.9 repeating = 1? Is our numerical system flawed?

reply to post by cucuto89


Eh, it gets a little mushy in the brain when you try to visualize infinities. I never let it go that far when I had to learn these concepts.

Just understand this:

They are always moving and growing.

1/infinity = 0

.999... = 1

Don't visualize why, or how you can mess with inifinities. You can't anyway. What you are seeing is just a representation of a theoretical number/idea.

All the other stuff is correct though:

.999... * 10 is 9.999... which is 10.

9.999... x 2 = 19.999999999999 = 20

It goes on.

The best way to view infinities is with limits.

Limits
f(x)=1/x



1/x = 0

It is constantly getting closer to 0, it never technically reaches, but it comes so close that it doesn't exist anymore.

I'm sorry I did not read the whole thread interesting to say the least.

Infinities can be added and subtracted... Be prepared, it will drive you mad.

"Dangerous Knowledge"

video.google.com...

reply to post by divideby3


Great post. I love when people add sarcasm and then instead of using any knowledge they just post someone else's work.

It was a pretty stupid thread (obviously some don't understand infinities). I agree with you there.

I'm going to make a stupid statement as this thread contradicted itself about 40 times in the 4 years it's been discussed.

"

Hi All,

I have Long looked at this "problem"...
I am now discussing it with my algebraic kids,
and my calculus teacher (college).

Anyhow.. it was shown to me in 8th grade,
and I agree with at least on posting here...
"you don't understand infinity"... (nor do i !!!)

Anyhow... the basics, as i see them...

1/3 = .333...
2/3 = .666...

1/3 + 2/3 = 1
.333... + .666... =
.999

that seems pretty simple.
And...
it is.

People say "those numbers are not real"
or..
"those are approximations".

I say... you (and me, and WE... ALL...)
can not understand infinity...
and that...
is ok.
for now.

M

Now I have such a headache. I read almost all posts here. But here comes my unlogic conclusion which I think is logic enough for me.

I got a choco bar. I am a nice guy and have 2 friends I share with. I always share. And I want to be fair.

1 choco bar / 3 doesnt equal 3 * 1/3......I tell you why. (all this after a long nights headache)

My explanation is that no matter how exactly I cut them into pieces, one will be bigger. Even if it's just an infinit bit bigger.

1 choco bar = (1/3) + (1/3) + ((1/3) + a lil more)

Just saying that my friends will get the exact same size 1/3 choco bar, and while I am gonna be fair and don't want to favour any of my 2 friends, I take the third a lil bit bigger.

I am always fair.

[edit on 28-9-2008 by Benarius]

Fine, I will give 2 arguments to why 0.999...=1

I)

Don't know if anybody said this before:
If 1 does not equal 0.9999999...., then there must be a number between it.
Let me explain: if x!= y Then we can easily look at (x+y)/2. This is a number between x and y and does not equal x and y (because if x=(x+y)/2, then it follows trivially that x=y).

So if 0.99999... != 1, what is the number between the two??? Indeed, there is no number between the two, that means...

II)

There are people here that like to use the following (false) argument (*):
1-0.9=0.1
1-0.99=0.01
1-0.999=0.001
...
The pattern says the following: No matter how far you go, there is always a 1 left. So 0.999... can not equal 1.
This is a very understandable mistake, but a mistake nevertheless. I will show this by deriving that 1=0.999.... with the same technique!!!
0.999... = 0.999... (DUH!!)
10*0.999... = 9.999...
Of course look at the following pattern:
10*0.9=9
10*0.99=9.9
10*0.999=9.99
...
The pattern clearly suggests that 10*0.999... = 9.999...
10.0.999... - 0.999... = 9.999...-0.999... = 9
The explanation to the second equation is again easy:
9.9-0.9=9
9.99-0.99=9
9.999-0.999=9
...
So we clearly see that 9.999... - 0.999... must equal 9
(10-1)*0.9999... = 9
I suppose this is very obvious
9.0.999... = 9
If we know divide both sides by 9, we get...
0.999... = 1
tada!!!

I suppose this suggest that argument (*) is false, however I will not explain why, because it's not really that interesting (:

And another new theory that arose from my previous post.


0.1 = 0.01 = 0.001 = 0.0001 = 0.00001 .....and so on.

Here is my proof.

