Does 0.999... equal 1?

Actually, that's wrong.

f(x) = 1/x CANNOT = 0
It can be an incredible small number, but can never reach zero.

That is what is taught in calculus courses.

Which is a useless piece of knowledge because for all intents and purposes it does reach zero. You can argue theoretically that x cannot be infinity and therefore the evaluation of 1/x cannot be zero. But if you pick a very large number... I.e one that is 10 orders of magnitude higher than the figures with which you are working... then your practical answer will be zero.

Thus while mathematically your statement is correct, in practice it is best to think of 1/infinity = 0.

 

[Shredderbeam]

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?

 

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?

Yes. If, as I said, your value for infinity was at least 10 orders of magnitude greater than the units with which you were working, then it would have no ramifications whatever. In fact the assumption would have to be made otherwise the calculation would likely be impossible. Most complex ballistics and rocket science equations are calculus based (and therefore limits are important.)

Allow me to demonstrate. Say we are working in meters for our missile trajectory:

10 orders of magnitude higher (than the base unit) would be 1 x 10^10. Thus the equation would be 1/10000000000 meters which is 0.1 nanometers.

If you are working in meters do you actually care about 0.1 of a nanometer? Of course not. You care about hitting Russia and not Japan, not about hitting molecule A and not molecule B.

 

Furthermore, consider this:

You argue that 1/infinity is not zero. But what is zero?
Let us define numbers in terms of a single dimension... a number line.
How do you know what 10 units positive is and what 5 units negative is?

Well you mark out evenly spaced units and you count them. When you count something you have to start somewhere and finish somewhere. So to determine where +10 is you start at zero and count to 10. To determine where -5 is you start from zero and count back 5 units to -5.

But how then do you count zero? You start at zero ... and stop at zero? But that is not a length, so how can you measure it? How is it anything at all, other than a reference point for all the other lengths?

You might say well both the start and finish point are at zero and thus it has zero length. But what is zero length? Can you physically show it to me in real life? In reality? Can the absence of something be measured as a form of the something which is missing?

The answer is that zero is an abstract concept with only loose connection with reality. When we say zero we mean a small enough quantity that it is not to be considered on paper. To be super specific we use the term nil.

So we may conclude that a length of zero, and indeed zero itself, in practice is not actually nothing but a very very very small amount that is: 1/(an effective infinity)

 

[Suby]

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.

 

[Shredderbeam]

So we may conclude that a length of zero, and indeed zero itself, in practice is not actually nothing but a very very very small amount that is: 1/(an effective infinity)

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.

 

When I plot a graph, and place two points on x = 4, the horizontal difference between them is zero.

You are using it in an abstract way. The absolute distance between the points is still > 0. To have an absolute distance between two points that is equal to zero is impossible. There would only be one point. You cannot measure a distance on only one point.

 

[orgcandman]

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.

 

[bosna]

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

 

[fatass1]

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

Why not?

 

[bosna]

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

 

[Schnell]

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...

 

[top score]

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

 

[fatass1]

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

 

[rscheater13]

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

 

[fatass1]

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.

 

[Dorito]

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?

 

[fatass1]

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

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.

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.

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.

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

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.

 

[WhereIsTheGod]

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

 

[Shredderbeam]

You are using it in an abstract way. The absolute distance between the points is still > 0. To have an absolute distance between two points that is equal to zero is impossible. There would only be one point. You cannot measure a distance on only one point.

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.