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Even Great Mathematicians Guess Wrong

April 13, 2011


How Hamilton spent ten years on an impossible search

William Hamilton, Sir William Rowan Hamilton, was one of the great mathematician-physicists of the 1800’s. He invented a key way to look at physical systems—now called Hamiltonian mechanics—that is used in most theories of modern physics. He also discovered the famous Cayley-Hamilton theorem, a matrix satisfies its own characteristic polynomial. Actually he presented a special case and Arthur Cayley stated the general case —another Matthew effect?

Today I want to talk about perhaps his most prized invention—quaternions—and how we can all guess wrong.

His Search

Hamilton searched for a way to define a useful algebra for reasoning about three-space. His intuition, apparently, was that complex numbers are a powerful tool for studying the two-dimensional plane: each point {(x,y)} corresponds to a unique complex number {x + iy}. The beauty of this correspondence is that it allows you to add, subtract, and multiply points in the plane. This power, especially of multiplication, is immense. It simplifies many proofs, gives us new intuition about points, and in general is extremely powerful.

He hoped to do the same for points in three-dimensional space. That is, he wanted to be able to map the point {(x,y,z)} to generalized complex numbers

\displaystyle  x + iy +jz

so that you could again add, subtract, and multiply points. He reasoned by analogy that such an algebraic formulation of three-space would yield wonderful new insights, and would make thinking about difficult problems easier.

Hamilton spent over ten years of his life trying to figure out how to make this work. That is, he sought a way to define how to multiply

\displaystyle  (a+ib+jc)(r+is+jt)

so that the right properties of algebra were preserved. Of course, one could define the product to be anything, but an arbitrary definition would be of little use. The definition of multiplication had to fit together with addition and subtraction; otherwise, the ability to multiply vectors would not be useful. It was not demanded that the algebra be commutative—Hamilton already understood that matrices are fundamental to Nature and their multiplication is not commutative.

All his attempts failed. No matter how he defined multiplication the resulting algebraic system was unusable. The reason was: he guessed wrong. It is in fact possible to prove that there is no such algebra. None.

His Success

The story is that Hamilton, while walking with his wife, suddenly realized that while he could not handle triples he could handle quads. And the notion of quaternion was born. He carved the equations into a bridge during this famous walk—on Monday 16 October 1843.

Prior to that day, during his search, his son used to ask:

Well, Papa, can you multiply triples? Which he answered each day before his great discovery: No, I can only add and subtract them.

Hamilton considered quaternions to be one of his great discoveries—he felt that the ability to have an algebra of four-space was absolutely fundamental. Let’s return to this in a moment, but first discuss how he could have saved over a decade of work.

Some Identities

Real numbers, complex numbers, and quaternions are closely related to algebraic identities. In particular, it is not hard to see that famous identities arise in a natural manner from the existence of real numbers, complex numbers, then quaternions, and finally octonions. A trivial example is this identity for the reals:

\displaystyle  a^{2}b^{2} = (ab)^{2}.

The more interesting identity

\displaystyle  (a^{2} + b^{2})(c^{2} + d^{2}) = (ac-bd)^{2} + (ad+bc)^{2}

underlies an important one for complex numbers. Let {N(x+iy) = x^{2} + y^{2}} be the norm of the complex number {x+iy}, OK technically the norm-squared. Then the above says that

\displaystyle N((a+ib)(c+id)) = N(a+ib)N(c+id).

This states that the norm is multiplicative. Note that this identity really can be guessed and checked, but a better method is to use complex numbers. The norm function can be defined as follows:

\displaystyle  N(z) = z \bar z,

where {\bar z} is the conjugate of {z}. Then

\displaystyle  \begin{array}{rcl}  	N(wz) 	&=& (wz) {\overline {wz}} \\ 			&=& w z {\bar w} {\bar z} \\ 			&=& w {\bar w} z {\bar z} \\ 			&=& N(w)N(z). \end{array}

This is the way to really “see” why the identity

\displaystyle  (a^{2} + b^{2})(c^{2} + d^{2}) = (ac-bd)^{2} + (ad+bc)^{2}

is true—don’t you agree?

Here is the four square identity that arises from the quaternions:

Even though quaternions are not commutative, a similar norm proof can be given for this more involved identity.