1 = 0.3 + 0.3 + 0.3 + 0.1
1 = 0.33 + 0.33 + 0.33 + 0.01
1 = 0.333 + 0.333 + 0.333 + 0.001
1 = 0.3333 + 0.3333 + 0.3333 + 0.0001

My formula would be this:

1 = (1/3) + (1/3) + ((1/3) + X)

Therefore X can be 0.01 or 0.000001 or 0.00000000001

Depending how many digits aftwe the point we allow.

Anyway. I am sure I am missing some or all basics of a giftet mathematician or magician. Tell me what you think and help me share my choco bars fairly. Thanks.

P.S. Can we say that a fraction like 1/3 is actually impossible ? Can we say that every fraction that has a infinity after the point is not a legal fraction ? I can share a choco bar with 3 friends, making 4 exact same size pieces. But I can't share a choco bar with 2 friends making exact same size 3 pieces. This seems so strange. 1/4 = 0.25, but 1/3 just doesn't make sense....AAAAAAAAAAAAAaaaaaaaaaaaaaaaaaaarrrrrrrrrrrrrrrrrrggggggggghhhhhhhhhhhh

Congratulations!

You've discovered...

Calculus!

reply to post by Johnmike


Thanks. I gogled calculus and am so intruiged by it and how I survived 40 years without it. Functions was the last in the mathematical hierarchy which I mastered 30 years ago. Started reading a calculus for beginners course. With so much time and choco stains on my hand I will be able to have my problem solved soon. Or shall I just make a new friend ?

P.S. Can we say that a fraction like 1/3 is actually impossible ?

It's not impossible. If I have three apples and I give one of them to each of my two friends, each of us will have precisely 1/3 of the apples I started with.

reply to post by nataylor


Very True !

I am just bugged by the 1/3 of 1 whole object.

reply to post by Benarius


It's amazing. Basically...

[f(x + h) - f(x)]/h^2 as the limit of h--> 0

What you're basically doing is taking two points on a curve, distance h from one another. Draw a line between them, that's the average slope. But you want the slope at one point. So you keep moving this secant line so that it's between two points that get closer and closer together. Finally you make it so that they're infinitely close to one another, therefore infinitely close to zero (but not zero - this is a limit, an independent concept you need to understand).

Therefore, tangent line.


And the 1/3 problem is just our number system. The .3 literally has to repeat infinitely, and that doesn't work in your mind because you can't think infinitely like that. For every decimal place, you need 1/3 of that more than the last decimal place. Like .333, a third of .0001 off, so .3333, and continue doing that. Infinite. So 1/3 is literally a number that can't be written in our base-ten number system easily, so we just use .3 with a line over it or write 1/3.

[edit on 30-9-2008 by Johnmike]

reply to post by Johnmike


[f(x + h) - f(x)]/h^2 as the limit of h--> 0

Wonderfull. I like your explanation of the limit. I take it when h is the limit, no mather how small (.000000000000000001 or smaller) it is there. No matter what. Can't be zero. And I assume that this formula can help me write a function calculating coordinates of those 2 points on the curve, angles or other properties. Not that I need to.

And the 1/3 problem has short curcuited my brain as it tried to cram all those infinite 3's inside it. Who said the human brain is only using 5% of its capacity. Mine just puffed smoke at 110% and 3's coming out of my ears.

Meanwhile I downloaded Maths: A student survival guide. Refreshing some old tricks. I remember how much fun it was back then.

If you have any intresting, out of the ordinary, phylosophical math threads or links to books, please let me know.

Thanks.

Originally posted by Benarius
[f(x + h) - f(x)]/h^2 as the limit of h--> 0

Wonderfull. I like your explanation of the limit.

No, I mean, the limit of h as it goes to 0. As h gets infinitely close to, but not exactly, zero. That formula is for a derivative.

f(x) means the function with x plugged in. So if f(x) = 2x + 4, f(a) = 2a + 4, and f(5) = 2(5) + 4 .

The limit is just when something gets infinitely close to but doesn't touch something.

What that formula is doing is taking the distance between two points (x and x + h) where h is the distance between these points. If h is 2, and x is 6, you're taking the distance between the points where x=6 and x=6 + 2=8.

You can draw a line between two points and find the average slope between them, easy. Change in Y over the change in X. But what if you want to find the instantaneous slope, that is, the slope at one point, instead of between two points? You can't use change in y over change in x because for change, you need two points!

