino" and the number 2 in any language. The most commonly played domino games are Domino Whist, Matador, and Muggins (All Fives). Other popular forms include Texas 42, Chicken Foot, Concentration, Double Fives, and Mexican Train. In Britain, the most popular league and pub game is Fives and Threes.

Dominoes have sometimes been used for divination, such as bone throwing in Chinese culture and in the African diaspora.

Construction and composition of domino sets
European-style dominoes are traditionally made of bone, silver lip ocean pearl oyster shell (mother of pearl), ivory, or a dark hardwood such as ebony, with contrasting black or white pips (inlaid or painted). Some sets feature the top half thickness in MOP, ivory, or bone, with the lower half in ebony. Alternatively, domino sets have been made from many different natural materials: stone (e.g., marble, granite or soapstone); other woods (e.g., ash, oak, redwood, and cedar); metals (e.g., brass or pewter); ceramic clay, or even frosted glass or crystal. These sets have a more novel look, and the often heavier weight makes them feel more substantial; also, such materials and the resulting products are usually much more expensive than polymer materials. Modern commercial domino sets are usually made of synthetic materials, such as ABS or polystyrene plastics, or Bakelite and other phenolic resins; many sets approximate the look and feel of ivory while others use colored or even translucent plastics to achieve a more contemporary look. Modern sets also commonly use a different color for the dots of each different end value (one-spots might have black pips while two-spots might be green, three red, etc.) to facilitate finding matching ends. Occasionally, one may find a domino set made of card stock like that for playing cards. Such sets are lightweight, compact, and inexpensive, and like cards are more susceptible to minor disturbances such as a sudden breeze. Sometimes, the tiles have a metal pin (called a spinner or pivot) in the middle.

The traditional domino set contains one unique piece for each possible combination of two ends with zero to six spots, and is known as a double-six set because the highest-value piece has six pips on each end (the "double six"). The spots from one to six are generally arranged as they are on six-sided dice, but because blank ends having no spots are used, seven faces are possible, allowing 28 unique pieces in a double-six set.

However, this is a relatively small number especially when playing with more than four people, so many domino sets are "extended" by introducing ends with greater numbers of spots, which increases the number of unique combinations of ends and thus of pieces. Each progressively larger set increases the maximum number of pips on an end by three; so the common extended sets are double-nine (55 tiles), double-12 (91 tiles), double-15 (136 tiles), and double-18 (190 tiles), which is the maximum in practice. Larger sets such as double-21 (253 tiles) could theoretically exist, but they seem to be extremely rare if not nonexistent, as that would be far more than is normally necessary for most domino games, even with eight players. As the set becomes larger, identifying the number of pips on each domino becomes more difficult, so some large domino sets use more readable Arabic numerals instead of pips.

Chinese dominoes
In China, early "domino" tiles were functionally identical to playing cards. An identifiable version of Chinese dominoes developed in the 12th or 13th century.

The oldest written mention of domino tiles in China dates to the 13th century and comes from Hangzhou where pupai (gambling plaques or tiles) and dice are listed as items sold by peddlers during the reign of Emperor Xiaozong of Song (r. 1162–1189). It is not entirely clear that pupai means dominoes, but the same term is used two centuries later by the Ming author Lu Rong (1436–1494) in a context that clearly describes domino tiles. The earliest known manual on dominoes is the Manual of the Xuanhe Period which purports to be written by Qu You (1341–1427), but some scholars believe it is a later forgery.

The traditional 32-piece Chinese domino set, made to represent each possible face of two thrown dice and thus have no blank faces, differs from the 28-piece domino set found in the West during the mid 18th century, although Chinese dominoes with blank faces were known during the 17th century. Each domino originally represented one of the 21 results of throwing two six-sided dice (2d6). One half of each domino is set with the pips from one die and the other half contains the pips from the second die. Chinese sets also introduce duplicates of some throws and divide the tiles into two suits: military and civil. Chinese dominoes are also longer than typical European ones.

