TITLE: What's the Clifford algebra?
QUESTION [31 upvotes]: I'm reading a book on Clifford algebra for physicists. I don't quite understand it conceptually even if I can do most algebraic manipulations. Can some-one teach me what the Clifford algebra really is? (Keep in mind that I don't know abstract algebra, nothing except some group theory.) 
Does it make sense to write the sum of a scalar and a bivector in the Clifford product? Both are very different things.

REPLY [3 votes]: When I first read about 'geometric algebra' and then went down the rabbit hole into exterior algebra, Clifford Algebra and so on, I was struck by mixture of numbers and geometries. I was also excited to see that this was a different way of thinking and offers new insight to how physics and engineering can be taught.
Through most of STEM education, we are pushed up through calculus (all levels) and pick up a few statistics and linear algebra courses. All the while, there is the use of vectors to represent mechanics and fields. What geometric algebra (and Clifford Algebra) does is to extend these concepts and show the inner workings of a larger mathematical basis where vectors take on new meaning. For example, in engineering, we use the cross-product to multiply two vectors and obtain things like a torque vector from a force vector and radius vector. This is limited to 3D. When you use geometric algebra, you find that the ‘wedge product’ is applicable and it is not limited to 3D.
To understand how different geometries can be used in the same equation, you have to consider generalizing the vector. A vector in 3D is the addition of three independent components. Each component has a scalar number, $x$, associated with a unit vector, $v$. Note that the unit vectors keep the numbers separate. When written out, the resultant vector might look like
$$
R = x_1 [v_1] + x_2 [v_2] + x_3[v_3]
$$
In this equation, you would not add $x_1$ to $x_2$ as they are not associated with the same unit vector. Also, the unit vectors are each geometrically distinct. This follows vector algebra.
Another interpretation is that there are different types of addition. Numerical addition is when numbers are added together. Geometric addition is when geometries are added together. To be clearer in the introduction, the geometric addition should be a different symbol, such as circle-plus, $\oplus$, to visually distinguish it from numerical addition, $+$. If this was done, the equation above would be (circle-plus is used)
$$
R = x_1 [v_1] \oplus x_2 [v_2] \oplus x_3[v_3]
$$
This could be extended to higher dimensional vectors.
So if you are comfortable with adding different vectors together, then you can apply this idea to add different geometries together, such as points, vectors, bi-vectors, etc. There is a whole set of geometries that can be constructed by multiplying vectors (not as dot and cross products) with extended multiplication provided by the wedge product. With this start, you can begin to see the impact of numbers and geometric elements and how they can play a role in the same equations.