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I know imaginary numbers solve $x^2 +1=0$, but what is the motivation for quaternions?

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Computer graphics. Hamilton was a visionary. –  mistermarko Aug 28 at 8:11
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Frankly, after the complexes, the idea that solving equations is the primary motivation for number systems breaks down. You can be forgiven for thinking that since apparently that seems to be the primary motivation given in schools... –  rschwieb Aug 28 at 10:13
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He wanted something that his students couldn't search on Wikipedia for their homework assignment. –  Asaf Karagila Aug 28 at 10:20
    
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@Asal: When the donkey claims something came from the horse's mouth, I'm not sure if that's a good thing. :-) –  Asaf Karagila Aug 28 at 17:55

6 Answers 6

up vote 23 down vote accepted

Hamilton (and Graves) wanted to generalize $\mathbb C$ - if viewed as $\mathbb R^2$ with a multiplication that turns it into a field with a multiplicative absolute value. They were looking for something similar in $\mathbb{R}^n$ for $n>2$. It turns out that Hamilton spent 13 years in vain with $n=3$ although it was essentially known since Diophantus that what he was looking for was impossible. He finally figured out that he could succeed for $n=4$ if he gave up commutativity.

(This is a short summary of chapter 20 of Stillwell's wonderful "Mathematics and Its History".)

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Just for information, if memory serves me correctly the last line (which is pretty illegible here) reads "...and carved it on a stone of this bridge". –  Asal Beag Dubh Aug 28 at 13:07
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Why $n=3$ isn't possible? –  metacompactness Aug 28 at 13:37
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@metacompactness: "Multiplicative absolute value" above means that you want multiplication to be defined such that $|\mathbf x_1|\cdot|\mathbf x_2|=|\mathbf x_1{\color{red}\cdot}\mathbf x_2|$ holds for all $\mathbf x_i \in \mathbb{R}^3$ no matter how $\color{red}\cdot$ is defined. So, if $(a_1,a_2,a_3){\color{red}\cdot}(b_1,b_2,b_3)$ where defined to be $(C_1,C_2,C_3)$ with the $C_i$ being arbitrary terms depending on the $a_i$ and $b_i$, then by squaring $(a_1^2+a_2^2+a_3^2)(b_1^2+b_2^2+b_3^2)=C_1^2+C_2^2+C_3^2$, but e.g. $15=(1^2+1^2+1^2)\cdot(0^2+1^2+2^2)$ is not the sum of three squares. –  Frunobulax Aug 28 at 14:05
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@Frunobulax does that argument really work? If the $C_i$ are truly arbitrary functions of the $a_i$ and $b_i$, then they don't necessarily have to be integers. –  echinodermata Aug 28 at 20:31
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@metacompactness: Suppose each number was expressed as $a_0+a_1i+a_2j$. Then letting $ij=a_0+a_1i+a_2j$ and multiplying on both sides by $i$ using field axioms shows that $j$ is a complex number. So unless we drop some of the field axioms we cannot get anything beyond complex numbers. –  Shahab Aug 29 at 9:33

With the complex numbers in hand, it's natural to wonder what other systems of numbers containing the real numbers one might have. Before constructing the quaternions, Hamilton tried in vain to construct a $3$-dimensional system; it turns out that this is impossible, and you can see a reproduction of Hamilton's own proof in this recent entry: Why should I consider the components $j^2$ and $k^2$ to be $=-1$ in the search for quaternions? . If you require that your "number system" be a division algebra, it turns out that there are just four such systems: the real numbers, the complex numbers, the quaternions, and the octonions, so each of this is quite special.

(NB there are plenty of interesting "number systems containing the real numbers", i.e., finite-dimensional algebras over $\mathbb{R}$; in other words division algebras are not the only interesting thing we can ask for.

Remark Let me add that Baez' paper "The Octonions" is a superb starting place for exploring these ideas.