Dominoes in Europe and North America
Modern dominoes first appeared in Italy during the 18th century, but they differ from Chinese dominoes in a number of respects, and there is no confirmed link between the two. European dominoes may have developed independently, or Italian missionaries in China may have brought the game to Europe. Having been established in Italy, the game of dominoes spread rapidly to Austria, southern Germany and France. The game became fashionable in France in the mid-18th century. The name domino does not appear before that time, being first recorded in 1771, in the Dictionnaire de Trévoux. There are two earlier recorded meanings for the French word domino, one referring to the masquerades of the period, derived from the term for the hooded garment of a priest, the other referred to crude and brightly colored woodcuts on paper formerly popular among French peasants. The way by which this word became the name of the game of domino remains unclear. The earliest game rules in Europe describe a simple block game for two or four players. Later French rules add the variant of Domino à la Pêche ("Fishing Domino"), an early draw game as well as a three-hand game with a pool.

From France, the game was introduced to England by the late 1700s, purportedly brought in by French prisoners-of-war. The early forms of the game in England were the Block Game and Draw Game. The rules for these games were reprinted, largely unchanged, for over half a century. In 1863, a new game variously described as All Fives, Fives or Cribbage Dominoes appeared for the first time in both English and American sources; this was the first scoring game and it borrowed the counting and scoring features of cribbage, but 5 domino spots instead of 15 card points became the basic scoring unit, worth 1 game point. The game was played to 31 and employed a cribbage board to keep score. In 1864, The American Hoyle describes three new variants: Muggins, Bergen and Rounce; alongside the Block Dominoes and Draw Dominoes. In Muggins, the cribbage board was dropped, 5 spots scored 5 points, and game was now 200 for two players and 150 for three or four. Despite the name, there was no 'muggins rule' as in cribbage to challenge a player who fails to declare his scoring combinations. This omission was rectified in the 1868 edition of The Modern Pocket Hoyle, but reprints of both rule sets continued to be produced in parallel for around twenty years before the version with the muggins rule prevailed. From around 1871, however, the names of All Fives and Muggins, became conflated and many publications issued rules for 'Muggins or All Fives' or 'Muggins or Fives' without making any distinction between the two. This confusion continues to the present day with some publications equating the names and others describing All Fives as a separate game.

In 1889, dominoes was described as having spread worldwide, "but nowhere is it more popular than in the cafés of France and Belgium. From the outset, the European game was different from the Chinese one. European domino sets contain neither the military-civilian suit distinctions of Chinese dominoes nor the duplicates that went with them. Moreover, according to Michael Dummett, in the Chinese games it is only the identity of the tile that matters; there is no concept of matching. Instead, the basic set of 28 unique tiles contains seven additional pieces, six of them representing the values that result from throwing a single die with the other half of the tile left blank, and the seventh domino representing the blank-blank (0–0) combination. Subsequently 45-piece (double eight) sets appeared in Austria and, in recent times, 55-piece (double nine) and 91-piece (double twelve) sets have been produced.

All the early games are still played today alongside games that have sprung up in the last 60 years such as Five Up, Mexican Train and Chicken Foot, the last two taking advantage of the larger domino sets available.

Some modern descriptions of All Fives are quite different from the original, having lost much of their cribbage character and incorporating a single spinner, making it identical, or closely related, to Sniff. Most published rule sets for Muggins include the rule that gives the game its name, but some modern publications omit it even though the muggins rule has been described as the unique feature of this game.

Dominoes is now played internationally. It is recognized as an "ingrained cultural activity within the Caribbean" but is also popular with the Windrush generation (who have Caribbean heritage) in the UK.

In the U.S. state of Alabama, although rarely prosecuted, it was illegal to play dominoes on Sunday within the state until the relevant section of the Alabama Criminal Code was repealed, effective April 21, 2015.

Tiles and suits

Dominoes (also known as bones, cards, men, pieces or tiles), are normally twice as long as they are wide, which makes it easier to re-stack pieces after use. A domino usually features a line in the middle to divide it visually into two squares, called ends. The value of either side is the number of spots or pips. In the most common variant (double-six), the values range from six pips down to none or blank. The sum of the two values, i.e. the total number of pips, may be referred to as the rank or weight of a tile; a tile may be described as "heavier" than a "lighter" one that has fewer (or no) pips.

Tiles are generally named after their two values. For instance, the following are descriptions of the tile bearing the values two and five: A tile that has the same pips-value on each end is called a double or doublet, and is typically referred to as double-zero, double-one , and so on. Conversely, a tile bearing different values is called a single.
 * Deuce-five
 * Five-deuce
 * 2-5
 * 5-2

Every tile which features a given number is a member of the suit of that number. A single tile is a member of two suits: for example, belongs both to the suit of threes and the suit of blanks, or 0 suit.