John Baez, in a delightful page which I have already quoted from above, relates that Hamilton’s friend John Graves took this “alchemy” one step further. Here is the eight square identity that arises from the octonion numbers:

But this is as far as it goes. There is a fun quote from the top of Baez’ page that I must include:

The real numbers are the dependable breadwinner of the family, the complete ordered field we all rely on. The complex numbers are a slightly flashier but still respectable younger brother: not ordered, but algebraically complete. The quaternions, being noncommutative, are the eccentric cousin who is shunned at important family gatherings. But the octonions are the crazy old uncle nobody lets out of the attic: they are nonassociative.

Why Hamilton Failed

Let us use the notation {\square_{k}} to stand for the set of numbers that are sums of {k} squares. Then the above show that

\displaystyle  \square_{k}\square_{k} = \square_{k},

for {k=1,2,4,8}.

The key insight that would have saved Hamilton a decade of hard work is simple: if there were a way to multiply triples, that had reasonable properties, then it would follow that

\displaystyle  \square_{3}\square_{3} = \square_{3}.

But this is impossible. Note that {3 = 1+1+1} and {5 = 0+1+4} are in {\square_{3}}, but their product {15} is not. To see this, assume that

\displaystyle  15 = a^{2} + b^{2} + c^{2}.

Now if this is true then it must be true modulo {8}. It is easy to check that the square of every integer modulo {8} is either {0}, {1}, or {4}. The punchline is that there is no way for {7}, which is {15 \bmod 8}, to be obtained as a sum of three numbers chosen from {\{0,1,4\}}.

To verify this, note we can assume that {4 \ge a \ge b \ge c \ge 0}. We argue by looking at how many of {a,b,c} are {4}. If none are {4}, then the maximum we can get is {3}, which is not {7}. If one is {4}, then {b^{2} + c^{2} \equiv 7}. But this is impossible, since the maximum this can be is {2}, which again is not {7}. If two are {4}, then {c^{2} \equiv 7}, which is clearly impossible. Finally, if three are {4}, then this also is clearly impossible. A question: should I have included this paragraph—is the proof too simple?

The Quaternion Scandal

As I stated earlier Hamilton thought quaternions were the answer, the answer to almost all questions. He wrote extensively on them, pointing out the many advantages of basing physics on them. The world quickly divided into two groups: those who thought quaternions were the answer to all, and the those who thought they were unneeded, useless, and dominated by general linear algebra. Two quotes, which I do not have to label, should make this point:

The invention of the calculus of quaternions is a step towards the knowledge of quantities related to space which can only be compared, for its importance, with the invention of triple coordinates by Descartes. The ideas of this calculus, as distinguished from its operations and symbols, are fitted to be of the greatest use in all parts of science.

and

Quaternions came from Hamilton after his really good work had been done; and, though beautifully ingenious, have been an unmixed evil to those who have touched them in any way, including James Maxwell.

These are both from an interesting article by Simon Altmann on quaternions: “Hamilton, Rodrigues, and the Quaternion Scandal.” There are two parts of the “scandal:” Who really invented quaternions? And how important are they? The first question raises another example of the Matthew effect. Olinde Rodrigues published the notion in 1840, while Hamilton discovered them in 1843. Rodrigues did not name them—he was not the famous mathematician that Hamilton was—and he never pushed quaternions like Hamilton did. Besides that, Rodrigues was a Socialist banker, who was certainly not the mathematician Hamilton was.

The other part of the scandal was: were quaternions really the answer to all? The battle was over using quaternions or using more general vector methods. Eventually vectors methods mainly won, although quaternions are used quite a bit today in modern computer graphics. They are quite good at representing orientations and rotations of objects in three dimensions—see this for more details. I wonder what the pure mathematician in Hamilton would have thought about that.

Open Problems

Hamilton guessed wrong: he thought for over ten years that there might be a way to define multiplication on triples that would make them into an algebra. He could have saved himself a great deal of work if he had thought: what happens if such an algebra does exists? He might have quickly realized that it would imply, in our notation, that

\displaystyle  \square_{3}\square_{3} = \square_{3}.

Since this is easily seen to be impossible, he would either have given up or tried four dimensions years earlier.

I wonder sometimes what complexity conjectures have we guessed wrong on, and are there short proofs that they are wrong? Noticing that

\displaystyle  \text{ algebra} \rightarrow \text{ identity } \rightarrow \text{ impossible}

is not trivial. But once it is explicitly stated, it is not too hard to prove.

