humor me.
okay, you want to prove to me that 0.999… is equal to 1.000… then do this for me…
draw a line. Label one point on the line 0.9. one inch later on that line mark a point 1.000… (add as many 0’s as you would like.
now, draw a line for 0.99. Now draw a line for 0.999. Now draw a line for 0.9999. Now one for 0.99999. Continue to keep drawing lines please.
Call me when you draw an additional line on 1.000
The inability for the thickness of the ball in my pen that applies the ink to create thinner and thinner lines as I draw them does not justify your point.
okay, then change the scale – make the two points 10 feet apart, and start drawing lines…
not convinced – okay 100 feet apart…1000 feet? How much space do you need to be convinced that a line will never be placed on the exact same locaton?
wait wait wait…I reread your first post. I’m on your side here, I thought you said “Okay you want me to prove that 0.99999 = 1?” Sorry. Pretend that’s what you were trying to say, then reread my post, then I’ll make sense.
0.99999…(to infinity) does not = 1.
If you say they are the same, then you are just saying “infinity is = 1”. And that’s just not true. Infinity is a concept ONLY, and not to be considered a number. “1” is just a number.
Apples are fruits. Oranges are round, like apples. Does that mean Oranges are Apples?
Saying an infinite number (like .99999…) is like 1 is comparing apples to oranges. They are two different figures in a huge way.
kinoons, you’re being silly. Obviously nobody has the time to draw the infinite number of lines that your demonstration would require.
Look at the proofs in the 2nd, 3rd, 4th and 5th posts in this thread. They are very simple and should erase all doubt. I mean, this is not a point of controversy in mathematics. It’s gradeschool stuff.
And for your task:
Present us with a list of 10 numbers between 0.9999… and 1. There should be an infinite number there for you to choose from, if they are really different numbers. Or, to go back to you number line - if they occupy different places, then there must be some space between them. Draw 10
lines in that space that are on neither 0.999… nor 1.
[sub]Yeah, Saltire brought this one up first, I know…[/sub]
I would just like to mention that, by defition, “0.000…01” does not go on for infinity. It has an end (the “…01”), and thusly, you cannot say that .999… has a difference of .000…01 from 1.
Infinity cannot be confined between two points.
But wait a minute, if 0.999…= 1 then 0.999…8= 0.999… then 0.999…7= 0.999…8 then 0.999…6= 0.999…7 etc. etc. Then if you did it infinite times, you’d soon come to 0 so are you saying that 1=0?
Burnt Toast- Read SPOOFE’s post above. You can’t put infinity between those two numbers. If you’re putting it between those two numbers, that means that it has two ends and you’ve restricted an infinite number.
kinoons - You can put two numbers on the same point on the number line if they are equivalent. For example, I could put 100/100 on the same point on the number line as 1, because they are the same number. Because .9rep = 1, (the proofs shown are the ones I was taught), then .9rep CAN go on the same spot on the number line as 1.
oops, ok then. Carry on.
You’ve hit upon it precisely!
Listen people, I force you to see the truth. All the doubters in here just need to READ the posts. The proofs are there plain and simple. And none of this mumbojumbo about “the philosophy of the numberline”. Point of the matter is that all numbers on the number line are rational or irrational. A number like 0.0000000etc.0001 is, first of all, absurd because you’re adding a number less than the smallest ratio in an infininte string of ever decreasing ratios. (explain to me how you’ll do that and I’ll concede you’re right).
It’s just so frustrating when people talk about 0.9999 etc. and 1 being different numbers. AAAURGH!
For all intents and purposes, this new number (if I’m understanding your definition of it correctly)0.0000000etc.01 is equal to zero. IT HAS TO BE!
why?
because you NEVER NEVER NEVER Reach the “one”… it’s as if it doesn’t exist. Really, it’s nonsensical to write the statement 1.000000000etc. - 0.99999999999999etc. = 0.00000etc.00001, because you won’t ever get there to do that subtraction at the very end. There IS no very end. The difference between the numbers is zero, thus the numbers are the same.
Taking this absurdity further: there are an infinite number of ways to write zero with this concept.
0.00etc.0120685 is also equal to zero, because you’ll NEVER NEVER NEVER reach the 120685. Anything you tack on the end of a repeating decimal, for all intents and purposes, doesn’t exist. You can NEVER NEVER NEVER get to it.
Im surprised no one has mentioned these previous threads:
And yes, 0.999rep equals 1.0
You still misunderstand. Read again. You’re going to keep stopping and looking, and yes, you’ll find a difference if you do. But the repeating bar means you DON’T stop and look. You have to keep going forever, and the difference disappears if you do that. Capisce?