In some versions the doubles can be treated as an additional suit of doubles. In these versions, the belongs both to the suit of sixes and the suit of doubles. However, the dominant approach is that each double belongs to only one suit.

The most common domino sets commercially available are double six (with 28 tiles) and double nine (with 55 tiles). Larger sets exist and are popular for games involving several players or for players looking for long domino games.

The number of tiles in a double-n set obeys the following formula:

 * $$\frac{(n+1)(n+2)}{2}$$

This formula can be simplified a little bit when $$n$$ is made equal to the total number of doubles in the domino set:

$$\frac{(n)(n+1)}{2}$$

The total number of pips in a double-n set is found by: $$\frac{n(n+1)(n+2)}{2}$$ i.e. the number of tiles multiplied by the maximum pip-count (n)

e.g. a 6-6 set has (7 × 8) / 2 = 56/2 = 28 tiles, the average number of pips per tile is 6 (range is from 0 to 12), giving a total pip count of 6 × 28 = 168

Rules
The most popular type of play are layout games, which fall into two main categories, blocking games and scoring games.

 * Most domino games are blocking games, where the objective is to empty one's hand while blocking the opponents’. In the end, a score may be determined by counting the pips in the losing players' hands.
 * In scoring games, the scoring is different and happens mostly during game play, making it the principal objective.
 * A popular version played predominantly in Singapore, referenced as Hector's Rules, allows for playing double tiles on opponents' hands and awards a bonus play of an additional tile immediately after playing a double tile.
 * If an opponent lays all their tiles on their turn, the game is a tie.

Blocking game
The most basic domino variant is for two players and requires a double-six set. The 28 tiles are shuffled face down and form the stock or boneyard. Each player draws seven tiles from the stock. Once the players begin drawing tiles, they are typically placed on-edge in front of the players, so players can see their own tiles, but not the value of their opponents' tiles. Players can thus see how many tiles remain in their opponents' hands at all times.

One player begins by downing (playing the first tile) one of their tiles. This tile starts the line of play, in which values of adjacent pairs of tile ends must match. The players alternately extend the line of play with one tile at one of its two ends; if a player is unable to place a valid tile, they must continue drawing tiles from the stock until they are able to place a tile. The game ends when one player wins by playing their last tile, or when the game is blocked because neither player can play. If that occurs, whoever caused the block receives all of the remaining player points not counting their own.

Middle Eastern version
A common variant of the blocking game that is played in the Middle East features four players with slightly altered rules. The stock is divided equally on all players, each having seven tiles in hand. After drawing the tiles, the player with the double-six tile starts by downing that tile on the table and the game then proceeds counter-clockwise.

Since there is no boneyard, a player without a matching tile passes their turn. A player that is unable to play is called a downed or sitting player. A less common alternation of the middle eastern game requires the player to the left of the sitting player to transfer one of their tiles (not necessarily playable) tile to the downed player. In this variant, if the transferred tile can be played, they have to down it.

Similar to a normal blocking game, the game ends when a player empties their hand or the game is blocked. If the game is blocked, the player with the lightest hand receives points equal to the sum of all losing players' hands. A set of games ends when any player reaches a set amount of points, in which they win. If no player reached a winning score, the winning player from the previous round starts the next game with any tile in their hand and the game proceeds normally.

Latin American version
Another variant of the blocking game is the Latin American version and is played in teams of two. The stock is divided equally among all players, each having seven tiles in hand. Players sitting on opposite ends of the table are part of the same team.

The game ends when one of the players has no tiles left or when the game is blocked. In the first case, the team of the player without any tiles left earns the sum of the points left in the opposing teams' hands. When the game is blocked, the team with the least points in its hands earns the points left in the opposing teams' hands. If both teams have the same points, the team that started wins the round.

Scoring game
Players accrue points during game play for certain configurations, moves, or emptying one's hand. Most scoring games use variations of the draw game. If a player does not call "domino" before the tile is laid on the table, and another player says domino after the tile is laid, the first player must pick up an extra domino.

Draw game
In a draw game (blocking or scoring), players are additionally allowed to draw as many tiles as desired from the stock before playing a tile, and they are not allowed to pass before the stock is (nearly) empty. The score of a game is the number of pips in the losing player's hand plus the number of pips in the stock. Most rules prescribe that two tiles need to remain in the stock. The draw game is often referred to as simply "dominoes".

