Complex numbers are multivectors | Geometric algebra episode 4

Transcript

Today we revisit our old friends, the complex numbers. We will find them hiding inside of Clifford algebra in a surprising and very satisfying way. In fact, I would argue that today we are finally going to fully understand what the imaginary unit really is. We will also discover that complex multiplication is just a special case of the geometric product. But first, a quick recap.

We started from a two-dimensional vector space with basis vectors E1 and E2. On top of this, we built a fourdimensional geometric algebra, also called a Clifford algebra, where multiv vectors are composed from scalar, vector, and bvector parts. We defined the geometric product between all of these multiv vectors. You can easily calculate it by distributing one multiffactor over another and then just applying a few simple properties of the product. We discovered that the product of two vectors can always be written as a real number plus a bi vector.

It probably still feels a bit weird to just add scalers and bctors together. Don't worry. Today we will find out exactly what this means. The main object we need to study is the basis bi vector. Remember that it's the wedge product between E1 and E2.

And we represent it geometrically as a little piece of the 2D plane. I typically draw it as a little square, but its shape doesn't actually matter. The only important properties of a B vector are the plane that it lives in and its size. When you create a B vector by wedging two vectors together, the size of the resulting B vector is always simply the area of the parallelogram spanned by those two vectors. So in our case, the size of the basis by vector is one.

That's why we call it a unit by vector. Similar to unit vectors which have a length of one. We already learned that the geometric product of two orthogonal vectors equals their wedge product because the dotproduct part is zero. So E1 wedge E2 is the same thing as E1 E2. I usually write it as E1 E2 because it's shorter and I am lazy.

Now for no apparent reason, let's square this by vector. We learned how to do that in the previous video. You can use the properties of the geometric product of basis vectors. For example, we can swap the order of two basis vectors provided that we introduce a minus sign. The square of a unit vector equals 1.

So we can replace this with one and this as well. The result apparently is minus1. H. So we have an object that squares to -1. That's very suggestive, isn't it? It should remind you of the imaginary unit I.

So let's be bold and just call this thing I. We will see just how far we can carry the analogy between the unit by vector and the imaginary unit. Next, let's see what happens when we multiply any given vector by i on the right. Once again, just write your vector as a linear combination of E1 and E2. Distribute, apply the same tricks as always, and we get this answer.

If you draw this out on a plane, you will see that the vector has been rotated over 90°. Wow. So, not only does our unit by vector square to -1, it also rotates vectors by a right angle. That's exactly what the imaginary unit also does. Isn't that amazing? By the way, if you multiply by I on the left and you work everything out, you will get a rotation by 90° in the opposite direction.

This is a stark reminder that the geometric product is not commutative. Anyway, we should really take a step back now so that we can fully appreciate what just happened. We started from a vector space and we invented a new product that had to satisfy a number of properties. Crucially, we demanded that all vectors should square to a real number. And we figured out that this real number is the square of the length of the vector.

In particular, the basis vectors having a length of 1 square to positive 1. So we never explicitly defined anything that squares to a negative number. And yet as if by magic we find that we can combine our two basis vectors into an object that squares to -1. It just shows up as one of the basis blades. We didn't have to invent it.

It's naturally part of our algebra. It seems that imaginary numbers are inevitable. In geometric algebra, objects and operations on objects are the same thing. We've already seen plenty of instances in the past of passive objects that become active and start manipulating and transforming other objects. Here we have another excellent example.

The unit by vector I plays two different but related roles. On the one hand, it's a passive object, a little plane segment with size one. But on the other hand, it's also an active transformation. It takes vectors and rotates them by 90°. This is very typical.

Objects are operators. This makes geometric algebra extremely efficient. We only need to define one set of elements to act as both geometric shapes and the operations on those shapes. But most importantly, we finally have a geometric interpretation for the imaginary unit I. In the videos about complex numbers, I hinted that it performs a 90° rotation.

But now we can say with confidence that it simply is a 90° rotation. And because it's a B vector, it also knows the plane in which it rotates. So it contains even more information than the complex numbers do. Okay, are you ready for the next surprise? Let's bring this formula back. It expresses the geometric product of two vectors as a real number plus a B vector.

But we now know that this unit by vector is actually the imaginary unit I. So we have a real number plus an imaginary number and that my friends is a complex number. Remember when you first saw this formula for the geometric product? Most people have the same reaction. Why on earth would we add two completely different kinds of objects together? And why specifically the dot product plus the wedge product? It feels arbitrary. But now now we see that this weird formula is simply a complex number in disguise.

Well actually I should say it's the other way around. Complex numbers are multiv vectors in this guys. The real part is a scalar an element of grade zero in our algebra. The imaginary part is a b vector which has grade two. So we have a multiv vector in which only the even graded parts are present.

