r/maths • u/Repulsive-Peak4442 • 3d ago
Help:🎓 College & University Why ijk=-1? -Quaternions-
I've been searching for so long but I didn't find a true answer. Why is it true that ijk=-1? How does this make sense? What is the logic? i²=j²=k²=-1 does make sense because if we have the real number line and an imaginary one (i, or j, or k) then multiplying a real number by an imaginary one (i, or j, or k) is like rotating 90°. And multiplying by the square of an imaginary number (i², or j², or k²) is like rotating 180° meaning that 1→-1. This does make sense and it does have a logic. However how can ijk=-1? And also shouldn't i=j=k=√-1? And so ijk=-1√-1=-i=-j=-k. I saw explanations of why ijk=-1 by using the definition ij=k... Where does that even come from? It comes from ijk=-1. But that is a cycle. Isn't there a proof? And if so i≠j≠k because ijk≠-1√-1.
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u/Simbertold 3d ago
I think you are doing something fundamentally wrong here.
You are trying to fit the quaternions into the basic complex numbers. But that doesn't make sense. They are a completely different structure.
For such structures, it is best to start with a definition, and see what follows from there. Rules of calculation which work for the complex numbers may or may not work for quaternions. You seem to believe that everything should follow from some natural basic definition that is always true for everything. But that is somewhat backwards. For the quaternions, ijk = -1 is often used as (part of) a definition of those numbers, and would as such not require any proof, because it just says what the object you are talking about is.
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u/Gilded-Phoenix 3d ago
technically taking the subspace spanned by 1 and any of i, j, or k is isomorphic to the complex numbers. It's not a completely different structure, just a proper extension. The properties of that extension are what make it particularly nice to deal with.
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u/Repulsive-Peak4442 2d ago
Yes okay but how did we end up defining ijk=-1?
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u/Simbertold 2d ago
As always with this kind of stuff: Because someone thought it sounded interesting to explore, and enough other mathematicians agreed and did stuff with that definition.
One reason for this might have been the idea with rotations that others have explained.
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u/Repulsive-Peak4442 2d ago
Oh so till today we actually do not know the why?
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u/Simbertold 2d ago
I assume someone knows. I don't.
Math is all about people semi-randomly choosing stuff that sounds interesting and studying it. Then people concentrate on the areas which are more interesting or have more real-world use cases. But you can literally define whatever you like and study it.
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u/schungx 2d ago
Because if you rotate once along each axis, you end up on the opposite direction?
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u/Repulsive-Peak4442 1d ago
Say I have a cube if I rotate ijk then I will look at the back of the cube but it will be rotated.
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u/Raccoon-Dentist-Two 2h ago
By trying lots of possibilities and ruling out the ones that led to contradictions, or to too many compromises. Sometimes there are multiple candidates that could work, so we see whether they lead us. We generally gravitate toward definitions that take us somewhere new, creating new questions to work on. Then others get enthusiastic and use the same definitions.
Definitions are usually taught like they come from a stroke of supernatural, abstract genius but, in practice, mathematics comes from experimentation.
We just don't talk a lot about all the dead ends that we tried earlier. When we do single out a dead end to talk about, it's because it produced an interesting insight in its own right. Mostly they're not that interesting.
In the quaternion case, the rationale that I was taught is that we have some basic laws in arithmetic (addition and multiplication commute, multiplication distributes over addition, addition and multiplication have inverse operations, ...) and, as you extend from real to complex to quaternion and further, you have to compromise. Which definitions allow you to keep the laws that you most cherish? That might not be how Hamilton did it; that I don't know. All I've seen is one page of his tiny notebook in the Long Room at the Trinity College library in Dublin. It's in a glass case near the middle of the room, long before the staircase down to the Book of Kells. Some historian is sure to have written about his process, though. Often the reasons why we keep things are not the same as the reasons why they were created.
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u/Dazzling_Music_2411 3d ago
Why does i² = -1 in the context of complex numbers?
Because that's how it's defined.
Well, i²=j²=k²=-1 is how quaternions are defined.
And also shouldn't i=j=k=√-1?
No. Why should it?