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44 Comments leave one →
  1. April 13, 2011 8:53 am

    I think the scientific progress needs not only positive results, but also negative, like the impossible search by Hamilton.

  2. somebody permalink
    April 13, 2011 8:57 am

    And then there is of course Clifford/geometric algebra based in part on Hamilton’s work. See here,

    http://en.wikipedia.org/wiki/Geometric_algebra http://faculty.luther.edu/~macdonal/GA&GC.pdf

    Hamilton’s 10 years will turn out to have been well spent.

  3. April 13, 2011 9:19 am

    > A question: should I have included this paragraph—is the proof too simple?

    Not for me, alas. Still an interesting post to me – I read Pynchon’s _Against the Day_ a few years ago and quaternions were a significant subplot in it, though I only vaguely knew of them.

  4. April 13, 2011 11:13 am

    Regarding the 15 = a^2 + b^2 + c^2 proof. I think it’s too complex. Omit the modulo argument, a, b, c must be 15. So you have to work with a, b, c element of {0, 1, 4, 9}. QED.

    • David Marquis permalink
      June 2, 2011 5:07 am

      I would rather check sums of {0,1,4} than sums of {0,1,4,9}. Sure its not a huge difference in this case but when you need to prove things like equalities not holding or the irreducibility of a polynomial mods save lots of time.

  5. Frédéric Grosshans permalink
    April 13, 2011 12:01 pm

    According to some, the “quaternion scandal” is linked with Alice in Wonderland’s mad hatter’s tea party.
    http://www.maa.org/devlin/devlin_03_10.html .

  6. April 13, 2011 12:10 pm

    We could also have quoted Hamilton’s letter written the next day to J.T. Graves: “And here there dawned on me the notion that we must admit, in some sense, a fourth dimension of space for the purpose of calculating with triples ..”

    This might raise the question, why can’t you regard triples as quaternions with the first (or last, say) coordinate set to zero? The answer is that this property is not preserved under quaternion multiplication.

    Indeed, the argument in the post basically establishes that there is no simple three-dimensional space that is preserved under quaternion multiplication. I wonder if this is the real surprise—i.e. that Hamilton may have focused on triples out of the feeling that “if it were possible for d > 3 then surely for d=3…” Perhaps someone more familiar with his letters can enlighten?

  7. April 13, 2011 1:11 pm

    Following the pattern, what about 16? If no one’s claimed it yet, I suggest sixteenions :-)

    Perhaps sometimes, we’re too focused on wanting to find an answer so that contemplating its non-existence becomes increasingly undesirable. And the longer we spend avoiding it, the more incentives there are to continue avoiding it.

    Paul.

    • Thomas Dybdahl Ahle permalink
      April 13, 2011 1:33 pm

      I think sixteenions can’t exist, because you would have to give up one more algebraic property, and there are none left.

      • o0o permalink
        April 14, 2011 2:04 am

        Well, the sedenions are still power-associative, although that ain’t much to work with!

    • Lucas permalink
      April 13, 2011 2:05 pm

      It exists already.

      http://en.wikipedia.org/wiki/Sedenion

    • o0o permalink
      April 13, 2011 2:58 pm

      Those are known as ‘sedenions’, and they not too useful for reasons similar to the above argument: they are not division algebras.

      The only normed division algebras are the reals, the complex numbers, the quaternions, and the octonions. Check out John Baez’s octonion pages linked above: for all other algebras, you have to give up division or norms, for any sane definition of those two words.

      You can do without these other algebras, because there is a homeomorphism between Clifford algebras Cl(n+8) and Cl(n), but I don’t understand the ‘why’ of that fact very well at all.

      • April 13, 2011 4:19 pm

        John Baez’s pages are really good. I’ve just skimmed them so far, but I really like when writers include the history. It helps me to get a sense how and why the knowledge accumulated. Definitely on my reading list :-)

        On the lighter side: who really needs division? It’s kinda messy anyways :-)

        Paul.

      • Paul Beame permalink
        April 14, 2011 12:00 am

        For some highly speculative material on the subject of quaternions and octonions, see Gil Kalai’s blog post on Michael Atiyah’s lecture last fall.