Adaptations of both games can accommodate more than two players, who may play individually or in teams.

Line of play

The line of play is the configuration of played tiles on the table. It starts with a single tile and typically grows in two opposite directions when players add matching tiles. In practice, players often play tiles at right angles when the line of play gets too close to the edge of the table.

The rules for the line of play often differ from one variant to another. In many rules, the doubles serve as spinners, i.e., they can be played on all four sides, causing the line of play to branch. Sometimes, the first tile is required to be a double, which serves as the only spinner. In some games such as Chicken Foot, all sides of a spinner must be occupied before anybody is allowed to play elsewhere. Matador has unusual rules for matching. Bendomino uses curved tiles, so one side of the line of play (or both) may be blocked for geometrical reasons.

In Mexican Train and other train games, the game starts with a spinner from which various trains branch off. Most trains are owned by a player and in most situations players are allowed to extend only their own train.

Scoring
In blocking games, scoring happens at the end of the game. After a player has emptied their hand, thereby winning the game for the team, the score consists of the total pip count of the losing team's hands. In some rules, the pip count of the remaining stock is added. If a game is blocked because no player can move, the winner is often determined by adding the pips in players' hands.

In scoring games, each individual can potentially add to the score. For example, in Bergen, players score two points whenever they cause a configuration in which both open ends have the same value and three points if additionally one open end is formed by a double. In Muggins, players score by ensuring the total pip count of the open ends is a multiple of a certain number. In variants of Muggins, the line of play may branch due to spinners. In the common U.S. variant known as Fives players score by making the open ends a multiple of five.

In British public houses and social clubs, a scoring version of "5s-and-3s" is used. The game is normally played in pairs (two against two) and is played as a series of "ends". In each "end", the objective is for players to attach a domino from their hand to one end of those already played so that the sum of the end tiles is divisible by five or three. One point is scored for each time five or three can be divided into the sum of the two tiles, i.e. four at one end and five at the other makes nine, which is divisible by three three times, resulting in three points. Double five at one end and five at the other makes 15, which is divisible by three five times (five points) and divisible by five three times (three points) for a total of eight points.

An "end" stops when one of the players is out, i.e., has played all of their tiles. In the event no player is able to empty their hand, then the player with the lowest domino left in hand is deemed to be out and scores one point. A game consists of any number of ends with points scored in the ends accumulating towards a total. The game ends when one of the pair's total score exceeds a set number of points. A running total score is often kept on a cribbage board. 5s-and-3s is played in a number of competitive leagues in the British Isles.

Card games using domino sets
Apart from the usual blocking and scoring games, games of a very different character are also played with dominoes, such as solitaire or trick-taking games. Most of these are adaptations of card games and were once popular in certain areas to circumvent religious proscriptions against playing cards. A very simple example is a Concentration variant played with a double-six set; two tiles are considered to match if their total pip count is 12.

A popular domino game in Texas is 42. The game is similar to the card game spades. It is played with four players paired into teams. Each player draws seven tiles, and the tiles are played into tricks. Each trick counts as one point, and any domino with a multiple of five dots counts toward the total of the hand. These 35 points of "five count" and seven tricks equals 42 points, hence the name.

Competitive play
Dominoes is played at a professional level, similar to poker. Numerous organisations and clubs of amateur domino players exist around the world. Some organizations organize international competitions. Examples include the Anglo Caribbean Dominoes League (ACDL) in the UK which includes over 40 clubs including the Brixton Immortals.

Dominoes in Unicode
Since April 2008, the character encoding standard Unicode includes characters that represent the double-six domino tiles. While a complete domino set has only 28 tiles, the Unicode set has "reversed" versions of the 21 tiles with different numbers on each end, a "back" image, and everything duplicated as horizontal and vertical orientations, for a total of 100 glyphs. Few fonts are known to support these glyphs.

Historical domino competitions

 * Col. Henry T. Titus vs. Capt. Clark Rice for the naming of Titusville, Florida.

Jeffery Sampson, Jr. became the #1 ranked player in the world in dominoes based on the amount of ass whoopings he gave out in 2016 at the pool hall in Gibsland, LA. He won over 100 games in one day on May 24, 2016.

