The question Alternative notation for exponents, logs and roots complains that we represent strongly related concepts with vastly different notation (e.g. $x^y = z, \sqrt[y]{z} = x, \log_x z = y$) and asks if there is any alternative that would be better for pedagogical purposes.

I am wondering if there is a more general solution to simplify mathematical notation, either by reducing the number of notations or by using similar-looking notations for similar concepts (like the "triangle of power").

For example, we could do away with addition, subtraction and powers by defining $\color{blue}{\underline \phi} = \ln(\phi)$ and $\color{blue}{\overline \phi} = \exp(\phi)$:

$$\begin{align*} \color{blue}{\underline{\overline x \ \overline y \ \overline z}} & = x + y + z \\ \\ \color{blue}{x \ y \ z} &= x \cdot y \cdot z \\ \\ \color{blue}{\overline{i \pi}} &= e^{i \pi} \\ \\ \color{blue}{\overline{2 \ \underline x}} &= x^2 \\ \\ \color{blue}{\overline{- \underline 2}} &= \frac 1 2 \\ \\ \color{blue}{\overline{\overline{- \underline 2} \ \underline x} = \overline{\underline x / 2}} &= \sqrt{x} \\ \\ \end{align*}$$

Would something like this work in practice, or is there a mathematical need for the kind of motley notation we currently use?

Has anyone proposed a simpler notation such as this or a notation that is more graphically intuitive? What research has been done towards improving or standardizing mathematical notation?

  • 5
    $\begingroup$ Esperanto comes to mind when reading this... $\endgroup$
    – Conrad
    Mar 8, 2022 at 3:29
  • 2
    $\begingroup$ It doesn't seem simple to me that $\overline{2\underline x}x =\overline{3\underline x}$. Maybe I could get used to it. But writing $\underline{\overline{\overline{2\underline x}}\overline x}$ in place of $x^2+x$ seems horrendous. $\endgroup$
    – MJD
    Mar 8, 2022 at 4:35
  • 4
    $\begingroup$ You're replacing each addition with a multiplication and two transcendental operations. This is never going to be simpler. $\endgroup$
    – MJD
    Mar 8, 2022 at 4:47
  • 3
    $\begingroup$ The computer programming language APL came from a project by the mathematician Kenneth Iverson to develop a more streamlined, uniform, "linear" mathematical notation. $\endgroup$ Mar 8, 2022 at 8:23
  • 2
    $\begingroup$ Mathematics is no more artificial than English as both developed organically across millennia and changed dramatically in doing so (try reading 1700 English in original not in modern transcription and same with mathematics of Newton etc) so your proposal is on the same par with Wilkins universal language of those times etc - ultimately an ideological approach that is absolutist (my way is THE way) in the name of this or that so it is a self serving way of getting attention $\endgroup$
    – Conrad
    Mar 8, 2022 at 13:50