Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

0.999...= 1

en.wikipedia.org

Post reply on HN
Print view View original

571–580 of 647 posts

Re: 0.999...= 1

#571

Post by schoen »

austincheney wrote: post #545
ISO-morphism wrote: post #527

Earlier quoted context omitted.

200 years ago we didn't need timezones because we couldn't communicate or travel fast enough or often enough for them to matter. With railroads and telegraphs the level of granularity required for "some time 300 miles away" went from days to hours and minutes. No one in the U.S. west wanted to be told "The sun is at its zenith around 8 a.m. for you." For most people talking about time in their day to day lives it's f…

> 200 years ago we didn't need timezones because... We still do not. China and India are examples of large geographies spanning across a vast amount of longitude and yet each are a single time zone. Time zones are a political entity only, an unnecessary complexity. The absence of time zones will not halt business or communication over long distances. > For most people talking about time in their day to day lives it's…

These two national policies aren't equally difficult: India extends across about 29° of longitude, while China extends across about 61°. The natural size of a time zone is 15° of longitude, so without political considerations India would include about 2-3 time zones, while China would include about 4-5.

I've heard there's some pushback against the Chinese policy in that some people in the west keep an unofficial local time which is widely understood and quoted (though presumably not for things that are sufficiently official or relevant to other regions). Apparently there's currently an ethnic conflict over the time zone status in Xinjiang:

https://en.wikipedia.org/wiki/Xinjiang_Time

Maybe this conflict has now been pushed underground by force?

> In 2018, according to Human Rights Watch, a Uyghur man was arrested and sent to a detention center because he set his watch to Xinjiang Time.

Re: 0.999...= 1

#572

Post by btilly »

mjd wrote: post #565
tom-thistime wrote: post #326

Earlier quoted context omitted.

Yeah, can we have an example of an irrational number whose decimal representation repeats or terminates? Or a rational number whose decimal representation doesn't repeat?

My phrasing was bad. I should have said "the mistaken belief that an irrational number is * defined to be * one whose decimal representation doesn't repeat”. Usually we define it like this: an irrational number is one that isn't a quotient of two integers. Starting from that definition, we then prove the _theorem_ that the decimal representation a number repeats if and only if the number is rational. It's much easier…

I am still not in agreement.

The proof that the usual definition is equivalent to the representation is fairly straightforward and easy, no matter which side you picked as the definition. And once the equivalence is established, all other proofs proceed naturally. It therefore matters a lot that we pick one as a definition and know which one we picked, but not so much which one we picked.

Now in fact the quotient definition is by far more interesting mathematically. There is also a clear foundational reason to prefer it, namely that you can easily construct and prove things about the rational numbers long before you construct the real numbers. However it is unlikely that anyone who is confused about the definition of a rational number has a clear understanding of how the reals are constructed, so that is not a particularly important consideration for them.

Furthermore the fact that foundational considerations argue for one construction over another has little bearing on what is pedagogically preferable. As a famous example, the easiest way to rigorously define logarithms is through the integral of 1/x. However explaining logarithms that way to someone who doesn't know them is a pedagogical disaster.

Re: 0.999...= 1

#573

Post by empath75 »

brutt wrote: post #304
tom-thistime wrote: post #121

Earlier quoted context omitted.

(STATEMENT OF PERSONAL IGNORANCE [SOPI]: Anyone who actually understands this stuff please correct my mistakes below. Thanks.) In the real numbers, which are not always simple or intuitive, 0.99... = 1. That's true and I seem to understand the proof. But the real numbers aren't the only system that might be sitting behind "0.99..." and "1" when I write those symbols down and talk intuitively to people in my family. T…

0.999... + 0.000...1 = 1 0.000...1 = 1/∞ 0.999... = 1 - 1/∞ 1/∞ is zero or not?

What on earth is .000...1

If it terminates it’s not an infinite series. In any case, if you add any finite number to .99999... it’ll be equal to 1 + that number.

Re: 0.999...= 1

#574

Post by empath75 »

Gehinnn wrote: post #467
TuringTest wrote: post #294

Earlier quoted context omitted.

I find that "infinite as an endless process" concept intuitively very heplful as well. However, reading Gödel, Escher, Bach [1] showed me that there's another, more static logical interpretation of infinity which also comes handy. In an infinite process, you can always take "one more step" to create the item after that. Let's assume there exists a "final" mathematical object that goes after every finite item in the g…

This thought can be made much more concrete when talking about sets. Clearly we can understand a collection of things as a set. Clearly numbers are things, so we can talk about the set of all numbers. But how many elements are in this set? A clever answer could be: It has as many elements as there are natural numbers! As subsets are also a thing, we could ask next how many subsets the set of all numbers has. A clever…

That’s actually not true, the power set of the naturals is uncountable.