No such number. That terminology is not defined. How many zeros are there before you get to the 1? An infinite number? Then you never get to the 1.
I think kinoons is confusing limits with infinately long numbers. What he is describing here is a limit that approaches one. But limits apply to functions, not numbers. Numbers are absolute, and 0.999rep absolutely equals one.
Perhaps kinoons could convince me if he could draw a line between 0.999rep and one.
You’re confusing things a bit here, Spoofe: 0.000…1 is being used as notation for “the difference between 1 and 0.99999…”, while no one here has been able to demonstrate that such a difference exists. Because the 1 falls after infinitely many zeros, it’s effectively not there.
There are several concepts that get confused here, namely the question of “reality” vs “theoretical mathematics.” The number .9999… with an infinite number of decimals, all 9s, is not a number in “reality” (since you can’t have an infinite number of decimals, you’d run out of paper even if you used every scrap of paper on the planet.) It is, however, a mathematical construct.
So, as a mathematical construct, if you think of .999… as the limit of an infinite sequence:
.9
.99
.999
.9999
etc
Then obviously no term of that sequence is equal to 1. And equally obviously, the sequence never reaches its limit because you can’t physically write an infinite number of places. So the sequence gets closer and closer to 1 (the hundred billionth term is really really close to 1) but never actually touches 1. However, the limit of the sequence is indeed exactly 1.
Thus, there is a difference between any term of that sequence and 1. So, if we have .9999[a billion more 9’s]99 and 1, there are numbers like .9999[a billion more 9’s]9912 between that number and 1.
When we write .999… we are using a shorthand to mean that the 9 repeats forever, infinitely, without ever stopping. This is not physically possible to write, of course, nor even to imagine. We are using this as a way of saying the “limit of the sequence of an infinite list of 9s” and that number is identically 1.
It’s analogous to .33333… being exactly 1/3. If you’re writing the .3333’s, you have to stop somewhere, and as soon as you stop, that number is different from 1/3. But if you somehow imagine the 3’s as going on forever without end, then you’ve got exactly 1/3.
And I should stress that this is a mathematical concept only. As soon as you start talking about “drawing a line”, you’re out of the theoretic mathematics universe and into a real universe. In the real world, .9999[a billion 9’s]99 doesn’t exist. If you draw a line a mile long, label 0 at one end and 1 at the other end, the difference between .9999[a billion more 9’s]99 and 1 is still smaller than the width of an electron. You can’t draw that difference in reality, only in the mind.
One of the problems with trying to imagine the number line as a bunch of points is that the points are dense. That is, there is no “space” between one point and another – any space is filled up with uncountably many numbers.
Please do note the other topics on this subject that have already been pointed out. This has been discussed to death on several occasions.
Is anyone going to note that this is one of Zeno’s paradoxes writ mathematically? Oops, I guess I did. Well, I am not the most responsible Doper, so I forgot to get the link, but just yesterday there was a post about Zeno’s paradox (spelled Xeno incorrectly). The basic point was that Greek’s hadn’t discovered the mathematics associated with limits, and therefore were considering the basic concept of infinite space between two coverging objects in a purely philosophical/conceptual way (e.g., if I throw a stone at a tree, at time t, it leaves my hand; at time t+1 it is half way between origin and tree; at time t+2, it is half way between half way and tree - i.e., 3/4 - and so on, which conceptually suggests that there is always another distance to halve, however small, between the stone and the tree). Yet, paradoxically according to Zeno, we observe that the stone DOES HIT the tree.
Well, that is the crux of this discussing. Kinoons, you are dealing with the conceptual paradox of being able to imagine a distance - however infinitely small - between .999… and 1. However, some of the other posters are trying to state that, like the stone, .999… does hit 1 eventually. The math that describes this is the math of limits and it outlined in many of the proofs in this thread.
Bottom line: conceptually, one can imagine convergence without concurrence, but mathematically and in real life, the stone does hit the tree.
WordMan
PS: One poster in the Zeno’s thread has a great line about how the next time somebody challenge’s you with Zeno’s paradox, just punch them in the face…
OH, PLEASE!!!
quote:
Evno gets into the act with
I’m not a math person, but I play one on T.V. And I have a question:
If .99999999… = 1, then does -.11111111… = 0?
Surely we can see not many math people here!
I know .9rep=1 since I was in elementary school. Don’t you people see the proofs.
Thanks to those people who still have clear heads! JC Princeton, it’s effortless to convience those “not a math person” a math truth.