Does 0.999... equal 1?

If I understand you, Sythe, you're saying that in the real world, the ramifications of assuming that it is equal to zero are nonexistent? As in, if this sort of mathematics was involved in plotting the trajectory of artillery, then it could be safely assumed to be equal without having any perceivable effect on where the shells hit?

 

I cant understand what sythe is saying really but its doesnt equal 1 because 1/3 isnt exactly 0.333 so if we take 3x(1/3) it will not equal 0.999 exactly.

 

I wouldn't say that zero can only be an infinitesimally small number, but rather, that it could be that, or smaller. It is still possible for zero to be nothing. When I plot a graph, and place two points on x = 4, the horizontal difference between them is zero. This may or may not apply in reality, depending on whether there is a minimum length for object. If there is not, then the argument could be made that the difference will never equal absolute zero, I suppose.

 

For all those people who disagree, I challenge you to jump off a 50 story building and come infinitely close to the ground but have that not equal the ground Since you're just falling half the distance each time, you'll never reach a point where the distance between you and the ground is 0, right?

Seriously though, it _must_ equal 1. Those people who can't understand this should spend some time with calculus. I understand that this (on the surface) doesn't make sense. However, you need to understand that .0(bar)X where X represents any number, cannot exist. Why? Because that means that infinity ends somewhere, which can't be true (that's the whole point behind infinity).

For those arguing that 1/3 != .3(bar) you need to take some 3rd grade math classes again. 1 divided by 3 is equal to .3(bar).

Think about it another way:
if .9(bar) is not 1, what number is between .9(bar) and 1?
Maybe you'll say, .9(bar)1. Fine, I'll say, but what about .9(bar)2? and then you'll counter with .9(bar)3 and we'll make a good game of it. Until we come to .9(bar)9 which is just .9(bar). Therefore, since we can't find a number which is between .9(bar) and 1, I guess they must be equal

BTW: this also means that .9(bar)1 is less than .9(bar). This is because if the 9s go out infinitely, there is no actual place that the 1 can be. There are just too many 9s

Gödel's Incompleteness Theorem is pretty good reading material, also, for those people who claim that .9(bar) = 1 because our math system is wrong.

 

well if it were made into 3/3 yes, but put as a decimal no

 

Why not?

 

well theres a .001 missing. so that doesnt truley make one. but im our messed up form of math 3/3=1 so 0.999 repeating does in some form = 1

 

I can't believe you are still discussing this. You're not convincing anyone, everyone here has made up there minds already.

I guess this thread could support the idea of infinity though...

 

i am justgoing to say does 99.9=100. no it dosent then why does
.999=1.00

 

99.999... is equal to 100 for the same reason that .999... is equal to 1

x=99.999...
10x=999.999...
10x-x=999.999...-99.999...
9x=900
x=100

 

if you store .999... as in int in java, it would be 0 i believe, lol

 

First of all your using the wrong type for decimals. It turns to 0 as because it gets truncated. Second, no computer can store a value of infinite length. The whole universe is smaller than infinity.

 

To say .999... = 1 you are giving infinity a definite value, which is impossible. Infinity isn't a number, it's a concept.

To say it does equal you are saying after an infinite number of 9s, it turns into a 1, yet it is impossible to add infinite 9s because infinity isn't a number.

So in other words, you're saying the 9s go on forever, very good, but at what point does it turn into a 1?

 

What NUMBER of 9s would it take to fill the gap between them? None. The 9s would go forever, which is the concept of infinity.

Adding infinite 9s means there is no stopping point.
You can sum infinite geometric series. The series .9*10^n from 1 to infinity is very easy to do.
The Geometric Series Theorem is very useful here.

From: http://web01.shu.edu/projects/reals/numser/t_geom.html
Using this theorem i can very easily prove the infinite sum .999...=1

This is the sum.

Using a property of sums.

Using the Geometric Series Theorem.

Simple math.
There you have it.

You already said infinity is not a number. Now your asking me when infinity occurs? They equal at no point but if the series goes forever then it would turn to 1.


EDIT: The starting n value on those sumations should be n=0 not n=1. I dont feel like retaking those pictures so im not going to.

 

Dorito is arguing about the numeral tokens. .9... and 1 have the same value, but different representations.

 

Exactly! So, as the question was proposed in an abstract manner (0.99... = 1 as opposed to 0.99... miles = 1 mile), and, in the abstract, there is no smallest number, then the answer to that question would be no.