Sums and products

As mentioned in the previous post, instead of a full binary tree, we will attempt use a simplified expression tree. One possible design decision is to only include the Sum and Product operators and not have a node for subtraction and division.

Representing subtraction

To represent subtraction, we will have to determine if a term within a sum is negative. We implement this by having all products have a numerical coefficient, apply subtraction when this coefficient is negative.

Representing division

We will use our Exponent operator to represent division by the presence of a negative power (either in numeric form or a product with a negative coefficient).

Drawbacks of this approach

Consider the following:

• $4+(-3x)$ vs $4-3x$
• $\frac{1}{3}xy$ vs $\frac{xy}{3}$
• $\frac{1}{3}x{y}^{-2}$ vs $\frac{x}{3y^2}$

Each set contain two different expressions one might want to work with representing the same mathematic content. Our current approach as it stands will be unable to represent both.

For the case of $4+(-3x)$, Mathlify current does not support it without workarounds as we believe the latter $4-3x$ is what most will considered as more "simplified".

For the next two cases, we supply a fraction_mode flag on each product, and typeset the left sided expression when this flag is false and the right sided expression when set to true.

This has worked so far in our investigations, but may warrant changes in the future. At the moment our code has a workaround for the edge case of expressions like $x^{-1}$. Ordinarily, Mathlify automatically simplifies the internal tree structure by removing singleton sums and products. For example, the sum containing the $x$ term should be just a Variable node, while the product containing just a factor of $x^2$ is better represented by just a Exponent node.

However, our fraction_mode flag only lives in a Product node at the moment, so converting the singleton product $x^{-1}$ into an Exponent node removes the flag. Thus, at the moment, our workaround is to only do the Product simplification of singletons if the fraction_mode is set to false. In the future (especially when we move back to JavaScript from the current Rust based experiments), we may want to have the fraction_mode flag at the root of our expression.

(Edit) Implementing quotients after all

After the original post, I was thinking about how working with (algebraic) fractions form a big part of every students' algebra journey. With that in mind, I have decided to implement the Quotient class with the relevant methods to handle them.