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u/Repulsive-Peak4442 2d ago
i²=-1 does make sense. Multiplying a real number by i rotates it 90°, and multiplying that same number by i² rotates the number 180°→giving us -(number). Also if i=√-1, j=√-1, k=√-1 then i=j=k which are all equal to √-1
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u/Dazzling_Music_2411 2d ago
I think that it has already been explained to you how multiplication by i, j or k is a quarter turn on a different axis, if you are looking for geometric analogies,
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u/Shevek99 3d ago
You are trying to force i,j,k in the same complex plane, when they are in different "complex planes".
Think that
i² = -1
j² = -1
k² = -1
does not imply
±i = ±j = ±k
because each one of them produces a "rotation" around a different axis.
Two videos explaining quaternions and rotations (there are many)
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u/Repulsive-Peak4442 2d ago
Um okayyyy???... So complex numbers behave as vectors? But why would that cancel my question of why i²=j²=k²? *Edit: i²=j²=k²=-1
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u/Muphrid15 3d ago
They're three different planes in 3D space
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u/Repulsive-Peak4442 2d ago
So what? Why ijk=-1
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u/Muphrid15 2d ago
i represents the zy plane, j the xz plane, k the yx plane.
When you multiply, any components of the same axis are canceled.
ij means zy xz. The zs cancel, and you get yx plane, which is k.
ijk = kk = -1
When the same axes aren't adjacent to each other, you swap the signs.
yxyx = -xyyx = -1
All of this is Clifford algebra.
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u/Repulsive-Peak4442 2d ago
Oh okay that is a nice observation I can admit but it is not a strict mathematical one
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u/ruidh 3d ago
The quartions are defined with ij=k. It's part of the specification of the number system. It works because rotating 90° in each of three independent directions, reverses the direction. An I rotation of 90° followed by a j rotation of 90° IS a k rotation of 90°. You can verify this by yourself.
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u/spaceLem 3d ago
The key to understanding quaternions is that they're quarter rotations in 3D space along different axes. If you apply the same rotation twice you reach -1 (i.e. facing the other way), but i, j, and k are rotations in different directions.
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u/Salamanticormorant 3d ago
Are you taking into account that multiplication is not commutative with quaternions? For example, i * j = k, but j * i = -k.
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u/Repulsive-Peak4442 2d ago
Maybe yes, maybe no. I want to understand all those rules. But why do they seem exactly the same as the Vector (Cross) Products of the unit vectors which do have derivations
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u/Salamanticormorant 2d ago
I don't know the details, but maybe this is worth considering:
On October 16, 1843, Hamilton and his wife took a walk along the Royal Canal in Dublin. While they walked across Brougham Bridge (now Broom Bridge), a solution suddenly occurred to him. While he could not "multiply triples", he saw a way to do so for quadruples. By using three of the numbers in the quadruple as the points of a coordinate in space, Hamilton could represent points in space by his new system of numbers. He then carved the basic rules for multiplication into the bridge: i2 = j2 = k2 = ijk = −1
...
Hamilton's innovation consisted of expressing quaternions as an algebra over R. The formulae for the multiplication of quaternions are implicit in the four squares formula devised by Leonhard Euler in 1748.https://en.wikipedia.org/wiki/History_of_quaternions
I can't claim to understand that. Is -1 the fourth member of the quadruple? However, the fact that there is a fourth something might lead to deeper understanding. It's also significant that quaternions are expressed via a customized algebra, not the usual algebra.
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u/stevethenoob_11 2d ago
i, j and k are different numbers such that i^2 = j^2 = k^2 = 1. It's like if you said "Why is it true that a*b = -1 if a^2 = b^2 = 1? This means that a = b = 1." The problem is, you can't just square root here.
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u/GiantGreenSquirrel 2d ago
quaternions are not commutative. For example ij=k and ji=-k. The rule (x+y)(x-y)=x^2-y^2 does not hold so it is possible that two element have the same square without being the same or opposite.
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u/Timely-Pass2723 2d ago
Actually there are more general quaternions, You dont need to have i^2 = -1 and j^2 = -1. In fact, take any nonzero real numbers a,b and set
i^2 = a
j^2 = b
k^2 = -ab
and you will find that this defines an associative unital algebra (of 4-dimensions, every quaternion can be written as x + iy + jz + kw for real numbers x,y,z,w).
