Ramble, Part 47: The Fourfold Way
Mar. 6th, 2008 12:11 pm![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
stoutfellowWe have seen how the study of geometry changed in the course of the nineteenth century. Algebra, starting from a very different point, followed a very different path, but ultimately arrived at a similar place. The groundwork for the first step on that path had been laid by two significant events, both previously described: the representation of the complex numbers as points on the plane, at the hands of Wessel, Argand, and Gauss, and the axiomatization of algebra by George Peacock. Enter William Rowan Hamilton....
The representation of the complex numbers as points on the plane had proven very fruitful, not only in reconciling mathematicians to the "imaginary" numbers, but also in terms of the understanding of geometry; trigonometry, in particular, had been revolutionized by this development. Hamilton, an Irish mathematician/physicist, asked the following question: if the real numbers provide algebraic structure to the line and the complex numbers to the plane, what sort of numbers correspond to the three-space in which we live? He spent considerable time searching for such an "algebra of space", to no avail. The story goes that while strolling through the streets of Dublin, he was finally struck by inspiration. He was crossing Trinity Bridge; he whipped out a pen-knife and carved the essentials of his realization onto the bridge itself. (The carving has long since worn away, but a plaque commemorates the incident - one of the great "AHA!" moments of history, along with Archimedes' bath and Kekulé's dream.)
There were two hurdles Hamilton had had to leap to reach his goal. One was the realization that three dimensions were not enough, that the algebra he sought was four-dimensional. The other was that it was necessary to alter Peacock's axioms - specifically, to abandon the commutative law of multiplication. Modeling his new "quaternions" on the complex numbers, he wrote them in the form a+bi+cj+dk, where a,b,c,d were real numbers and j,k were two new imaginary units. Addition and multiplication were to be carried out as usual, subject to the rules i2=j2=k2=-1, ij=-ji=k, jk=-kj=i, ki=-ik=j. Subtraction was obviously possible; it transpired that division by any nonzero quaternion was also possible. (Care needs to be taken, since multiplication of quaternions is not commutative. If you wish to divide p by q, you must decide whether you want the solution to qx=p or to xq=p; the answers are usually different.)
The shift from the real numbers to the complex numbers required some mental rebalancing; among other things, the concept of order - of numbers being larger or smaller - had to be discarded. The quaternions were even more problematic. One of the most central properties of the complex numbers is the Fundamental Theorem of Algebra, imposing limits on the number of zeroes a polynomial can have. In particular, every nonzero complex number has exactly two square roots. Obviously the analog is untrue in the quaternions, since i, j, k (and their negatives) are all square roots of -1. But it's worse: if a, b, c are three real numbers whose squares sum to 1, then the square of ai+bj+ck is -1. That is, in the quaternions, -1 has infinitely many (in fact, uncountably many) square roots.
The idea of a hierarchy of types of number - integer, rational, real, complex - became harder to hold onto at this point; other, seemingly unrelated, developments elsewhere were to lead to its demise, as we will see in the next couple of Ramble posts.
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The representation of the complex numbers as points on the plane had proven very fruitful, not only in reconciling mathematicians to the "imaginary" numbers, but also in terms of the understanding of geometry; trigonometry, in particular, had been revolutionized by this development. Hamilton, an Irish mathematician/physicist, asked the following question: if the real numbers provide algebraic structure to the line and the complex numbers to the plane, what sort of numbers correspond to the three-space in which we live? He spent considerable time searching for such an "algebra of space", to no avail. The story goes that while strolling through the streets of Dublin, he was finally struck by inspiration. He was crossing Trinity Bridge; he whipped out a pen-knife and carved the essentials of his realization onto the bridge itself. (The carving has long since worn away, but a plaque commemorates the incident - one of the great "AHA!" moments of history, along with Archimedes' bath and Kekulé's dream.)
There were two hurdles Hamilton had had to leap to reach his goal. One was the realization that three dimensions were not enough, that the algebra he sought was four-dimensional. The other was that it was necessary to alter Peacock's axioms - specifically, to abandon the commutative law of multiplication. Modeling his new "quaternions" on the complex numbers, he wrote them in the form a+bi+cj+dk, where a,b,c,d were real numbers and j,k were two new imaginary units. Addition and multiplication were to be carried out as usual, subject to the rules i2=j2=k2=-1, ij=-ji=k, jk=-kj=i, ki=-ik=j. Subtraction was obviously possible; it transpired that division by any nonzero quaternion was also possible. (Care needs to be taken, since multiplication of quaternions is not commutative. If you wish to divide p by q, you must decide whether you want the solution to qx=p or to xq=p; the answers are usually different.)
The quaternions became a central tool of nineteenth-century physics, as Hamilton had hoped, due to their close relation to the various "multiplication" operations that appeared in the newly-developing vector analysis. The quaternion a+bi+cj+dk can be thought of as the sum of the scalar a and the vector v=bi+cj+dk. From this viewpoint, if we multiply the quaternions a+v and b+w, the product has scalar part ab-v·w and vector part aw+bv+v×w. Thus, quaternion multiplication subsumes the ordinary multiplication of scalars, the multiplication of vectors by scalars, the scalar (dot) product of vectors, and the vector (cross) product of vectors. Quaternions were eventually superseded by the more flexible vector analysis of Josiah Willard Gibbs, though, and now are mainly of theoretical interest.
The shift from the real numbers to the complex numbers required some mental rebalancing; among other things, the concept of order - of numbers being larger or smaller - had to be discarded. The quaternions were even more problematic. One of the most central properties of the complex numbers is the Fundamental Theorem of Algebra, imposing limits on the number of zeroes a polynomial can have. In particular, every nonzero complex number has exactly two square roots. Obviously the analog is untrue in the quaternions, since i, j, k (and their negatives) are all square roots of -1. But it's worse: if a, b, c are three real numbers whose squares sum to 1, then the square of ai+bj+ck is -1. That is, in the quaternions, -1 has infinitely many (in fact, uncountably many) square roots.
The idea of a hierarchy of types of number - integer, rational, real, complex - became harder to hold onto at this point; other, seemingly unrelated, developments elsewhere were to lead to its demise, as we will see in the next couple of Ramble posts.
Previous Next
Ramble Contents
![[identity profile]](https://www.dreamwidth.org/img/silk/identity/openid.png){kind=small}
no subject
Date: 2008-03-06 08:24 pm (UTC)Or, at least not so ignorant.
Love, C.
no subject
Date: 2008-03-07 03:08 am (UTC)The first parts in particular might be of interest.
Love, C.
no subject
Date: 2008-03-07 08:03 pm (UTC)no subject
Date: 2008-03-08 06:56 pm (UTC)Ah, he has a web page at http://quaternions.com , and it looks like he's trying to do EM and gravity.
no subject
Date: 2008-03-08 10:20 pm (UTC)