0.999...=1? Seriously?

[lotanddaughters]

The article found on Wikipedia.org, on the topic "0.999..." seems to accept that 0.999...=1. I don't. The civilized world bases its numerical system on groups of ten:

1 2 3 4 5 6 7 8 9 10

Using this method, 0.9 = 9/10, of course. But what if a civilization decided to base its numerical system on groups of what we call "twelve"? It might look something like this:

1 2 3 4 5 6 7 8 9 A B 10

"10", to this hypothetical civilization, would be the equivalent to our "twelve".

.6 would be equal to 1/2.

What if a civilization based their numerical system on groups of what we call "five"? It might look something like this:

1 2 3 4 10

Their .4 would be the equivalent to our .8, or 8/10.

In conclusion, 0.999...=1 just because we decided to assign "9", the ninth number, as our highest valued digit? In the numeric system of the first of the two suggested hypothetical civilizations, their .B is equivalent to our 11/12. Their .B is closer to 1 than our .9, or 9/10.

If you think that 0.999...=1, then you must also agree that 0.888...=1, because when using the second suggested hypothetical civilization's numeric system, that's the best you've got.

That's why I say, "0.999... does not equal 1, no matter how hard it tries".

[Martingale]

The article found on Wikipedia.org, on the topic "0.999..." seems to accept that 0.999...=1. I don't. The civilized world bases its numerical system on groups of ten:

1 2 3 4 5 6 7 8 9 10

Using this method, 0.9 = 9/10, of course. But what if a civilization decided to base its numerical system on groups of what we call "twelve"? It might look something like this:

1 2 3 4 5 6 7 8 9 A B 10

"10", to this hypothetical civilization, would be the equivalent to our "twelve".

.6 would be equal to 1/2.

What if a civilization based their numerical system on groups of what we call "five"? It might look something like this:

1 2 3 4 10

Their .4 would be the equivalent to our .8, or 8/10.

In conclusion, 0.999...=1 just because we decided to assign "9", the ninth number, as our highest valued digit? In the numeric system of the first of the two suggested hypothetical civilizations, their .B is equivalent to our 11/12. Their .B is closer to 1 than our .9, or 9/10.

If you think that 0.999...=1, then you must also agree that 0.888...=1, because when using the second suggested hypothetical civilization's numeric system, that's the best you've got.

That's why I say, "0.999... does not equal 1, no matter how hard it tries".

Fascinating, you have proved that all mathematicians are wrong. Good for you.






This follows easily from the standard definitions.

[lotanddaughters]

Fascinating, you have proved that all mathematicians are wrong. Good for you.






This follows easily from the standard definitions.

Okay, let me grant you that .

By accepting that, you must also accept that .

Do you agree?

[Martingale]

By accepting that, you must also accept that .

Do you agree?

No I Don't.

[Martingale]

What if a civilization based their numerical system on groups of what we call "five"? It might look something like this:

1 2 3 4 10

Their .4 would be the equivalent to our .8, or 8/10.

do you know what .44 (base 5) would be in our base 10 system?

[Grasshopper]

The article found on Wikipedia.org, on the topic "0.999..." seems to accept that 0.999...=1. I don't....

(1) Let x = 0.999...

Then,

(2) 10x = 9.999...

10x - x = 9.999... - 0.999...

9x = 9

x = 1

From (1) and (2), x = 1 and x = 0.999...

Therefore 1 = 0.999...

Or try this.

0.999... = 9 (1/10) + 9 (1/10)² + 9 (1/10)³ + ...

This is a geometric series. Since |(1/10)| < 1, it follows that:

ar + ar² + ar³ + ... = ar/(1-r).

In this case, a=9, r = (1/10), and ar/(1-r) = (9 x (1/10))/(1 - (1/10) ) = 1.

Therefore 0.999... = 1

Or you can always go the common sense approach:

Assume 0.999... < 1.

Then there exists a real number a, a is not equal to zero, such that 0.999... + a = 1.

Find that real number.

EDIT- suppose someone should actually respond to your argument.

0.999... is just a group of symbols to represent a number. For base 5, the number that plays that role is 0.444...

0.444... = 4(1/5) + 4(1/5)² + 4(1/5)³+...

Which is a geometric series, so,

0.444... = (4 x (1/5))/(1- (1/5)) = 1

Don't get so hung up on NAMES and symbols, and instead concern your self with the structure itself. We can CALL numbers whatever we want to. Their structure is unchanged. Your argument is invalid, and stems from confusing the symbols and names we give numbers with their actual structure.