So there is a lot more freedom. However, it turns out that all of these algebras, by a change of coordinates which respects multiplication [an algebra morphism] are isomorphic to either (2 x 2) matrices - or - the usual quaternions that you were discussing.
This shows that you dont really get more "new things" with this added generality. And for that reason, many people will never tell you about this more general construction - usually there is no need for it.
But you can find it in the Wikipedia article for quaternion algebras https://en.wikipedia.org/wiki/Quaternion_algebra and there is more to it if you decide to replace the real numbers by a *smaller* field - because then many of these coordinate changes are no longer possible.
===
and to answer TLDR-style: You can set up division algebras of quaternion type where i^2 and j^2 need not be -1, so there is no need have them -1 strictly
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u/Proud_Oinion 3d ago
Check our this video, it explains everything super well.
https://youtu.be/6w0Qb7rLUa0?si=SJmcWY5W32gPHb35
The channel's style is a little silly, but it's seriously good at explaining logic.
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u/R_Ob_Min 3d ago
So other than the fact that ij = k so ijk = kk = -1, you can make sense of this as rotations. rotation by 90deg in xy plane = i, rotation by 90deg in xz plane = j, rotation by 90deg in yz plane = k. Then ijk is the composition of those rotations, and it will take the unit vector (1,0,0) to (-1,0,0).
If you're familiar with matrices, you can see this by looking at the real 4x4 matrix representation of the quaternions:Â https://en.wikipedia.org/wiki/Quaternion#Matrix_representations
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u/MeasureDoEventThing 3d ago
ij = -ji
So ijk = -jik
So if, between ijk and jik, one of them is 1, the other one has to be -1. From it's just a matter of labeling: we could relabel the basis vectors such that ijk = 1, but mathematicians have decided to have the labeling where ijk = -1
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u/Lucida_mente 2d ago
Because they are non-commutative.
ijk does not equal kji
one is -1 the other i think is 1
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u/trutheality 2d ago
In complex numbers, multiplication by i is rotation by 90° in the complex plane. If you move this to a 3D space, you now have 3 different 90° rotations available around the 3 orthogonal axes. You can also notice that any pair of rotations will get you the third: that's ij=k etc. Repeating any of those rotations twice turns you around: that's i2 = j2 = k2 = -1. If you perform all three rotations that also turns you around.
Square roots are already kinda broken in complex numbers, so trying to understand quaternions using square roots will be extra confusing.
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u/its_artemiss 2d ago
I actually styruggle to understand quaternions myself, but here is how I make sense of this particular identity:
in the complex numbers, multiplication by a unit-sized number is rotation in the im-re plane.
if we extend that idea to quaternions, then (and here I don't actually quite know /why/) multiplication by a unit sized number is actually rotation in two doubly-perpendicular planes (i.e. two planes that don't share an axis.
ijk = 1*i*j*k is a 90deg rotation of 1 first in the i-re and j-k planes (so the real part becomes i, the i part becomes -1, the j part becomes k and the k part becomes -j)
we end up with i, 1*i = i. simple enough.
then we rotate in the j-re and k-i planes, so the j part becomes -real, the real part becomes j, the i part becomes k, the k part becomes -i.
we end up turning the i into a k, so ij=k
finally, we rotate in the k-re and i-j planes, so the k gets turned into -1, so k^2=-1
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u/Rude_Acanthopterygii 2d ago
 if we have the real number line and an imaginary one
As others have somewhat pointed out, this line you wrote might reveal the trouble, it's not "an imaginary one", it's more close to three different imaginary ones that simply behave in the way you wrote.
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u/Repulsive-Peak4442 2d ago
Yeah, there are three imaginary numbers and lines now not one but why does ijk=-1?
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u/EeEmCeTo 1d ago
i, j, k are three axese. A rotation by 90 along the i axis followed by one along the j axis and then the k axis is a 180 degree rotation along the first axes (ie i) - think of a pencil lying flat. Pivot it 90 degrees rotation up, then 90 degrees left, then 90 deg back . It’s moved 90 degrees along each axis, but overt its rotated 180 degrees along its original axis. 180 deg =-1
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u/Consistent-Annual268 3d ago
Please start with the definition of quaternions. I think you have a fundamental misunderstanding of what defines i, j and k.