>A different definition involves what Terry Tao refers to as ultralimit, i.e., the equivalence class [(0.9, 0.99, 0.999, …)] of this sequence in the ultrapower construction, which is a number that falls short of 1 by an infinitesimal amount.
Can we now finally admit that .999... != 1?
.999...! isn't a number that falls short of 1 by an infinitesimal amount.
It's just a representation of a number, in this case, 1.
.333...! represents 1/3
.666...! represents 2/3
.999...! represents 3/3 or 1
If you argue that .999...! isn't 1,
then you have to argue that .333...! doesn't equal 1/3.
Which is stupid, because .333...! is a commonly agreed upon way to represent 1/3 in decimal form.
>>7850796
>Which is stupid, because .333...! is a commonly agreed upon way to represent 1/3 in decimal form.
But the question isn't whether we agree to let it represent 1/3, the question is whether it -is- 1/3.
>>7850801
You can't play this game in math.
>>7850801
representation is identity
>>7850796
>.333...! represents 1/3
>.666...! represents 2/3
>.999...! represents 3/3 or 1
>If you argue that .999...! isn't 1,
>then you have to argue that .333...! doesn't equal 1/3.
It's pretty easy really. Said fractions merely don't have an accurate decimal expansion in the reals.
>>7850790
>Can we now finally admit that .999... != 1?
If we're in nonstandard analysis, you'll be right.
However, the most common field of work is the field of standard real numbers, and the way decimal representation is defined in that has 0.9999....=1 as a simple consequence.
>>7851078
>dude, representation is identity
>"No there's multiple representations!"
fuck off
>>7851264
This.
The important thing to remember is that there is *no non-zero infintesimal* in the real numbers. It *does* exist in the hyperreals, but without qualification, decimal numbers generally represent real numbers (rational if they repeat or terminate). That's why a number like 0.000...1 (if it were to exist) could only equal 0 as a real number.
>>7850790
Ultralimits are used to construct nonstandard extensions of the real numbers, called "hyper-reals".
[math] .999\ldots = 1 [/math] as real numbers. This is absolutely unambiguously true.
Perhaps, by abuse of notations, the same symbols could be used to represent distinct hyper-reals. But this would be strictly abuse of notation.
[math]
{
Proving that 0.333... equals 1/3 : \begin{eqnarray*}
{x}&=&0.333...\\
10x&=&3.333...\\
10x&=&3 + 0.333...\\
10x&=&3 + x \quad\text{(from first equation)}\\
9x&=&3\\
x&=&\frac{3}{9}=\frac{1}{3}\quad\square
\end{eqnarray*}
Proving that 0.999... equals 1: \begin{eqnarray*}
x&=&0.999...\quad\text{(1)}\\
10x &=&9.999...\\
10x &=&9 + 0.999...\\
10x &=&9 + x\\
9x &=& 9\\
x &=& \frac{9}{9} = 1 \quad\square
\end{eqnarray*}}
[/math]
>>7850796
>.999...!
>.666...!
>.333...!
>!
>>7851519
Fuck TeX and 4chan's usage of mathjax
Why not latex? either way, screenshotted it. Here is the proof now stfu
>>7851572
Can someone help me see where the analogy of this reasoning must break down for the hypernatural number defined as the equivalence class of (.9, .99, .999, ...) in the ultrapower?
>>7851739
What are you talking about?
There aint no fallacy in the reasoning.
>>7851931
In the real numbers .999 = 1.
In the hyper-reals, the hyper-real defined as the equivalence class of (.9, .99, .999, ...) in the ultra filter is not equal to 1 (i.e the equivalence class of (1, 1, 1, ...) in the ultra power), as follows by Los's theorem.
So there is some fallacy in the analogue of >>7851572 applied to the specified hyper-real.
>>7851993
Uhm, i gues so, I don't know i'm just an engineer never heard of that theorem.
[math]
0.999...=\sum\limits_{i=1}^\infty9 *10^{-i}
[/math]
>>7850831
They do though, just not in base 10
https://www.youtube.com/watch?v=3cI7sFr707s
I'll just leave this here, watch it until you actually understand math. :)
>>7852440
It worls perfectly in other bases too.
What you don't understand is that the reasoning only works for rationals, when we now that the decimals are repeated to an infinity.
>>7852475
Well, I define the real number using the number line and the supremumaxiom. But i read more on about cauchy sequences and guess what.
0.999... does converge into a rational number, while sqrt(2) does not.
>>7850790
>Can we now finally admit that .999... != 1?
We never needed to. Almost all deniers are mathematically illiterate, and all deniers don't fully understand the concept.
