Does our notation that uses Pi over Tau by default hinder our ability to understand some concepts? [closed]

Possibly an awkward question, however, I am trying to understand why some mathematicians "prefer" to use Tau over Pi and why it is said that our notation is wrong? Is there consensus over this topic? Am I widely misunderstanding it, and can someone give a brief explanation? Thanks.

2 Answers

why it is said that our notation is wrong?

No. A notation can be more useful / convenient than some other notation or less useful / convenient. But just some notation it not wrong in itself.

Of course, one could use some notation in the wrong way, but that applies both to π and to τ or whatever symbol you are using.

Using τ=2π as an abbreviation instead of using 2π would safe you typing / writing some 2's, and maybe sparing some ()'s here and there.

However π and τ are so close semantically that it makes not much sense to "waste" a symbol for that: Most variables in mathematics are just one letter, hence you don't want to attach a new meaning to it. For example, in relativity τ is common and you see π then and when; hence using τ=2π might be confusing. Some authors propose a new symbol like $$\tau\!\pi$$ but that topic is mostly bike-shedding, IMO.

Most publications / books list notations in their appendix, and if you like τ over 2π and it adds value to your publication, just include it in the appendix and clarify your notation. But also notice that humans are perceiving written text not like a collection of letters but rather like an image, hence even sparing some glyphs might reduce legibility.

About preference: I think Tau is prefered because it makes computations using the trigonometric circle more intuitive. For example, since $$\tau = 2\pi$$, a quarter of the circle becomes $$\frac{\tau}{4}$$ instead of $$\frac{\pi}{2}$$. However, as said above, no notation is wrong or right, that is completely arbitrary.