      • April 14, 2011 8:54 am

        Dear Paul and all. Thanks for the links to my post. Let me mention that beside the description of speculative application of Octonions in physics in Atiya’s lecture the post described Xavier Dahan and Jean-Pierre Tillich’s Octonion-based Ramanujan Graphs with High Girth. Their paper http://arxiv.org/PS_cache/arxiv/pdf/1011/1011.2642v1.pdf uses Octonions in place of Quaternions for the construction of Ramanujan graphs and describes a wonderful breakthrough in creating small graphs with large girth.

  8. Roy Maclean permalink
    April 13, 2011 2:25 pm

    If you are willing to allow zero divisors then you can get the tricomplex numbers, a commutative algebra with its own analytic function theory.

    http://en.wikipedia.org/wiki/Tricomplex_number

    http://arxiv.org/abs/math/0008120

  9. April 13, 2011 4:48 pm

    Another perfect example of negative results. Most people would never publish negative results.

  10. Allen Knutson permalink
    April 13, 2011 8:47 pm

    The famous Lord who declares quaternions an unmixed evil goes on to rail “Vector is a useless survival”. I think Hamilton’s insight has been pretty well borne out on that score.

  11. Computronium permalink
    April 14, 2011 12:59 am

    >A question: should I have included this paragraph—is the proof too simple?

    For every mathematician here, there’s probably a dilettante like me that’s struggling hard to understand whatever small portion they can of every post. Including a simple proof every now and again is throwing us a bone. Some might think throwing bones to dilettantes isn’t worth the effort, but I sure appreciate it.

  12. PaulC permalink
    April 14, 2011 4:13 pm

    Great post. I read a similar account of Hamilton’s discovery of quaternions a while back when I was quixotically trying to derive a simpler form of Leslie Greengard’s Fast Multipole Method in three dimensions. I’m not even sure how I got to quaternions from there, except that I knew I needed a 3-D equivalent of complex numbers. I didn’t make much progress, but I wound up reading biography of Hamilton with a memorable quotation from Tait that I was able to track down, likening quaternions to an elephant’s trunk:

    “A much more natural and adequate comparison would, it seems to me, liken Coordinate Geometry to a steam-hammer, which an expert may employ on any destructive or constructive work of one general kind, say the cracking of an eggshell, or the welding of an anchor. But you must have your expert to manage it, for without him it is useless. He has to toil amid the heat, smoke, grime, grease, and perpetual din of the suffocating engine-room. The work has to be brought to the hammer, for it cannot usually be taken to its work. And it is not in general, transferable; for each expert, as a rule, knows, fully and confidently, the working details of his own weapon only. Quaternions, on the other hand, are like the elephant’s trunk, ready at any moment for anything, be it to pick up a crumb or a field-gun, to strangle a tiger, or uproot a tree; portable in the extreme, applicable anywhere—alike in the trackless jungle and in the barrack square—directed by a little native who requires no special skill or training, and who can be transferred from one elephant to another without much hesitation. Surely this, which adapts itself to its work, is the grander instrument. But then, it is the natural, the other, the artificial one.”

    See, e.g. http://www.economicexpert.com/3a/Arthur:Cayley.htm Tait’s book is also online.

    What I like about this is that it’s probably true in a sense–quaternions being a curiosity of nature, and matrices a general-purpose tool–but it gets the final outcome wrong. There are still things that elephants can do better than machines, but it’s probably not worth learning the care and feeding of elephants when some machine will do an adequate job at the same task. Likewise I wonder about quaternions. I have heard that they are useful in computer graphics, but given that you will probably use other linear transformations, it’s not obvious that there is much point to learning them specially.

    Before this, I had only really seen them used as an example in introductory group theory classes, usually without any motivation. The history and controversy is very interesting and it’s good to see it highlighted here.

    • October 10, 2012 10:36 am

      PaulC did you get anywhere with simplifying FMM in 3d with quaternions? I would like to do it too. But I’m not sure if quaternions are the best choise so I would like to find appropriate Clifford algebra and use it. I’m still at very beginning so I study Clifford algebra than I have to study Clifford analysis and finaly than I might be able to “simplify” FMM in 3 or maybe in n dimensions. So if this message gets to you somehow I would love to hear from you and see what have you done!

  13. proaonuiq permalink
    April 15, 2011 6:34 am

    Who invented the quaternions ?

    There is a nice chapter about this in Kline´s monumental history of mathematics which includes the discovery of octonions by Graves and Cayley, but omites Rodrigues, the so called invisible mathematician. In any case it seems that Rodrigues was following some research started by Euler related with SO(3) and Hamilton was following a different more algebraic thread.