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

title: Dominoes


text:
Dissociation constant

In chemistry, biochemistry, and pharmacology, a dissociation constant (KD) is a specific type of equilibrium constant that measures the propensity of a larger object to separate (dissociate) reversibly into smaller components, as when a complex falls apart into its component molecules, or when a salt splits up into its component ions. The dissociation constant is the inverse of the association constant. In the special case of salts, the dissociation constant can also be called an ionization constant. For a general reaction:

in which a complex breaks down into x A subunits and y B subunits, the dissociation constant is defined as

where [A], [B], and [Ax By] are the equilibrium concentrations of A, B, and the complex Ax By, respectively.

One reason for the popularity of the dissociation constant in biochemistry and pharmacology is that in the frequently encountered case where x = y = 1, KD has a simple physical interpretation: when [A] = KD, then [B] = [AB] or, equivalently,. That is, KD, which has the dimensions of concentration, equals the concentration of free A at which half of the total molecules of B are associated with A. This simple interpretation does not apply for higher values of x or y. It also presumes the absence of competing reactions, though the derivation can be extended to explicitly allow for and describe competitive binding. It is useful as a quick description of the binding of a substance, in the same way that EC50 and IC50 describe the biological activities of substances.

Molecules with one binding site
Experimentally, the concentration of the molecule complex [AB] is obtained indirectly from the measurement of the concentration of a free molecules, either [A] or [B]. In principle, the total amounts of molecule [A]0 and [B]0 added to the reaction are known. They separate into free and bound components according to the mass conservation principle:

To track the concentration of the complex [AB], one substitutes the concentration of the free molecules ([A] or [B]), of the respective conservation equations, by the definition of the dissociation constant,

This yields the concentration of the complex related to the concentration of either one of the free molecules

Macromolecules with identical independent binding sites
Many biological proteins and enzymes can possess more than one binding site. Usually, when a ligand $L$ binds with a macromolecule $M$, it can influence binding kinetics of other ligands $L$ binding to the macromolecule. A simplified mechanism can be formulated if the affinity of all binding sites can be considered independent of the number of ligands bound to the macromolecule. This is valid for macromolecules composed of more than one, mostly identical, subunits. It can be then assumed that each of these $n$ subunits are identical, symmetric and that they possess only a single binding site. Then the concentration of bound ligands becomes

In this case,, but comprises all partially saturated forms of the macromolecule:

where the saturation occurs stepwise

For the derivation of the general binding equation a saturation function r is defined as the quotient from the portion of bound ligand to the total amount of the macromolecule:

K′n are so-called macroscopic or apparent dissociation constants and can result from multiple individual reactions. For example, if a macromolecule M has three binding sites, K′1 describes a ligand being bound to any of the three binding sites. In this example, K′2 describes two molecules being bound and K′3 three molecules being bound to the macromolecule. The microscopic or individual dissociation constant describes the equilibrium of ligands binding to specific binding sites. Because we assume identical binding sites with no cooperativity, the microscopic dissociation constant must be equal for every binding site and can be abbreviated simply as KD. In our example, K′1 is the amalgamation of a ligand binding to either of the three possible binding sites (I, II and III), hence three microscopic dissociation constants and three distinct states of the ligand–macromolecule complex. For K′2 there are six different microscopic dissociation constants (I–II, I–III, II–I, II–III, III–I, III–II) but only three distinct states (it does not matter whether you bind pocket I first and then II or II first and then I). For K′3 there are three different dissociation constants — there are only three possibilities for which pocket is filled last (I, II or III) — and one state (I–II–III).

Even when the microscopic dissociation constant is the same for each individual binding event, the macroscopic outcome (K′1, K′2 and K′3) is not equal. This can be understood intuitively for our example of three possible binding sites. K′1 describes the reaction from one state (no ligand bound) to three states (one ligand bound to either of the three binding sides). The apparent K′1 would therefore be three times smaller than the individual KD. K′2 describes the reaction from three states (one ligand bound) to three states (two ligands bound); therefore, K′2 would be equal to KD. K′3 describes the reaction from three states (two ligands bound) to one state (three ligands bound); hence, the apparent dissociation constant K′3 is three times bigger than the microscopic dissociation constant KD. The general relationship between both types of dissociation constants for n binding sites is

Hence, the ratio of bound ligand to macromolecules becomes

where is the binomial coefficient. Then the first equation is proved by applying the binomial rule

Protein–ligand binding
The dissociation constant is commonly used to describe the affinity between a ligand (such as a drug) and a protein ; i.e., how tightly a ligand binds to a particular protein. Ligand–protein affinities are influenced by non-covalent intermolecular interactions between the two molecules such as hydrogen bonding,  electrostatic interactions, hydrophobic and van der Waals forces. Affinities can also be affected by high concentrations of other macromolecules, which causes macromolecular crowding.

