Wow, talk about your mathematical snobbery. (I’m always getting into arguments with mathematicians on these points, because they INSIST they know what the best way to say something always is).
Really, our two statements (yours of teh series, mine with the decimal) are ENTIRELY equivalent. Honestly. If I define a repitend to be infinitely repeating digits, by God, that’s what it is.
I like your proof, in any case. This has become sorta fun, looking at all the ways you can come up with one.
By the by, your “starting with zero” series is the subject of another thread that’s been zipping around tonight. I’d parse a URL, but I’m lazy. Look for it yourself all ye with curiousity.
Course, I’ve never said that was the “best way”, just the way I prefer. The algebraic answer is easier for most people to digest, and in a lot of cases that’s what matters. I just thought I’d chime in with this as an alternate version.
I hope that these proofs have been enough to convince everyone, but I’m gonna add one anyway. Not really anything rigorous, but just a logical argument.
In order for two real numbers (however they may be represented) to be different, you should be able to describe some number that would exist between them. Stated another way, there should be some value you can add to one of the numbers that will increase it, but not push it higher than the other number. You should also be able to subtract something from one number without going below the other.
Any number you add to .9rep makes it larger than 1. Any number subtracted from 1 makes it smaller than .9rep.
so I’m curious – mathmatics is supposed to be an absolute. 1=1 – therefor wouldnt 0.9999999… = 0.9999999… ?
in any pratical application of the numbers it would be so close to 1 that there would be no point to call it anything but 1, but, it still is not absolutely completely equal to one?
Here’s the way I first wrapped my head around the concept when it was presented to me:
What’s 1 - 0.9999999…? It’s 0.000000…1, where the one is preceded by an infinite number of zeros. Effectively, you never reach the 10[sup]-n[/sup] that is the difference between 1 and 0.9999999999. So the difference is an infinite decimal expansion of 0.00000…, which is zero.
This has been beaten to death, I don’t know why I’m jumping in but I always want to…
Consider the axiom of completeness for the real numbers (that is, look it up if you don’t know what it means). Any convergent sequence of real numbers converges to something that is a real number. .9rep is just a convienient way to represent a series, which has a sequence of partial sums, that converges.
So, to what real number does it converge? Use the axiom that says that for all real numbers A and B, one and only one is true: A>B, A<B, or A=B. Since there’s no number in between .9rep and 1, then the first two can’t be true. That leaves the last, .9rep=1.
Well I thought I had it, but;
How would you notate the difference between .9999… and one? In other words, The difference between .9999 and 1 is .00001, right?
How would one write the difference between .9999… (to infinity)and 1?
.0000…1 ?
Peace,
mangeorge
but 0.999… and 1 are not the same thing… they cannot be. Any two values cannot at the same place in a number line. As soon as you deviate from the value of 1.000… you are at a different number that may be so incredibly close to one that it is pratically the same, but it is not exactly the same.
They are the the exact same thing. .999… does not deviate from 1 by even the slightest amount, as has been proven several different ways in this thread. They are dead-on, precisely the exact same number.
True only if you stop at a certain number of decimal places and look where you are. But the repeating bar means you NEVER stop. After an INFINITE number of decimal places, the difference from 1 DOES vanish.
guys, your applying a mathematical proof to a philosophical debate – the number line is not only a mathematical concept, but a philosophical one as well. Every number on the number line has its own position on the number line. No matter how you manipulate the math to try and prove that two different values can occupy the same location on the number line cannot be true. It is akin to saying that two different particles of matter can occupy the same space. It is simply not possible. The two pieces of matter can get so absolutely close to one another that the difference cannot be measured, but the two particles cannot be at the same place at the same time.
please, please do keep adding the 9 to the end of 0.999… and I’ll keep adding the 0 to 1.000 – the two numbers will never be the same, they can not be. It is fundamentally not possible for any two numbers to occupy the same location on the number line.
They are not two numbers, they are the same number, just written differently. They do occupy the same point on the number line because there is no difference between them.
Okay, this arguement is not getting anywhere…you seem to have compeletely ignored my above analogy – two numbers cannot occupy the same point on the number line just as two particles of matter cannot occupy the same space. Every time you say that 1.000… is the same as 0.999… there is an additional 0.000…01 difference between the two numbers.
Your analogy is flawed. You are taking the fermi-exclusion principle and trying to apply it where is does not apply. There is nothing in philosophy nor in mathematics that prevents two different names from referring to the same thing. Just as 2 + 2 equals 4, two representations can refer to the same number. One and 0.999. . . are two representations for the exact same number.
It might help. If he realizes that zero can have more than one representation (like zero, 00.000…01, 0, 1-1, etc.) , mabye he’ll realize that one can have different representations too. You know, like 0.999. . . and 1 both represent the same number.
And when you say that, you demonstrate a misapprehension of the concept of ‘infinite’. If the ellipsis in your 0.000…01 stands for ‘an infinite number of zeroes’, then there is no ‘1’ at the end. Where would you put it? Go ahead and walk out to the end of that infinite line of zeroes to put the one out there.
Another proof:
If .99999… is not exactly 1, then there is some number epsilon such that 1 - epsilon = .9999…
But, no nonzero epsilon can be demonstrated to satisfy this.
Kinoons would say epsilon is 0.000…01, but that is only do to a bad definition of “…”