    Cayley, Hamilton, Euler.. A lindo read&guess post!.

  14. Rafee Kamouna permalink
    April 16, 2011 4:39 am

    The Godel sentence that Godel forgot:

    “The algorithm that recognizes all algorithms that do not recognize themselves.”

    Does it recognize itself?!?!

    Some say that the above sentence is worth 7 million dollars. Others say that this is the single true statement in all mathematics. Anything else is both true and false.

    Best,

    Rafee Kamouna.

    • MciprianM permalink
      April 17, 2011 5:30 am

      @Rafee Kamouna:
      Your sentence is of the type that high-school freshmen study in my country. First of all, your sentence is ambigous. It could be transformed in the following , more english friendly, version:
      P : “Does the algorithm that recognizes all algorithms that do not recognize themselves recognize itself?”
      Well the answer can only be “Yes, it does recognize itself”. Suppose the answer is “No, it doesn’t recognize itself”. Then “the algorithm that recognizes all algorithms that do not recognize themselves” ( let’s call this algorithm A from now on ) enters the category of algorithms that don’t recognize themselves and, by the sentence P above it is recognized by itself ( contradiction ). So the answer to P can’t be “No, it doesn’t recognize itself”.
      If it does recognize itself, then A does not fall into the category of algorithms that don’t recognize themselves ( let’s call this category C ). But your sentence doesn’t say that only algorithms from C are recognized by A. So this is ok. And the answer must be yes, because this is a yes or no question and the answer cannot be no ( Of course given that you prove that A exists ).

      This leads me to the formulation of the sentence, that, I think you intended:

      P’ :”Let C be the set of algorithms that do not recognize themselves and let A be an algorithm that recognizes all algorithms in C and only those. Does A recognize itself?”

      By similar reasoning as above we reach the conclusion that the answer cannot be “yes”, neither “no”. This is a contradiction. “Why does this contradiction appear?” one might ask.
      It is simple. The sentence above ( or the problem or whatever you might want to call it ) is wrong. It says “Let C be the set of algorithms that do not recognize themselves and let A be an algorithm that recognizes all algorithms in C and only those.”. But either A, either C doesn’t exist ( either both don’t exist ). Why? Proof by contradiction again: Suppose they exist. Then the answer to P’ can’t be neither yes, neither no. This is impossible. So either A or C don’t exist. Herefrom the “paradox”. The sentence is simply wrong.

      I hope this cleared your misunderstanding of your sentence.

  15. April 23, 2011 7:53 pm

    I would like to make a few small comments to the original article, which I much enjoyed:
    1. Rodrigues wasn’t looking for an algebraic structure; he was merely (!) the first person to give the correct algorithm for combining 3D rotations. Cayley noticed that the algorithm was equivalent to quaternion multiplication, and thought this was “interesting”.
    2. Altmann’s most interesting claim in the paper cited, imho, is that Hamilton failed to realize the full power of quaternions because he was fixated on the idea that i, j, and k should rotate through pi/2 (like i in C). To make the quaternions isomorphic to (or, strictly, a double cover of) SO3, you have to let i, j, and k rotate through pi. I’ve looked at some of Hamilton’s papers, and he does in fact make use of this, but never clearly explains the angle doubling, which would have been clear if he had read Rodrigues’s paper.
    3. Quaternions are used rather more extensively nowadays than Lipton suggests. They are provided by almost every game engine (yes: Microsoft does quaternions!) and, as far as I know, are used by NASA et al for spacecraft control.
    I’m an amateur in this area, so please correct me if I’m wrong – I’d like to know!
    Peter

    • anonymouse permalink
      April 29, 2011 6:00 pm

      I’ve seen quaternions used in graphics for controlling rotations, because it ends up being a more natural way of representing them for UI purposes. But once you’ve figured out where to rotate to, it’s all converted into matrices which is what the rendering hardware (or software) actually sees.

      • July 29, 2011 7:12 am

        It is true that quaternions are eventually converted to matrices for the hardware, but they are nonetheless worthwhile for some operations, such as interpolating between rotations for animation.