The formation of a ligand–protein complex can be described by a two-state process

the corresponding dissociation constant is defined

where, and represent molar concentrations of the protein, ligand, and protein–ligand complex, respectively.

The dissociation constant has molar units (M) and corresponds to the ligand concentration at which half of the proteins are occupied at equilibrium, i.e., the concentration of ligand at which the concentration of protein with ligand bound  equals the concentration of protein with no ligand bound. The smaller the dissociation constant, the more tightly bound the ligand is, or the higher the affinity between ligand and protein. For example, a ligand with a nanomolar (nM) dissociation constant binds more tightly to a particular protein than a ligand with a micromolar (μM) dissociation constant.

Sub-picomolar dissociation constants as a result of non-covalent binding interactions between two molecules are rare. Nevertheless, there are some important exceptions. Biotin and avidin bind with a dissociation constant of roughly 10−15 M = 1 fM = 0.000001 nM. Ribonuclease inhibitor proteins may also bind to ribonuclease with a similar 10−15 M affinity.

The dissociation constant for a particular ligand–protein interaction can change with solution conditions (e.g., temperature, pH and salt concentration). The effect of different solution conditions is to effectively modify the strength of any intermolecular interactions holding a particular ligand–protein complex together.

Drugs can produce harmful side effects through interactions with proteins for which they were not meant to or designed to interact. Therefore, much pharmaceutical research is aimed at designing drugs that bind to only their target proteins (negative design) with high affinity (typically 0.1–10 nM) or at improving the affinity between a particular drug and its in vivo protein target (positive design).

Antibodies
In the specific case of antibodies (Ab) binding to antigen (Ag), usually the term affinity constant refers to the association constant.

This chemical equilibrium is also the ratio of the on-rate (kforward or ka) and off-rate (kback or kd) constants. Two antibodies can have the same affinity, but one may have both a high on- and off-rate constant, while the other may have both a low on- and off-rate constant.

Acid–base reactions
For the deprotonation of acids, K is known as Ka, the acid dissociation constant. Strong acids, such as sulfuric or phosphoric acid, have large dissociation constants; weak acids, such as acetic acid, have small dissociation constants.

The symbol Ka, used for the acid dissociation constant, can lead to confusion with the association constant, and it may be necessary to see the reaction or the equilibrium expression to know which is meant.

Acid dissociation constants are sometimes expressed by pKa, which is defined by

This \mathrm{p}K notation is seen in other contexts as well; it is mainly used for covalent dissociations (i.e., reactions in which chemical bonds are made or broken) since such dissociation constants can vary greatly.

A molecule can have several acid dissociation constants. In this regard, that is depending on the number of the protons they can give up, we define monoprotic, diprotic and triprotic acids. The first (e.g., acetic acid or ammonium) have only one dissociable group, the second (e.g., carbonic acid, bicarbonate, glycine) have two dissociable groups and the third (e.g., phosphoric acid) have three dissociable groups. In the case of multiple pK values they are designated by indices: pK1, pK2, pK3 and so on. For amino acids, the pK1 constant refers to its carboxyl (–COOH) group, pK2 refers to its amino (–NH2) group and the pK3 is the pK value of its side chain.

Dissociation constant of water
The dissociation constant of water is denoted Kw:

The concentration of water, [H2O], is omitted by convention, which means that the value of Kw differs from the value of Keq that would be computed using that concentration.

The value of Kw varies with temperature, as shown in the table below. This variation must be taken into account when making precise measurements of quantities such as pH.

 * {| class="wikitable" style="text-align:center;"

! Water temperature ! Kw ! pKw
 * 0 °C
 * 0.112
 * 14.95
 * 25 °C
 * 1.023
 * 13.99
 * 50 °C
 * 5.495
 * 13.26
 * 75 °C
 * 19.95
 * 12.70
 * 100 °C
 * 56.23
 * 12.25
 * }
 * 100 °C
 * 56.23
 * 12.25
 * }
 * }

title: Dissociation constant