/sci/ - Science & Math

>>16222922
.999.... is equal to 1 in the hyperreals.

the sequence of .9, .99, .999, ... is not the same as the constant sequence .999..., .999..., .999..., ...

.999... is just 1 without even introducing the hyperreals.

the non-constant hyperreal sequence .9, .99, .999, ...
is "infinitesimally" smaller than 1
But .999... is just 1 since the real number .999... is equal to 1.

>>16221016
9*10^-n on (1,infinity) is not the only way to get .999...
As already stated, not all .999... are equal.
Some .888...s in base nine are equal to some .999...s in base ten.
9*10^-n on (1,infinity) in base ten and 8*10^-n on (1,infinity) in base nine can both be written as .999... in base ten or as .888... in base nine even though they are not equal (in the realm of hyperreals). And obviously "10" in the former means ten whereas in the latter "10" means nine.
>>16221066
>>16221066
>avoid conceding the point is asinine.
There is no need for me or >>16221034 to concede to you when we are right and you are wrong.
Will you AT LEAST admit that .999... is not equal to 1 in the realm of hyperreals, and that not all .999...s are equal to each other in the realm of hyperreals?

>>16221027
>.A=1
.A=.909090...

As I said, you just don't know how positional notation works.

>>16221034
I see the pattern. I discussed the pattern earlier. Proving why the pattern holds is legitimately more difficult than proving that .999...=1 which is how you should be writing it in any base. Writing .999... in base 10 as .888... in base 9 to avoid conceding the point is asinine.

>>16221027
>.A=1
.A=.91666...

As I said, you just don't know how positional notation works.

>>16221034
I see the pattern. I discussed the pattern earlier. Proving why the pattern holds is legitimately more difficult than proving that .999...=1 which is how you should be writing it in any base. Writing .999... in base 10 as .888... in base 9 to avoid conceding the point is asinine.

>>16220991
>So you just don't know how positional notation works
of course i do
a*base^2+b*base^1+c*base^0+d*base^-1+e*base^-2+...
and 0.9...=1 due to the fact that positional notation involves sums so, im amazed that you took from my writings that i attempted somehow to disprove the obvious, lots go over why 0.999... in base 9 is equal to 1.111... in base 9 though the analogue in base 10(here we take A=9+1 due to backporting notation from hexadecimal):
0.A=1.0, 0.AA=1.10, 0.AAA=1.110, 0.AAA=1.1110, and as such 0.AAA...=1.111...
the added zeroes at the end of the finite ones are to make the pattern that i said should be obvious to see with a bit of thought, well, visible

>>16220973
So you just don't know how positional notation works, huh?

1 in base 9=1 in base 10.

You are arguing that .999... in base 10 is >1 in base 10, you fucking idiot.

>>16220913
>So how would you write .999... in base 9?
.888...
obviously

>>16220913
>So how would you write .999... in base 9?
Depends which .999... you are talking about. There are many ways to reach .999... and not all of them are equal.
>muh series
They approach 1. Their limit is 1. A function (or series) never reaches its limit; it is bounded by its limit.

>>16220812
actually 0.999...=1.111... in base 9 in the same way that 0.AAA...=1.111... in base 10, for reasons that i'd hope should be obvious with a bit of thought

>>16220864
>There is no '9' symbol in base nine. In base nine, the number nine is written as '10'.
Yes. Thank you, Sherlock. So how would you write .999... in base 9? You understand decimal decimals can be written in other bases, right? As an example, .5 in binary is .1.

Fucking idiot.

>3/3=1
3*1/3=3*.333...
3/3=.999...
>9/9=1
9*1/9=9*.111...
9/9=.999...

>sum of 9/10^n on (1,infinity)=.999...
No. It's 1. The same way that 1/2^n on (1,infinity)=1.

Any geometric series of that form equals 1. Which makes perfect sense if you think about it for even half a second. The 1/2+1/4+1/8... geometric series is just .111... in base-2. And that's also 1, the same way .222... in base-3 is 1. And .333... in base-4 is 1. And so on and so forth. It's all fucking 1.

>>16220790
3/3=1
9/9=1
sum of 9/10^n on (1,infinity)=.999...
.999...=/=1
and in fact .999...=/=.999... more often than not
In fractions .999... is sometimes used as an approximation but it is just that.
>>16220812
There is no '9' symbol in base nine. In base nine, the number nine is written as '10'.

