For instance, most people are familiar with polynomials. Take a polynomial in x (with integer coefficients) and substitute x for 5t everywhere. So for instance, 2x^3 - 8x + 3 becomes 250t^3 - 40t + 3.

The Rees algebra Z[5t] (here Z is the integers) is then just the set of all polynomials you can get this way.

If Wikipedia introduced it like this, I don't think most people would have a problem understanding.

The concept is not (at least not always) actually that complicated, like others are implying. It's our communication that is lacking.

The only maintainers it causes a burden for are the compiler developers (and tech debt within the compiler). But this is pretty much unavoidable if the language is to evolve while remaining backwards compatible -- e.g. it's not like the c++ compiler has a lower rate of tech debt accrual.

- Have a play-by-play view so users can see plays they missed

- Make the "between innings" tabs clickable rather than forcing users to wait for the cycle (cycle by default, pin if user clicked a tab)

- show glove on right hand for outfielders that throw LH

- maybe show baserunners taking leads rather than keeping a foot on the base

Best line is N4 N5, Nx6+ K7, R4 N3+!, K2 N5, N8! Kx8, Rx5#. The site has black instead play Kx6 on the third move, allowing a faster mate.

1. N4 N5

2. Nx6+ K7

3. R4 N3+!

4. K2 N5

5. N8! Kx8

6. Rx5#

No, there is a weekly limit as well. Maxing out a single 5h window uses ~10% of the weekly limit

Of course, the homeomorphism to (R/Z)^2 does not respect the geometry (it is not conformal). If we want the map to preserve angles, we need our fundamental domain to be a parallelogram instead of a rigid square. The shape of the parallelogram depends on the coefficients of the cubic, and the isomorphism is uniquely defined up to choice of a base point O (mapping to the identity element; for elliptic curves, this is normally taken to be the point at infinity). You still get a group law on the parallelogram from vector addition in the same way, and this pulls back to the precise group action on the elliptic curve.

The real magic is that the resulting group law is algebraic, meaning that a*b can be written as an algebraic function of a and b. This means you can do the same arithmetic over any field, not just the complex numbers, and still get a group action.

First = 0 Second = 1 Toward = 2 Third = 3 …

This way, the semantic meaning of the words “first” (prior to all others) and “second” (prior to all but one) are preserved, but we get sensical indexing as well.

(Maybe I am too dogmatic)

As a mathematician, yes, I do. Pretty much everything becomes simpler if you treat zero as the first natural number.

> Incidentally, I find it amusing that in Europe, the ground floor is zero, while in the US, it's one.

I really think it makes no sense to number the ground floor as 1. This means if you want to know the height of the nth floor, you have to multiply the height of a story by (n-1).

There are a ton of other cases where you have to needlessly subtract 1 when people use 1-based indexing. To name a few

- Dates: why did the 21st century start on 1/1/2001? Why are the 1800s the 19th century? It doesn’t make any sense. If we indexed from zero, today would be 5/0/2023 (5 months, 0 days, 2023 years since the common era), in the 20th century. It all becomes so easy and intuitive. - Mathematical foundations: If we are using set theory to encode mathematics, how is the number 1 defined? As a set containing itself? This leads to paradoxes in many cases. As the set containing a different single element? Then we can call its contents 0. - Musical intervals: why do two thirds (ok, major and minor, but I’ll gloss over that) make a fifth? Does 3+3=5? The fact that musical intervals index from 1 significantly increases the cognitive burden for music theory. It becomes much easier when we index from 0. - Birthdays: Age is correctly indexed from zero, but it may seem counterintuitive that your first birthday is the day you are born. So when is your second birthday? The day you turn 1, of course. The word “third” sounds like 3, so it seems reasonable to me to introduce a new ordinal here (I like “toward”). - Computing: languages that use 1-based indexing are obscuring what is actually going on; they generally just subtract 1 internally from the user’s input. They have to, since indexing from 0 is fundamentally more efficient at the hardware level. 4 bits can only store 15 possible addresses if you throw away 0.

These are just a few examples and by no means an exhaustive list. Conversely, I have yet to know of a single instance where 1-based indexing makes more sense or simplifies things (aside from being more compatible with legacy features of our society).

After a while, when you think deeply about it, you start to feel that 0-based indexing is something closer to a fundamental truth, rather than simply a convention. Indeed, I propose that the only reason people find 0-based indexing counterintuitive is due to social conditioning.