My understanding is they actually are fractions of things called differential one-forms[1], but even most people who can do calculus don't get to differential geometry, so the sense in which they are fractions is not commonly understood. Michael Penn explains it here https://youtu.be/oaAnkzOaNwM?si=nwNNg4pl7WW4KvIO
A 1-form is a section[1] of the cotangent bundle[2] of a manifold. In other words, a rank 1 covariant tensor field.
At any given point p on an n-dimensional manifold, a 1-form defines an n-dimensional cotangent vector (in the language of bundles[3], a point in the fiber over p).
So how do we define fractions of sections or vectors?
In the article, Baez defines fractions of 2-forms on the plane as the pointwise ratio of coefficients of a basis vector, which he can do because, as he points out, the space of 2-forms at a point on a 2-dimensional manifold is a 1-dimensional vector space (more generally, for k-forms on an n-dimensional manifold, this dimension is n choose k, so only 1 for 0-forms [functions] and n-forms).
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My understanding is they actually are fractions of things called differential one-forms[1], but even most people who can do calculus don't get to differential geometry, so the sense in which they are fractions is not commonly understood. Michael Penn explains it here https://youtu.be/oaAnkzOaNwM?si=nwNNg4pl7WW4KvIO
[1] https://mathworld.wolfram.com/Differentialk-Form.html
A 1-form is a section[1] of the cotangent bundle[2] of a manifold. In other words, a rank 1 covariant tensor field.
At any given point p on an n-dimensional manifold, a 1-form defines an n-dimensional cotangent vector (in the language of bundles[3], a point in the fiber over p).
So how do we define fractions of sections or vectors?
In the article, Baez defines fractions of 2-forms on the plane as the pointwise ratio of coefficients of a basis vector, which he can do because, as he points out, the space of 2-forms at a point on a 2-dimensional manifold is a 1-dimensional vector space (more generally, for k-forms on an n-dimensional manifold, this dimension is n choose k, so only 1 for 0-forms [functions] and n-forms).
[1] https://mathworld.wolfram.com/BundleSection.html
[2] https://mathworld.wolfram.com/CotangentBundle.html
[3] https://mathworld.wolfram.com/FiberBundle.html