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+1 Just feel like remarking that insisting on having a division algebra is not a problem at all, if you don't insist that it contains the reals. For example over rationals there are infinitely many non-isomorphic non-commutative division algebras of dimensions 4,9,16,25,... (any square will do) with center $\Bbb{Q}$. The problems begin when you combine the requirements of division algebra and contains $\Bbb{R}$. –  Jyrki Lahtonen Aug 28 at 12:50
    
@JyrkiLahtonen Hmm that's interesting: can you point me toward consructions of these finite dimensional division algebras over $\Bbb Q$? With all the finite field extensions of $\Bbb Q$ available, I guess I never sat down to think about the division ring extensions :) –  rschwieb Aug 28 at 12:57
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@rscwieb: Class field theory includes results (Grunwald-Wang and Albert-Brauer-Hasse-Noether) telling that the division algebras over a number field are all cyclic division algebras. The structure of cyclic algebras as well as the condition for them to be division algebras together with examples are outlined in Matt Emerton's answer and an answer by yours truly. I picked up this bit of theory from N.Jacobson's Basic Algebra II, chapter 8. –  Jyrki Lahtonen Aug 28 at 13:05
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I'll point out that you can instead weaken the requirement of having a division algebra over the reals by asking for a composition algebra. Here we allow zero divisors in our rings, and we still require a quadratic form that respects multiplication, but we don't ask for it to be positive. What's interesting is that this weakening only allows for three more algebras over $\mathbb{R}$, namely, the split complex numbers, the split quaternions, and the split octonions, the lattermost of which has attracted a good deal of attention in the field of geometric structures in the last few years. –  Travis Aug 28 at 13:12

$\qquad\qquad\qquad\qquad\qquad$ What is the motivation for quaternions?


Hamilton knew that the complex numbers could be interpreted as points in a plane, and he was looking for a way to do the same for points in three-dimensional space. However, $[$he$]$ had been stuck on the problem of multiplication and division for a long time. He could not figure out how to calculate the quotient of the coordinates of two points in space.

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad$ — Quaternions: History


$\quad$ In $1843$, Hamilton knew that the complex numbers could be viewed as points in a plane and that they could be added and multiplied together using certain geometric operations. $~$ Hamilton sought to find a way to do the same for points in space. But $[$he$]$ had been stuck on defining the appropriate multiplication. According to a letter Hamilton wrote later to his son Archibald:

$\quad$ Every morning in the early part of October $1843$, on my coming down to breakfast, your brother William Edward and yourself used to ask me: "Well, Papa, can you multiply triples?" Whereto I was always obliged to reply, with a sad shake of the head, "No, I can only add and subtract them."

$\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad\quad$ — History of Quaternions: Hamilton's Discovery

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Two important applications come to mind:

  1. They model rotations of the sphere, much like the complex numbers model rotations of the circle.
  2. You can get the four squares theorem, that every integer $n$ can be represented as $n=a^2+b^2+c^2+d^2$ for some integers $a,b,c,d \geq 0$, using unique factorization-like properties of the quaternions. This has to do with thinking of the norm of a quaternion $N(a+bi+cj+dk):= (a+bi+cj+dk)(a-bi-cj-dk) = a^2+b^2+c^2+d^2$.

    Similarly, you can prove that in any commutative ring $R$, if $a$ and $b$ are each the sum of four squares, then $ab$ is the sum of four squares, using an algebraic identity that actually doesn't involve the quaternions, but is completely absurd to discover except by proving that for quaternions $N(\alpha\beta) = N(\alpha)N(\beta)$.

In short, they have both important geometric and algebraic properties, particularly at the intersection.

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A modern motivation is to model 3D orientation and rotation in computer graphics. If you use the simple model of Euler angles, you may run into a problem: gimbal lock. Quaternions (and rotation matrices) avoid this.

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Complex numbers do indeed enable us to solve certain previously unsolvable condition equations; but that is not the motivation for them. It is unfortunately common to confuse application with motivation.

Complex numbers extends the idea of quantity as algebraically-expressed-location to two dimensions after "negation" introduces it in one dimension.

The obvious next extension is to three dimensions; but alas, the same neat properties from one and two dimensions don't hold in three dimensions, as related by others above. That doesn't mean three dimensions is uninteresting; but just that this notational device for representation is less interesting.

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Do you mean the historical motivation? I would consider applications to be very motivating. –  mathematician Aug 29 at 9:17
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Historically, complex numbers were discovered/invented in order to solve cubic equations. –  Matt McNabb Aug 29 at 11:46

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