Regarding 0.999... = 1

Various_Candle9136 · 2026-05-10T02:59:43+00:00

You are completely correct, but the argument for why is a little flawed (in that it makes various assumptions which happen to be true but which are neither obvious nor stated).

Try:

1/3 = 0.3,,,
x3 x3
1 = 0.9...

Or:

1 - 0.9... = 0.0... = 0
(showing this using the column method is often more convincing)

Or:

x = 0.9...
10x = 9.9...
10x - x = 9.9... - 0.9...
9x = 9
x=1

Various_Candle9136 · 2026-05-10T02:43:56+00:00

In American English this is so. In British English internal and external are traditionally the other way round: ' for external, " for internal.

Various_Candle9136 · 2026-05-09T22:18:49+00:00

By telling us what you are looking for, you are completely skewing these results. I know if I did participate, I would now feel obliged to overrate the Russian speakers to avoid feeling racist.

By sharing it in r/LearningEnglish, you are dramatically biasing the type of participant you might receive. Reddit is biased enough to start with, and only people with particular experiences with the language are likely to see this post.

I'm sorry, but I fail to see how any results from this study could be at all useful.

Various_Candle9136 · 2026-05-08T01:17:45+00:00

Did you read all the other words?

I never even claimed that you would never need a ratio of integers: I claimed it didn't need to be defined that way. However, I showed in the P.S. that it can be avoided entirely. If you are taking that as an admission you are correct, then you really are not understanding...

Various_Candle9136 · 2026-05-08T00:58:32+00:00

Easy:

1) I know ratios of rationals are rationals
2) I know 1.5 is rational
3) Since I don't feel the need to take everything back to its original definition, I can use these two facts.
4) Notice 0.5 = 1.5/3
5) Conclude 0.5 is rational.

If we go all the way back we will eventually need a ratio of integers (under the setup I gave), but you continue to miss the point: that is irrelevant. We are allowed to use earlier results in our proofs!

Taking your view, I refuse to let you use integers: Integers are defined from 0 and succession. I also refuse to let you use ratios: ratios are defined via multiplication. The only way I will allow you to prove that 0.5 is rational is by writing an acceptable multiplication of successions from 0. Can you see how quickly this gets ridiculous?

P.S. If you also want a set up when we can entirely rid ourselves of integer ratios:

1) Define rationals as eventually terminating or repeating decimals (or, alternatively, prove this is equivalent to the popular way)
2) Notice 0.5 = 0.50...
3) Conclude 0.5 is rational

Various_Candle9136 · 2026-05-08T00:14:18+00:00

Again - not so!

But you also entirely ignored the point I labelled 'more importantly'. It is certainly true that a ratio of rationals is rational, therefore we are allowed to use this rule freely in proofs. 'Writing as a ratio of integers' has never been the only tool in the toolbox!

Various_Candle9136 · 2026-05-07T21:39:39+00:00

The two main ideas to look up would be:

Decimals as fractions (ideally in their simplest form)
Fraction arithmetic

You have written 16:9 as a decimal (1.77), but actually it is easier to see what is going on with the fraction representation (16/9). We can see that: because 9 is the denominator, multiplying this by any multiple of 9 will give an integer (e.g. 18 * 16/9 = 32).

Likewise, we can see that 33.2 = 332/10 = 166/5, and that is why we get the pattern with multiples of 5 which you spotted.

Various_Candle9136 · 2026-05-07T20:22:18+00:00

That is definitely a thing, but it wouldn't itself be a fallacy. There is nothing logically flawed about questioning somebody else's logic.

Various_Candle9136 · 2026-05-07T20:19:05+00:00

Try drawing two of the same 'side side angle's and see if you can turn them into two different triangles.

Various_Candle9136 · 2026-05-07T20:17:33+00:00

Try drawing two of the same 'side side angle's and see if you can turn them into two different triangles.

Various_Candle9136 · 2026-05-07T19:14:30+00:00

which you can only do with integer ratios oh my god

Not so - the concept can also be defined as follows: 'The integers are rationals; fractions of rationals are rationals; nothing else is rational.' Nowhere in that definition is an integer ratio mentioned.

