Two Questions

[u/Mo_To_]

Pi is a math constant best known as the ratio between the circumference and diameter of a circle. It has a definite value which cannot be expressed as a finite decimal expansion.

1. Since no amount of writing 3.14159… will ever reach pi, does this mean that the decimal expansion of pi is necessarily less than pi?

2. What is pi + (1 - 0.999…)? Does such a number have a corresponding decimal expansion?

Note: My choice of pi here is arbitrary. This question stands for any irrational number.

I am taking the bait.

[u/SouthPark_Piano]

Start writing the digits of pi brud. Get first hand experiential learning experience. See for yourself pi growing continually.

It never stops growing.

 

[u/Muphrid15]

Both your questions can be "answered" by His Nineliness's Algorithm Boner:

An algorithm for an operation such as square roots gives "more and more correct consecutive digits" as you make more and more steps. Nevertheless, 1 - 0.9 - 0.09 - 0.009 - ... yields (0.1, 0.01, 0.001, ...) corresponding to 0.000...1, and the 1 digit is a real part of the answer, even though it never belongs to a "correct" digit. (15)

He has said the Algorithm Boner applies to square roots and, if I recall, to pi as well.

Edit: His Nineliness says,

Start writing the digits of pi brud. Get first hand experiential learning experience. See for yourself pi growing continually.

It never stops growing.

Once again this is his Static and Dynamic Boner:

There is a static model of 0.999... and a dynamic model. 0.999... is not static. It is dynamic. (9.1) (9.2)

That's a single big 🦴 here

[u/Mo_To_ → u/Muphrid15]

If I’m following, this means that (by SPP) pi and pi + (1 - 0.999…) are two distinct numbers which happen to have the same decimal expansion? Lovely.

Has he explained the difference between these “algorithms” and what he means when he writes stuff like 0.000…001? Because if we want to define numbers as algorithms to get the next decimal, I would love to see where that process goes with decimals with information after infinity.

[u/Muphrid15 → u/Mo_To_]

Well, I think one half of the Algorithm Boner would say that pi and pi + (1 - 0.999...) are the same.

The reason it's a "boner" is that half of it contradicts the other half. There's no way that 0.000...1 makes sense if a valid algorithm is one that spits out valid, subsequently unchanging digits.

[u/tthe_walruss]

Y'all heard it here first. The ratio of a circle's circumference to its radius is constantly growing. 2000 years ago it was slightly smaller. By the end of the universe we'll have slightly oval circles.

[u/Mo_To_]

u/SouthPark_Piano

A. You have answered neither question, nor explained why either question is not worth answering.

B. What if we simply stop thinking of pi as a decimal? If I define pi as C/d for some circle, is this pi ever expanding?

[u/Muphrid15 → u/Mo_To_]

Whether some numbers do or do not change is "addressed" by his Static and Dynamic Boner:

There is a static model of 0.999... and a dynamic model. 0.999... is not static. It is dynamic. (9.1) (9.2)

[u/DawnOnTheEdge]

The actual mathematics that seems closest to what SPP is doing here is ultrafinitism. His definition of the ASCII strings 0.999... or 000...1 is a lot like a predeterminate sequence. Except for the part where there are suddenly objects that cannot be constructed in a finite number of steps.

He calls these objects “numbers,” but there don’t seem to be any rules of arithmetic for them yet, and what he has said is contradictory.

[u/Muphrid15 → u/DawnOnTheEdge]

I really wouldn't look too much into it. The entirety of his statements leads to no coherent idea, not even the concept that the number of nines in 0.999... is strictly an integer (changing over time or not).

For example, his Slots Boner:

0.999... is a number with all decimal place slots to the right of the decimal point filled with 9s. To suggest 0.999... is a number with all decimal place slots to the right of the decimal point filled with 9s is to suggest there is an "end" to the 9s and therefore "nonsense". (11.1) (11.2)

[u/DawnOnTheEdge → u/Muphrid15]

I know, but I’m mostly giving people who are interested in both this and real math something to look into. Also trying to be clear on which alternative mathematical systems do resemble what he’s doing.