When you multiply two such multiff vectors together the result is again a sum of only even graded elements. So we see that the elements with even grade form a sub algebra. It's closed under multiplication. The elements with even grade are a two-dimensional sub algebra of our fourdimensional Clifford algebra and those two dimensions are exactly what the complex numbers need. Even grades are always interesting.

Later in a three-dimensional vector space, we will see that the even-graded elements give us yet another interesting number system. By the way, there is no such thing as the oddgraded subalgebra. The elements of odd grade are not closed under multiplication and they don't include the neutral element. So only elements with even grades can produce an algebra. What is absolutely amazing is that the geometric product creates complex numbers.

When you multiply any two vectors together, the result must always have this form and so it must be complex. Next time we will see exactly what this complex number does and how it relates to the two vectors that we started from. Right now that connection is obscured by the fact that we are writing everything in terms of coordinates. We have a formula that relates the coordinates of the two vectors to the coordinates of the complex number. Its real and imaginary parts.

In the next video, we will take a geometric approach instead and then the connection will become totally clear and it will give us some extremely useful tools. What's also amazing is that complex multiplication is the geometric product. You simply limit it to the even subal algebra and it turns out to be exactly the same thing. This gives us more confidence that our weird formula for the geometric product makes sense. It's something we already knew and understood.

The geometric product basically generalizes complex multiplication to a larger set of objects. Finally, I want to say a few things about the complex conjugate. For any complex number a + b i you find its conjugate by simply flipping the sign of the imaginary part. So you get a minus b i. geometrically this reflects the point across the real axis.

I have always found this definition quite arbitrary. Why exactly do we flip only the imaginary part but not the real part? One clue for why this definition is more sensible and useful than I expected comes from considering a unit complex number. A number with a modulus of one. It lies on the unit circle. Its conjugate lies on the other side of the real axis also on the unit circle.

The point is that this new number has the opposite angle as the original. When you multiply the number with its conjugate, the angles sum to zero. This causes the product to be the number one, which is the neutral element for multiplication. This means that the conjugate is the inverse of the original number. They multiply to one.

That's exactly the definition of an inverse. So the importance of complex conjugation lies in the fact that it flips angles and that it inverts unit complex numbers and the inverse of a rotation rotates over the opposite angle. Right? So it makes complete sense that angle flipping is related to inverses. A rotation by 45° can be turned into the inverse rotation by simply conjugating the corresponding complex number. But now we can formulate all of this in the language of geometric algebra.

We represent the imaginary unit as the unit by vector E1 E2 which is really the same thing as E1 wedge E2. In order to flip the angle, we can just swap the order of the two vectors. The wedge product is anti-commutative. After all, E1 E2 represents a 90° rotation and E2 E1 represents a minus 90° rotation. Now watch this.

We take a full complex number and we write it in its geometric algebra form. A complex number is always the result of a geometric product between two vectors. The resulting complex number has a real part plus an imaginary part. The real part is the dotproduct of two vectors, so it's symmetric. And the imaginary part is their wedge product, so it's anti-ymmetric.

When we swap the order of the two input vectors, the symmetric part stays the same, but the anti-ymmetric part flips over. And that is exactly what complex conjugation does. So it turns out that conjugation isn't arbitrary after all. It corresponds to swapping the order of the two input vectors which inverts the angle of rotation. We will generalize this in upcoming videos where we will see that you can perform reflections, rotations, translations and other geometric operations by just multiplying a number of vectors together with the geometric product.

In order to invert such an operation, all you have to do is invert the order of the vectors in the product. This is called a reversion and it's why such products are known as verss. Don't worry, we will dive into the details later. I just wanted you to know that complex conjugation is just one very specific form of reversion. It's quite exciting to see all of these concepts come together like this.

inverses, reversion, complex conjugation, flipping the angle, symmetry, and anti-ymmetry. They all cooperate perfectly to make the algebra and the geometry work flawlessly. I hope you enjoyed how at every step we took in this video, we discovered something new, or rather, we rediscovered something familiar. The complex numbers are naturally sitting inside of two-dimensional geometric algebra. We finally understand that the imaginary unit I is a 90° rotation.

Next time we will understand more generally what complex numbers are by looking at their polar representation. And then after that when we move into the third dimension we will see that every 2D plane contains the complex numbers. In every plane as long as it goes through the origin you can always find two orthonormal vectors. They serve as a basis for all the other vectors in the plane. You then wedge them together into a unit by vector.

any such unit by vector squares to negative -1 and then you can add real numbers to scalar multiples of your unit by vector to obtain a complex number system inside your plane of choice. This is going to help us rotate in any plane in any higher dimensional space. Rotations, complex numbers, and two-dimensional planes are intimately connected to each other. And we need geometric algebra to fully understand and appreciate all these connections. I probably don't have to tell you that you can already watch all of those amazing connections right now on Patreon.

Thanks for subscribing, liking, sharing, and commenting. Always keep learning, guys. See you next time.