By defining h as the limit as h gets infinitely close to zero, you're making the distance between these two points infinitely small. By doing this, you can use change in y over change in x, but since the two points are infinitely close to one another, it's effectively one point. And therefore, the slope at a point! This is much easier to understand if you see it graphically. You can probably find explanations with graphing pretty easily, and I don't want to seem to patronizing since you really do have more wisdom by virtue of age.

Crap! I read this over, and I realized I said h^2. It's [f(x+h) - f(x)]/h!

Apperently your not familar with (479001600 480000000) (362880/2=181440)(1000000-181440=-818560)(sqr75=8.6602540378443864676372317075294-8.18561=.474645ect)
reverse factor deteroation through polygonic breakdown of circles.

That didn't make any sense to me. Except inscribing polygons in a circle.

Originally posted by Canopene
Apperently your not familar with (479001600 480000000) (362880/2=181440)(1000000-181440=-818560)(sqr75=8.6602540378443864676372317075294-8.18561=.474645ect)
reverse factor deteroation through polygonic breakdown of circles.

Hm. I'm confused enough at the concept of this thread enough, now you just added to it.

What are you trying to say here? The equation doesn't seem to make sense--to me anyways and the guy above.

Originally posted by Sublime620
Just understand this:

They are always moving and growing.

1/infinity = 0

.999... = 1

1/infinity = 0 is wrong, so is .999... = 1. Infinity is not a number and therefore the result cannot be a unique number.

Here is a conceptual explanation of your mistake:

Where did you get the idea that 1/infinity = 0?

The very sentence "1/infinity = 0" has no meaning. Why? Because
"infinity" is a concept, NOT a number. It is a concept that means
"limitlessness." As such, it cannot be used with any mathematical
operators. The symbols of +, -, x, and / are arithmetic operators, and
we can only use them for numbers.

To write 1/infinity and mean "1 divided by infinity" doesn't make any
sense. 1 cannot be divided by a concept. It can only be divided by
a number. Similarly, "infinity + 1" or "2 times infinity" are also
meaningless.

As another example, what does this mean: "1 / justice = 5"?

That's right! It is as meaningless as "1 / infinity = 0" because
justice is a concept, not a number.

In math, when you hear people say things like "1 over infinity is
zero" what they are usually referring to is something called a limit.
They are just using a kind of shorthand, however. They do NOT mean
that 1 can actually be divided by infinity. Instead, they mean that,
if you divide 1 by successively higher numbers, the result becomes
closer and closer to 0. If I divide 1 by a very large number, like a
billion, then I get one-billionth, which is a VERY small number, but
it isn't 0. Since there is no largest number, I can always divide 1 by
a bigger number. But that will just produce an even smaller number,
right? It will NEVER produce 0, no matter how high I go. But since the
answer to the division is getting closer to and closer to 0, we say
that "the limit of the expression is zero." But we have still not
divided anything by infinity, since that isn't a number.

To go back to your chocolate bar, what if you divide it among every
person living on earth? Each person would get roughly 1 six-billionth
of a chocolate bar. That's a very, very small amount, and you'd
probably need a microscope to see your piece, but it wouldn't be zero,
right? Ah, but you asked about dividing it up amongst an infinite
number of people. Well, we can't. Why? Because infinity isn't a
number, so you can't show me an infinite number of people. If you try
to, I will just add one more person, and then we'd realize that the
number you thought was "infinity" actually wasn't.

So, to finish up, you are perfectly correct in saying that "1/infinity
= infinitesimally small." But only if you realize that you REALLY mean
"1 divided by a REALLY big number is a REALLY small number."

The conclusive paragraph corrects your result this way: 1/infinity = h where h -> 0 (h approaches zero). h is not a unique number; it represents a set of very small numbers that differ in value according to what the concept of infinity calls for under different conditions. To explain this using the last paragraph, there is no unique result to 1/infinity:

1/ BIG NUMBER = SMALL NUMBER
1 / BIGGER NUMBER = SMALLER NUMBER

Now, you can't use another superlative, BIGGEST NUMBER, because infinity won't let you -- there is no biggest number, because if there was, then what is BIGGEST NUMBER + 1?

And so if you grasped the concept of infinity, you can write down your own examples that won't stall.






[edit on 10/4/2008 by stander]