>>16220581
>>16220790
Ooh, you can also drive the point home by moving from base-10 to base-9. I fucking defy you to tell me .999... in base-10 is anything but 1 in base-9 and further defy you to tell me that 1 in any base is anything but 1.

>>16220581
>.999... is only a valid concept if we are dealing with hyperreals.
Or geometric series. Or fractions.

Whether you define it as 3/3, 9/9, or the sum of 9/10^n for n=1 to infinity, it's the same shit. Literally fuck all about repeating decimals requires hyperreals.

>>16220058
.999... is only a valid concept if we are dealing with hyperreals. So no, .999... is not exactly equal to 1. It is infinitely close to 1, but there is a non-zero infinitesimal difference.
If we disregard the concept of hyperreals, then we disregard the concept of .999... itself.

>>16219264
.999... isn't approximately equal to 1. It's strictly equal to 1.

Using certain notations and hyperreals it can be approximately equal to 1, but that .999... and the normal .999... aren't the same .999...

>>16218636
It's just a way of keeping an accurate accounting of vanishingly small numbers. It can blow things up if it is in a denominator, but if it's additive like that you can take the limit to 0 and kill the term because it's... infinitesimal. It's approximately equal to in the same sense that 0.999... is approximately equal to 1.0.

>>16218649
>sls is a gay and nonsensical riff off sts for no reason.
This is made 1000% worse by the fact that only space nerds even know the "STS" term exists let alone what it means. To 99.999% of the public it has no other name than "The Space Shuttle"

>half melted flap with 999 missing tiles
>melted metal tore through the hinge
>still actuating
>IT FUCKING LANDED
I AM COOM

All you needed was common sense. That's it. Just enough to say "hmm, maybe I won't take the untested, experimental vaccine for an illness that I have a 99.999% chance of surving with no long term effects"

You didn't need tomes of research, scientific debate, statistics and to wrap a piece of plastic around your mouth for 2 years. You just had to not be retarded.

>>16197520
999

>>16201243
>is that impossible/unreasonable because they're the same number by definition(?)
So when you're in middle school your teacher explains to you something like "real numbers are these infinite strings of digits, but the ones that end with 999... are actually the same as the ones that end with 000..." or something like that.
That's not what real numbers are. Real numbers are usually constructed with Dedekind cuts or Cauchy sequences and the essential thing you need to know is that real numbers need to be separated by rationals. 0.999... = 1 because there are no rationals between them.
>But is there ever any situation where you would want or need to distinguish between the two
You can do stuff with infinitesimals for stuff like that.
You can also directly operate on strings of digits, but then you have the issue with arithmetics that your teacher has already proven pops up.

So 0.999... = 1
Like this is a proof that works out fine. I get that. I also get the image of the infinite limit where it eventually approaches 1 (or maybe I don't because I'm here asking about it). But is there ever any situation where you would want or need to distinguish between the two, or is that impossible/unreasonable because they're the same number by definition(?)

0.999...bros.....
I don't feel so good....

>>16194964
>that basically get washed off with exposure to water and resets the skin to a ground zero that one must build up all over again

long answer: 1/2
The chlorine in tap water doesn't kill all the bacteria on your skin, that's why we also use soap to mechanically remove them away. Even if you scrub your skin with soap, there are always bacteria left behind.

Rain water doesn't kill the bacteria on your skin, nor dislodge them all from the sebum in your skin either. The same goes for river or lake water.

Salt water is full of bacteria, so even thought the chlorine in it might kill, partially, the ones on your skin, it seeds it with new fresh ones that were in sea water, though those end up dying soon since your skin being exposed to the air, is not sea water where they are adapted to line in.

To attempt to kill all bacteria on your skin you need to use alcohol, antibacterial soap, antibiotic ointments, peroxides, UV radiation, etc, and even so, it will only kill 99.999(etc)% of all the bacteria.

What is the best option for your skin health? Non-chlorinated fresh water to rinse, and mild vegetable-based soap to wash the regions of your body where bacterial colonies tend to grown intensively and therefore, stink (axillae, genitals, gluteal cleft, scalp).