0.999...

[Agent Smith]

You really cannot deal with infinities as if they were numbers. And dots require a definition, or in many cases a convention we all agree on. I don't understand the sequence of nines in the denominator. If we use the decimal system to represent [imath] \pi [/imath] then
[math]\pi = 3.141592653589793238462\ldots =3+\dfrac{1}{10}+\dfrac{4}{100}+\dfrac{1}{1000}+\dfrac{5}{10000}+\dfrac{9}{100000}+\ldots[/math]You cannot even write [imath] 3+\dfrac{14159265\ldots}{1000000000\ldots} [/imath] - at least not if you want to be understood - let alone nines in the denominator. Note that I wrote dots although the numerator is a sequence of changing digits and the denominator is a regular sequence that only adds a zero from term to term.

Dots are always dependent on their context. That's why you cannot take [imath] 0.999\ldots = 1, [/imath] rip off the context and replace ones by dot nines. If you want to write unusual dots in [imath] \pi ,[/imath] I suggest

View attachment 37680

Infinity is a symbol you should try to avoid. It makes sense in sums [imath] \displaystyle{\sum_{k=0}^\infty a_k} := \lim_{n \to \infty} \sum_{k=0}^n a_k[/imath] as an abbreviation and sometimes as the result of a limit like [imath] \lim_{n \to \infty} n=\infty , [/imath] but even that is already an abbreviation for a complex logical statement, not something infinitely large. It is rather something growing beyond all finite bounds!

As you so correctly pointed to and I must admit, [imath]0.14159... = \frac{14159...}{1000...}[/imath] is not something I've seen in my mathematical life. Although [imath]0.14159...[/imath] has an infinite repetend, it's actually a finite number, sits somewhere between [imath]0.14[/imath] and [imath]0.15[/imath] and the trailing dots simply imply, in the Pythagorean sense, incommensurability. What is incommensurability?

 

[fresh_42]

As you so correctly pointed to and I must admit, [imath]0.14159... = \frac{14159...}{1000...}[/imath] is not something I've seen in my mathematical life. Although [imath]0.14159...[/imath] has an infinite repetend, it's actually a finite number, sits somewhere between [imath]0.14[/imath] and [imath]0.15[/imath] and the trailing dots simply imply, in the Pythagorean sense, incommensurability. What is incommensurability?

Yes, that's another issue. The digits depend on the number system that we use. Ok, [imath] \pi [/imath] is unpredictable in every system, but [imath] 1/7=0.\overline{142857} [/imath] looks a bit different if we use [imath] 7 [/imath] as a base, [imath] 1/7=0.1\,. [/imath]

The ancient Greeks are again another topic. AFAIK they only knew rational and irrational numbers that could be easily constructed as a length on a straight, and numbers they could not construct like [imath] \pi, \sqrt[3]{2} [/imath] or certain angles. They knew the difference between the rational side length [imath] 2 [/imath] and the irrational length of the diagonal [imath] \sqrt{8} [/imath] but had no idea about the properties of [imath] \pi [/imath] although Archimedes already had quite a good approximation. "Length on a straight" for them was always a ratio, a part of the yardstick: [imath] x:y [/imath].

Commensurability (mathematics) - Wikipedia

en.wikipedia.org

It took more than 2,000 years to answer their questions finally. The words we still use: rational, irrational, transcendent were born in their geometric understanding of mathematics. Pythagoras, however, was a charlatan, in my opinion, a numerologist.

 

[Agent Smith]

Yes, that's another issue. The digits depend on the number system that we use. Ok, [imath] \pi [/imath] is unpredictable in every system, but [imath] 1/7=0.\overline{142857} [/imath] looks a bit different if we use [imath] 7 [/imath] as a base, [imath] 1/7=0.1\,. [/imath]

The ancient Greeks are again another topic. AFAIK they only knew rational and irrational numbers that could be easily constructed as a length on a straight, and numbers they could not construct like [imath] \pi, \sqrt[3]{2} [/imath] or certain angles. They knew the difference between the rational side length [imath] 2 [/imath] and the irrational length of the diagonal [imath] \sqrt{8} [/imath] but had no idea about the properties of [imath] \pi [/imath] although Archimedes already had quite a good approximation. "Length on a straight" for them was always a ratio, a part of the yardstick: [imath] x:y [/imath].

Commensurability (mathematics) - Wikipedia

en.wikipedia.org

It took more than 2,000 years to answer their questions finally. The words we still use: rational, irrational, transcendent were born in their geometric understanding of mathematics. Pythagoras, however, was a charlatan, in my opinion, a numerologist.

Pythagoras, a charlatan?!

Per the linked article on commensurability, incommensurability(a, b) = the nonexistence of a real c, and integers m, n such that mc = a and nc = b i.e. a/b is irrational. If mc = a and nc = b then c = a/m = b/n, which means a = (m/n)b. So if m/n = 2/3, we know we can "commensure" a by breaking b into thirds, and taking two of these pieces. It would be exact.

For irrationals, there is no c and no m and no n that meets this condition. We can approximate though (like we do with [imath]\pi[/imath]), but <insert appropriate description>.

 

[fresh_42]

Pythagoras, a charlatan?!

Well, I didn't know him personally, but by all you can hear about him, he was one. I wouldn't put his name in the same sentence with the giants Euclid and Archimedes, or even folks like Thales, Eratosthenes, or Diophant. I even wouldn't call this famous theorem by his name, only because he knew that [imath] 9+16=25. [/imath] I would call it Law of Cosines! Even Pythagorean triples, e.g. [imath] (12709 , 13500 , 18541), [/imath] have been found on Babylonian clay tables about 1829 - 1530 BC, more than 1000 years before he lived!