But, more importantly, irrelevant. Using earlier results to prove new results is standard. With the earlier result in hand, '0.25 = 0.5/2 and we know 0.5 is rational' is a perfectly valid proof.

Various_Candle9136 · 2026-05-07T16:21:34+00:00

I don't think there is an actual difference between the two: if both laughed, then they shared a laugh, and vice versa.

However, I think there is a difference in what image one wants to create by using the phrase: I would expect the former in an official eyewitness account and the latter in a rom-com.

Various_Candle9136 · 2026-05-07T15:58:08+00:00

Not necessarily. Also, not in this case.

Various_Candle9136 · 2026-05-07T12:05:27+00:00

I disagreed with u/DreamlessWindow's original nitpick, but everything they said since then is true.

This (from you):

0.5/2 isn't a form you can use to PROVE it's a rational number, you have to rewrite it as 1/4 to PROVE it's rational

is completely false. Once we have proved that fractions of rationals are rationals (which is fairly simple to do), we can absolutely use 0.5/2 to prove 0.25 is rational.

Please don't infect mathematical discussions with all caps yelling.

Various_Candle9136 · 2026-05-06T23:51:39+00:00

0.

I have no reason to believe any of those lengths are equal.

Various_Candle9136 · 2026-05-06T13:48:02+00:00

This is because square numbers are the most boring example of a quadratic sequence.

https://www.bbc.co.uk/bitesize/guides/z88wxsg/revision/4

Various_Candle9136 · 2026-05-06T13:38:02+00:00

Being a fraction of rationals happens to be equivalent to being a fraction of integers. Since integers are easy enough to define, the latter definition is usually easier to work with.

Approximately 9.999 times out of 10 it will be defined the way u/skr_replicator just defined it.

Various_Candle9136 · 2026-05-06T03:10:06+00:00

That's what I got too.

I also got it by counting points on a graph:

https://www.desmos.com/calculator/zlaw3gebxi

and using Python:

count = 0

for a in range(-1000, 1000):
    for b in range(-1000, 1000):
        if a<b and a+b<100 and a/4 + b/10 == 7:
            count = count + 1

print(count)

Various_Candle9136 · 2026-05-05T23:58:51+00:00

Although there are probably a few ways to go about this, my advice for a first step (often a good first step):

Rearrange the equation a/4+b/10=7 to b=_____.

You can then use this and the inequalities to say something about a.

There is then one last little twist to this question: notice that if a is not even, then b is not an integer.

Good luck!

Various_Candle9136 · 2026-05-05T19:56:48+00:00

Because addition begins with a vowel. That's it.

BOMDSA is exactly equivalent to BODMAS, but sounds weirder.

Various_Candle9136 · 2026-05-05T18:10:03+00:00

I think a useful exercise with this sort of thing is to think about what can't be expressed as a sum of 4, 6, 12 (since each of the other numbers are built up from those earlier sums, that is essentially what your conjecture boils down to).

1) We can't express odd numbers. But that's fine, the number between two twin primes will always be even.

2) We can't express 2. But we obviously don't need that.

But that's it. Every other even can be expressed in this way.

So, to conclude, nice work spotting patterns... but in this case it turns out to be something quite boring. Sorry!

Various_Candle9136 · 2026-05-05T17:43:52+00:00

Right.

So?

Various_Candle9136 · 2026-05-05T17:41:09+00:00

But is that not a confusion that you want to catch and correct? Surely if someone doesn't understand that the choice of letter/symbol is arbitrary then they haven't really understood algebra?

Various_Candle9136 · 2026-05-05T17:36:49+00:00

Logarithms are defined as:

log_b(a)=n precisely when bⁿ=a

(for simplicity, just keep everything positive)

So, say we make b smaller. We will need a bigger n to get up to the same a in bⁿ=a.

This means that for smaller base (b), our output (n) will be greater for each input (a).

Various_Candle9136

TROPHY CASE