0.999... = 1

[Noa.]

Hehe my class went over .9...=1 when we were reviewing how to convert repeating decimals into fractions. Hell I wrote it on my binder.

 

[Spelt]

i like how susa is basically arguing both sides.
pick one

 

[Ch3s]

ALSO if .999... wasn't one, then the limit as change in x -> 0 (the derivative) of anything wouldn't exist, because it would infinitely approach the number from both sides, but never reach it, and the requirement is that it reaches it otherwise it is a hole. Also any two numbers have at least one real number between them (i found a mathematic proof somewhere but i dont remember where, it's just like a property of numbers in number theory), and .999... and 1 dont have any numbers between them. Anyways most of the logical arguments are defeated by the fact that without this assumption, big parts of calculus are estimations, so it is an important assumption

 

[#HBC | Gorf]

I was intent on looking into this till I saw this...

Divide a circle into 3 even parts. Exactly 1/3rd.

A circle is 360 degrees. 360/3 is 120 degrees. Make three 120 degree angles in a circle. OMG I DID IT!

It will only give the illusion of being complete. The incomplete part is likely unnoticeable at all, but existent. It's just infinitely small.

120+120+120=360. The amount of degrees in a complete circle.

You can divide anything into three. Including one. It's:
..._
0.3

Reason I write it like that is cuz idk how to write it any other way. So ignore the periods.

Hell... Maybe you can't even divide a nondivisible number of 3 by 3 EVENLY. But it can be divided.

 

[SuSa]

i like how susa is basically arguing both sides.
pick one

Anyways most of the logical arguments are defeated by the fact that without this assumption, big parts of calculus are estimations, so it is an important assumption

Kind of why I'm challenging it.

A circle is 360 degrees. 360/3 is 120 degrees. Make three 120 degree angles in a circle. OMG I DID IT!

If you kept reading, you'd have seen that in 2 in the morning that was a HORRIBLE example.

I forgot a circle was 360 degrees. I was thinking "1 circle" not "360 degrees".


Yea. I make stupid mistakes.

 

[A Pimp Named Slickback]

Integration IS estimation, taken to an infinite degree.

It is basically drawing rectangles under a curve and measuring the area as those rectangles become infinitely small.

If you don't think infinity can be approached, then you cannot do calculus.



Also SuSa, what, exactly, are you challenging?

 

[SuSa]

No matter how much smaller your rectangles get, they are all equidistant from infinity - as there is still forever to go.

Imagine each degree of a circle is a starting point.
No matter where you start you need to draw 360 degrees to make a circle.
When you finish the circle, keep tracing it infinitely.
No matter which point you started on - you have traveled the same distance.

 

[Spamus43]

Something that no one has pointed out yet, that usually gets mentioned fairly early whenever I see one of these "debates," is that infinite decimal expansions are only a product of notation. Let's go back to your original argument that 0.(9) = 1, the one where you add 1/3 + 1/3 + 1/3. You seem to claim that the argument is flawed, because 0.(3) is not exactly equal to 1/3; rather, it is just very close. But the infinite expansion of 1/3 is only needed because you chose to write it in base 10.

Any system of notation you choose will have numbers like this. For example, in base 3, 1/2 has an infinite decimal representation. Surely, though, you won't argue that 1/2 + 1/2 =/= 1, because 0.(1) + 0.(1) = 1.00000......1 is slightly more than 1. For one thing, you'd be wrong; in particular, you wouldn't (I hope) question the validity of an argument simply because of the notational system is uses.

Now, that's certainly not a proof, but I hope it does help to illustrate why your refutation is flawed.

That said, you clearly don't understand the concept of infinity. That's not meant in any kind of derogatory sense - it's not a easy thing to comprehend - but it's evident from what you've posted. Until you can get your head around that, no one can offer a proof that will satisfy you, and you'll continue to be wrong.

 

[Deleted member]

the reason the 1/3rd argument is flawed is that you use 0.(3) = 1/3 to prove 0.(9) = 1, which is circular.

 

[Spamus43]

the reason the 1/3rd argument is flawed is that you use 0.(3) = 1/3 to prove 0.(9) = 1, which is circular.

Right, I totally agree. It's not a proof, by any means, but it is a plausible argument, and its probably the easiest way to get yourself comfortable with the idea that 0.(9) = 1. All I was pointing out is that that particular line of reasoning (namely, "its not 1 , its just really close") is an artifact of notation.

 

[Ganonsburg]

Go from point a to point b which is 10 feet by going half of the remaining distance each step.

n=1 E infinity <- My notation for sigma notation, because I can't use it on these boards. Basically, I'm taking the sum of every term from n=1 to n=infinity.

n=1 E infinity ((1/2)^(n-1))/2 = (1/2)/(1-(1/2)) = (1/2)/(1/2) = 1.

At the infinitieth step, you will reach the end.

Similarly, using infinite sums, we can take .999... = .9 + .09 + .009 =

n=1 E infinity ((9/10)((1/10)^n)) = (9/10)/(1-(1/10)) = (9/10)/(9/10) = 1

I hope the notation at least makes sense to those who have taken calculus. For those of you that haven't, you can prove this using geometry via similar triangles. Loads of fun.