    • July 28, 2011 12:45 pm

      I was doing a web search that included Hamilton, Rodrigues, and quaternions and came across this post. The section “The Quaternion Scandal” was of particular interest to me because recently I became interested in quaternions and their relation to rotations. In this context, one of the papers I read was the one by Altmann referred to in the post, and what he said was not consistent with what I had read elsewhere. Although I am not a mathematician, I am interested in how mathematical ideas develop, and I went to some of the original sources, which now is easy thanks to the internet. This allowed me to see that some of Altmann’s conclusions are incorrect. His main premise is that Hamilton misunderstood the relation between quaternions and rotations, and that it was Rodrigues who had gotten it right earlier (by a few years) than Hamilton. The story however, is different. Rodrigues was indeed first, but Hamilton found the correct relation independently shortly after he introduced the quaternions, and published his results in a paper that Altmann must have overlooked.

      Altmann also hinted that what lead to the demise of quaternions was Hamilton’s misunderstanding of rotations, but the answer is quite different. There was nothing wrong with quaternions; the problem was that most of what could be done with them could be done more easily with vectors, which are direct descendants of quaternions. In fact, one of the definitions of quaternions that Hamilton used was quaternion = scalar + vector.
      Moreover, his definition of product of quaternions involves the scalar and vector products that we used today.

      The post also refers to the question of “Who really invented quaternions?”. Interestingly,
      at the end of the post there is a link to “Why Is Everything Named After Gauss?”. These two questions are somewhat related. Regarding the first question, it is clear from some of Gauss’s short notes published posthumously that he had introduced concepts that anticipated Hamilton quaternions. In addition, he had derived some of the rotation relations derived by Rodrigues before Hamilton. Neither of them were aware of Gauss’s unpublished results, and here we have another example of his amazing foresight. In this case, however, his name is not associated with any of these ideas.

      These matters are discussed in detail in a paper I submitted for publication. The link is here.

      To help the readers, I placed copies of papers by Hamilton, Gauss, and other relevant authors in this web folder.

      [KWR: made links clickable]

  16. Peter L. Griffiths permalink
    August 26, 2011 10:30 am

    Hamilton in his letter of 17 October 1843 to John Graves is very confused about the relationship between i, j, +1 and -1. His response to encountering ij is to exclaim, What are we to do with ij? Its square would seem to be +1 and this might tempt us to take ij=+1 or
    ij=-1. Hamilton does not seem to recognise that the product of unequal roots of a single common square is non-commutative. All multiplication whether of real or imaginary numbers is commutative. Hamilton seemed to have no idea that i, j, +1 and -1 each had n nth roots. The fact that two of these roots had a single common square did not make them interchangeable, let alone equal or commutative.

  17. Peter L. Griffiths permalink
    August 27, 2011 7:59 am

    Hamilton in his letter of 17 October 1843 to John Graves is very confused about the relationship between i, j, +1 and -1. He asks what are we to do with ij? In fact i and j are the unequal roots of a common square. It is these doubts of Hamilton which are the source of his fallacious theory of the non-commutative properties of the multiplication of imaginary numbers. There is no law of arithmetic which makes ij equal to anything but +1. All multiplication whether of real or imaginary numbers is commutative.

    • Frédéric Grosshans permalink
      March 6, 2012 11:51 am

      The whole point of quaternion is to go beyond real and imaginary numbers. Read at least wikipedia’s entry ( http://en.wikipedia.org/wiki/Quaternion ) before claiming Hamilton doesn’t understand anything.

      • Peter L. Griffiths permalink
        March 8, 2012 10:54 am

        There is no point in Hamilton introducing Quaternions when he clearly had virtually no understanding of real numbers, imaginary numbers and complex numbers. One particular question well beyond Hamilton’s capabilities is why do the roots of +1,-1, +i and -i add up to zero? The answer is that computing all these roots is a matter of going round the angles of a circle and finishing at the 360 degrees (=0 degrees) starting point.

  18. Peter L.Griffiths permalink
    February 26, 2012 11:56 am

    Further to my comment of 27 August 2011, on 16 October 1843 Hamilton alleged that i^2=j^2=k^2=ijk=-1. This equation is incorrect because every real, imaginary and complex number has only two square roots, so k^2=-1 is wrong unless k can equal either i or j.

  19. March 6, 2012 10:44 am

    Further to my comment of 26 February 2012, -1 can only have two square roots, nevertheless -1 can have three cube roots which are cos60+isin60, cos180+isin180 which equals -1, and cos300+isin300. Hamilton seemed to have no understanding of these matters.

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