Re: 0.999...= 1

#575

Post by empath75 »

tom-thistime wrote: post #205
_bxg1 wrote: post #204

Earlier quoted context omitted.

Why not 0.00...1?

Then the question is, is 0.00....1 equal to zero? We use the definition of equality above and we say yes. EDIT: The number above seems well defined. It's lim n->inf (10^-n). That's zero.

0.0000... is the repeating decimal representation of zero.

Re: 0.999...= 1

#576

Post by effie »

thom wrote: post #291
ta1234567890 wrote: post #235

Earlier quoted context omitted.

> As a static number, the concept of infinity makes no sense; but once reimagined as a process Super insightful. That's the key right there. The same concept can also be applied to the physical world. Things are not static, they are in constant flux, everything is a process in motion.

Yes, although for me this conception of infinity as a process also captures why there are probably no actual infinite things in the universe, only in silly games with numbers.

Could you elaborate? Why infinite process cannot be actual thing in the universe? It's not like we know the start and the end date of the universe...

Re: 0.999...= 1

#577

Post by naikrovek »

gowld wrote: post #548
naikrovek wrote: post #541

Earlier quoted context omitted.

I don't understand how they're the same number. I will never accept that they are the same. The difference between 0.9 repeating infinitely and 1 is infinitely small, but it isn't zero.

What is an "infinitely small" number? Is 9999..... the same as infinity? What is 1.0 - 0.99999.... = ? What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer?

> What is an "infinitely small" number?

What is an infinitely large number?

> What does it mean to say X is a number, if you can't subtract it from another number and get a number as an answer?

By that logic, 0.99 repeating isn't a number at all, and therefore can't be equivalent to 1, because you can't subtract it from 1. So my understanding that they are different is correct.

Re: 0.999...= 1

#578

Post by effie »

chithanh wrote: post #539
twiceaday wrote: post #440

Earlier quoted context omitted.

.999... and 1 exist on a continuous line. If they are different numbers, name a number between them.

The Wikipedia article says how, you need a definition of real numbers that includes nonzero infinitesimals (IOW does not satisfy the Archimedian property). So let there be an ω with 0.999... = 1 - 1/ω. Then a number between 0.999... and 1 would be 1 - 0.5/ω.

How do you know that 1/ω is not 0?

Re: 0.999...= 1

#579

Post by effie »

hota_mazi wrote: post #247
undecisive wrote: post #115

There is no proof that will ever satisfy a person dead-set against this. Ever since I brought this home from school as a child, my whole family ribbed me mercilessly for it. If you tell a person that 3/6 = 1/2, they'll believe you - because they have been taught from an early age that fractions can have multiple "representations" for the same underlying amount. People mistakenly believe that decimal numbers don't hav…

Not only can numbers have more than one representation, but they can also have zero! Looking at you, irrational numbers.

Irrationals have a unique decimal representation in the mathematical sense: given a definition such as $x^2 = 2 $, any digit of the decimal expansion of $x$ can be determined.

Re: 0.999...= 1

#580

Post by ltbarcly3 »

saagarjha wrote: post #76
ltbarcly3 wrote: post #18

Earlier quoted context omitted.

So, lets take .9, .99, .999 and so on. If a sequence of rational numbers converges, it converges to a real number. What number does .9, .99, .999, .9999 (and so on) converge to? Which is to say, is there a number that it gets closer and closer to at every step? Clearly it gets closer and closer to 1 at every step, so the sequence converges to 1. This is one of the many (equivalent) ways the real numbers are defined t…

I don’t think this reasoning is correct, because the limit of an expression does not have to be the same as the value at the point the limit is being taken. For example, if your sequence is “sin(1/x) * x“, then it’ll slowly appear to converge to one as x approaches infinity, but it cannot reach it. So there’s really no relevant conclusion you can make.

You're taking a map from R -> R, I'm taking a sequence of discrete values. It's not the same thing.

However, I was a bit fast and loose, the sequence .9, .99, .999, .9999 also gets closer and closer to 2 but it doesn't converge to 2, I should have said if you have a metric || and some number X such that for any d, there exists an N such that |X -An| N, then the sequence A converges to X. But I wasn't trying to write a proof.

https://en.wikipedia.org/wiki/(%CE%B5,_%CE%B4)-definition_of...

Post reply on HN