0.999 repeating

[mackdaddy]

Also, using fractions, 1/3=.33... 2/3=.66... so thus, 3/3=.99...

.333...=1/3
3*.333...=3*(1/3) =3*1/(1/3)
.999...=1

I know this is just going in loops but, how could one say 1/3 is equal to 0.333... without saying that 0.999... is equal to one?

 

[Bob Brown MSEE]

And if one says that this is still never equal zero, would that not be the same as saying 0.999... could never equal 1, or 1-0.999... could never equal 0?

Hi Mackdaddy,

If a person was trying to create an expression to represent a converging series, then your insights are very important as you learn to think about limits. But, as Hallsofivy said, this notation is not used to represent a series or a function. daon2, makes it clear that he thinks of this expression as a geometric location on a number line. I am saying that I think of 0.999... as 7 ascii charactors in search of a consistant definition.

 

[mackdaddy]

Hi Mackdaddy,

If a person was trying to create an expression to represent a converging series, then your insights are very important as you learn to think about limits. But, as Hallsofivy said, this notation is not used to represent a series or a function. daon2, makes it clear that he thinks of this expression as a geometric location on a number line. I am saying that I think of 0.999... as 7 ascii charactors in search of a consistant definition.

I'm really sorry but I'm in ninth grade and don't understand what you are saying. And I wasn't trying to compare 0.999... to a function, but was simply showing how the concepts , of a function approaching an axis and 0.999... approaching 1, are alike. So since it is said that 0.999... doesn't approach, but is 1, I am using a function that approaches an axis infinitely but never touches it, as a counter-argument.

for every 9 you add on to the end of 0.9 you get a little closer to one and you do this forever apparently reaching one in infinity 9's. But in the function f(x)=1/x the "y" value gets a little closer to 0 every time the x value increases, and it does infinitely, but yet it never becomes 0 so how can one be but not the other? Do you see what I'm saying?

 

[Bob Brown MSEE]

In a physical sense, the real number system is fictitious; no one has ever measured anything that was not a rational number. You may now understand Kronecker's comment that God invented the integers, but the rest are the work of man.

Hi Jeff, YES, YES, YES!
That is why I like to refer back to a (yet unspecified by me) definition of rational numbers. My preference is ordered pairs of God's integers [n,d], with a test for equality like...
Iff [n1,d1]=[n2,d2] then n1d2 - n2d1 = 0

The use of repeated decimal expressions are nice because of the 1 to 1 mapping to the rationals. Even more, it is easier to quickly see the "magnitude" when trying to order a list of expressions than it is for the [n,d] notation (which is more closely tied to the definition of rational numbers that I just proposed).

Why not confuse an expression with the notion that it represents? Why is that important?
I wrote the "Just for Fun" response (above), to illustrate why.
I believe that these repeating integers do, infact, represent the rationals in a 1-to-1 manner (as do repeating decimals). However, repeating integers exaggerate the uneasy and strange behavior of digits at infinity. There are certainly many that would argue, "It is obvious that \(\displaystyle \overline{285714}3.0\) is clearly not 1/7". They might, however, become convinced (given enough time) that it is an alternative way to represent 1/7 that is consistent and unique within the repeating integer scheme.

 

[mackdaddy]

I'm getting a brain ache thinking about this so I'm going to stop! maybe tomorrow.

 

[Bob Brown MSEE]

I'm really sorry but I'm in ninth grade and don't understand what you are saying. And I wasn't trying to compare 0.999... to a function, but was simply showing how the concepts , of a function approaching an axis and 0.999... approaching 1, are alike. So since it is said that 0.999... doesn't approach, but is 1, I am using a function that approaches an axis infinitely but never touches it, as a counter-argument.

for every 9 you add on to the end of 0.9 you get a little closer to one and you do this forever apparently reaching one in infinity 9's. But in the function f(x)=1/x the "y" value gets a little closer to 0 every time the x value increases, and it does infinitely, but yet it never becomes 0 so how can one be but not the other? Do you see what I'm saying?

Hi mackdaddy,

My comment is intended to be a compliment.
When you take Calculus, you will be exposed to Limits in a very detailed way.
Without that experience you have, on your own, come to some very good insights about the nature of limits. You have done so out of observations about the domains of functions that have asymptotes. VERY GOOD!!! You have what it takes to be a great mathematician. I believe that you will do well in Calculus.

Hallsofivy was making the important point that 0.9999... is intended to represent a number, not a process, limit or a function.
Although your insights are correct, it is a subtle point that is often missed by people on math boards.

 

[mackdaddy]

Hi mackdaddy,

My comment is intended to be a compliment.
When you take Calculus, you will be exposed to Limits in a very detailed way.
Without that experience you have, on your own, come to some very good insights about the nature of limits. You have done so out of observations about the domains of functions that have asymptotes. VERY GOOD!!! You have what it takes to be a great mathematician. I believe that you will do well in Calculus.

Hallsofivy was making the important point that 0.9999... is intended to represent a number, not a process, limit or a function.
Although your insights are correct, it is a subtle point that is often missed by people on math boards.

Oh ok thank you very much! I'm sorry I just misunderstood what you were saying but thank you.

 

[lookagain]

I am saying that I think of > > 0.999... < < as 7 ascii charactors in search of a consistant definition.

I see eight ASCII characters there: a zero digit, a decimal point, three digits that are nines,
and one ellipsis made up of three dots.

 

[Bob Brown MSEE]

I see eight ASCII characters there: a zero digit, a decimal point, three digits that are nines,
and one ellipsis made up of three dots.

True. Thanks for the correction.
So, you didn't see 1.
I always look forward to your deep insights,
I would love to get your comments on this topic -- be sure to check out the continuation in odds-n-ends,
"Thread: Final 0.999... question I promise"