0.999...≠1 and repeating decimals are not rational number.

[wij]

> "Chris M. Thomasson" <chris.m.t...@gmail.com> writes:
>
>> For instance, .999999999999999 is close to 1, however, its not
>> arbitrarily close to 1 like .999... is
...

If 0.999... is a number, say 0.999, then
0.9999≠1
0.99999≠1
0.999999≠1
...
So, the formal/general question is: ∀n∈ℕ, S(n)=1-1/10^n ≠1 ?
By mathematical induction, the answer is true, ∀n∈ℕ, S(n)≠1
Note that S(n) is rational, we have only addressed rational, there are still
lots more irrationals out there unaddressed.

... IMO, S(∞)≠1 is still true:
https://sourceforge.net/projects/cscall/files/MisFiles/NumberView-en.txt/download

Common question: 1/3=0.333...? Or are repeating decimals rational?
Quick answer: No. Because repeating(infinite, cyclic) decimals rely on the
existence of non-zero remainder, LHS and RHS are never equal.

[Mike Terry]

I think you would enjoy posting over at sci.math where this would be on
topic. There are plenty of people over there who enjoy debating this
very question!!

Mike.

[Jeff Barnett]

Certainly on topic there but all the people who say they agree with his
viewpoint are trolls or cranks. Do you think it would be an appropriate
setting for him to work in? Oh. Never mind, I get it now. Good idea.
--
Jeff Barnett

[Chris M. Thomasson]

1/3 * 3 = 1

.333... = 1/3 is true

In base 10, the decimal approximation of 1/3 is (.333...)

Therefore, (.333...) * 3 = 1

[wij]

Therefor, (.333...)*3 ≈ 1/3 (approximate)

Will the end of the mandelbrot set end?
What would it mean if the end of the zoom is 'zero' (blank?)
https://www.youtube.com/watch?v=0jGaio87u3A
---
If this is an exam question, score first, do what the teacher teaches.

[wij]

I consider myself a programmer, but seems that I have to fix the foundation of
calculus AS A PROGRAMMER! The foundation of calculus is full of shits.
The question (as in the title) is a done deal for me at age 14, no desire to
debate such a low level thing. Thanks for link anyway, I might try testing latter.

[Malcolm McLean]

On Friday, 6 August 2021 at 20:34:19 UTC+1, Chris M. Thomasson wrote:
>
> In base 10, the decimal approximation of 1/3 is (.333...)
>

All rationals can be expressed as either
N + a/b where N, a and b are whole numbers. (Normally the plus sign is omitted).

Or as a decimal, with a finite sequence of digits recurring

eg 2.5432432432....

These are both exact representations. In a sense the decimal representation is
getting closer to the true value all the time, but never gets there, unless the recurring
digit is 0. But that's because the notation isn't really designed to handle the
recurring case, it's tacked on to a system designed to represent to x decimal
places.

[Andy Walker]

Notations using "..." are somewhat ambiguous; eg, is

sqrt(2) == 1.414...

supposed to indicate that "4" recurs, or "14" does, or that this is merely the
start of a somewhat random sequence of digits? /We/ know it's the third, but
a learner might well not. Likewise with "e == 2.718281828...", which briefly
looks very repetitive. Further, "pi == 3.1416..." is correct to 4dp but wrong
if regarded as indicating the start of the string "3.1415926535897...". For
recurring decimals, I prefer a notation such as "1/3 == 0.(3)". Textbooks
[used to] use the notation with dots over the recurring part, but that's
harder to do in Ascii.

Once you have a suitable notation, you can start playing games with
the strings, as opposed to the numbers they represent. This is amenable to
exploration by students, both as a programming exercise and as an exercise
in alternative number systems. Further details if anyone expresses interest
[but see also

http://www.cuboid.me.uk/anw/Research/Hack

(about the game "Hackenstrings") -- it's rather old, and looks better on a
typical 'phone or tablet than on a typical PC screen as the pebbles are too
big at some resolutions, sorry].

--
Andy Walker, Nottingham.
Andy's music pages: www.cuboid.me.uk/andy/Music
Composer of the day: www.cuboid.me.uk/andy/Music/Composers/Haydn

[Chris M. Thomasson]

The Mandelbrot set settles into a finite shape, however, the border of
the set is infinitely complex. Zooming in gives one more places to
explore, forevermore.

> What would it mean if the end of the zoom is 'zero' (blank?)
> https://www.youtube.com/watch?v=0jGaio87u3A
> ---
> If this is an exam question, score first, do what the teacher teaches.
>

Zooming in on the border is very interesting to me, and it never ends.