I probably wouldn't win, and I accept that but I would still do it on principal, that principal being that fact is imperatively true regardless of how many people believe it and fact must not be silenced due to opposition.smithy_2045 said:
Poll: 0.999... = 1
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M'kay. The number 0.999... is equal to an infinite series 0.9 + 0.09 + 0.009 + 0.0009 and so on. If you know anything about slightly advanced maths, you'll know that the sum of an infinite geometric series is equal to a/(1-r) when |r| < 1 (explained below for those who aren't so good at maths)
In our example here, a, the first term, is 0.9, and r, the common ratio, is 0.1 (because each term is the previous term multiplied by 0.1).
So we have 0.9/(1-0.1) which equals 0.9/0.9 which equals 1.
Explanation of maths involved:
A geometric sequence is one where each term is the previous term multiplied by some number r. The first term is a, the second term is ar, the third term is ar[sup]2[/sup] and so on. The nth term is ar[sup]n-1[/sup].
The sum of a geometric series to n terms, which we shall call S[sub]n[/sub], is therefore equal to a + ar + ar[sup]2[/sup]...+ ar[sup]n-2[/sup] + ar[sup]n-1[/sup]
Multiplying by r, we get rS[sub]n[/sub] = ar + ar[sup]2[/sup] + ar[sup]3[/sup]...+ ar[sup]n-1[/sup] + ar[sup]n[/sup]
Subracting rS[sub]n[/sub] from S[sub]n[/sub] leads to S[sub]n[/sub] - rS[sub]n[/sub] = a - ar[sup]n[/sup]
This means S[sub]n[/sub](1-r) = a(1 - r[sup]n[/sup])
And S[sub]n[/sub] = a(1 - r[sup]n[/sup])/(1-r)
Now, to find the sum to infinity, n must be equal to infinity. If |r| > 1, r[sup]infinity[/sup] is infinite. If |r| < 1, r[sup]infinity[/sup] is equal to zero.
Thus, S[sub]infinity[/sub] = a(1 - r[sup]infinity[/sup])/(1-r) = a(1-0)/(1-r) = a/(1-r) when |r| < 1
Satisfied now?
In our example here, a, the first term, is 0.9, and r, the common ratio, is 0.1 (because each term is the previous term multiplied by 0.1).
So we have 0.9/(1-0.1) which equals 0.9/0.9 which equals 1.
Explanation of maths involved:
A geometric sequence is one where each term is the previous term multiplied by some number r. The first term is a, the second term is ar, the third term is ar[sup]2[/sup] and so on. The nth term is ar[sup]n-1[/sup].
The sum of a geometric series to n terms, which we shall call S[sub]n[/sub], is therefore equal to a + ar + ar[sup]2[/sup]...+ ar[sup]n-2[/sup] + ar[sup]n-1[/sup]
Multiplying by r, we get rS[sub]n[/sub] = ar + ar[sup]2[/sup] + ar[sup]3[/sup]...+ ar[sup]n-1[/sup] + ar[sup]n[/sup]
Subracting rS[sub]n[/sub] from S[sub]n[/sub] leads to S[sub]n[/sub] - rS[sub]n[/sub] = a - ar[sup]n[/sup]
This means S[sub]n[/sub](1-r) = a(1 - r[sup]n[/sup])
And S[sub]n[/sub] = a(1 - r[sup]n[/sup])/(1-r)
Now, to find the sum to infinity, n must be equal to infinity. If |r| > 1, r[sup]infinity[/sup] is infinite. If |r| < 1, r[sup]infinity[/sup] is equal to zero.
Thus, S[sub]infinity[/sub] = a(1 - r[sup]infinity[/sup])/(1-r) = a(1-0)/(1-r) = a/(1-r) when |r| < 1
Satisfied now?
So what you're saying is that you read the summary AND completely ignored the context of the sentence?emeraldrafael said:
I certainly can't see why you don't understand the reasoning behind the original claim.
The point there is that mathematics educators have a strong interest in understanding why learners accept or reject the claim given that it's demonstrably true. One of the reasons that students accept it is on appeal to authority. That has nothing to do with the actual truth or falsity of the claim, only why students typically accept or reject it.
They silenced it because it was disrupting the ability for other discussions to commence. At the time they outlawed it, there had been something like _hundreds_ of threads created and thousands of posts in the main thread discussing it.BlacklightVirus said:
More of a "Damn...we have created a monster." rather than a "Stop having discussion!"
That and the "discussion" (like we see so often on the intertubez) devolved into each side slamming eachother because the other was just _wrong_ and an idiot for it.
M'kay. The number 0.999... is equal to an infinite series 0.9 + 0.09 + 0.009 + 0.0009 and so on. If you know anything about slightly advanced maths, you'll know that the sum of an infinite geometric series is equal to a/(1-r) when |r| < 1 (explained below for those who aren't so good at maths)
In our example here, a, the first term, is 0.9, and r, the common ratio, is 0.1 (because each term is the previous term multiplied by 0.1).
So we have 0.9/(1-0.1) which equals 0.9/0.9 which equals 1.
Explanation of maths involved:
A geometric sequence is one where each term is the previous term multiplied by some number r. The first term is a, the second term is ar, the third term is ar[sup]2[/sup] and so on. The nth term is ar[sup]n-1[/sup].
The sum of a geometric series to n terms, which we shall call S[sub]n[/sub], is therefore equal to a + ar + ar[sup]2[/sup]...+ ar[sup]n-2[/sup] + ar[sup]n-1[/sup]
Multiplying by r, we get rS[sub]n[/sub] = ar + ar[sup]2[/sup] + ar[sup]3[/sup]...+ ar[sup]n-1[/sup] + ar[sup]n[/sup]
Subracting rS[sub]n[/sub] from S[sub]n[/sub] leads to S[sub]n[/sub] - rS[sub]n[/sub] = a - ar[sup]n[/sup]
This means S[sub]n[/sub](1-r) = a(1 - r[sup]n[/sup])
And S[sub]n[/sub] = a(1 - r[sup]n[/sup])/(1-r)
Now, to find the sum to infinity, n must be equal to infinity. If |r| > 1, r[sup]infinity[/sup] is infinite. If |r| < 1, r[sup]infinity[/sup] is equal to zero. (If |r| = 1, we end up with 0/0, and I don't wanna go there (it's not 1)).
Thus, S[sub]infinity[/sub] = a(1 - r[sup]infinity[/sup])/(1-r) = a(1-0)/(1-r) = a/(1-r) when |r| < 1
Satisfied now?
In our example here, a, the first term, is 0.9, and r, the common ratio, is 0.1 (because each term is the previous term multiplied by 0.1).
So we have 0.9/(1-0.1) which equals 0.9/0.9 which equals 1.
Explanation of maths involved:
A geometric sequence is one where each term is the previous term multiplied by some number r. The first term is a, the second term is ar, the third term is ar[sup]2[/sup] and so on. The nth term is ar[sup]n-1[/sup].
The sum of a geometric series to n terms, which we shall call S[sub]n[/sub], is therefore equal to a + ar + ar[sup]2[/sup]...+ ar[sup]n-2[/sup] + ar[sup]n-1[/sup]
Multiplying by r, we get rS[sub]n[/sub] = ar + ar[sup]2[/sup] + ar[sup]3[/sup]...+ ar[sup]n-1[/sup] + ar[sup]n[/sup]
Subracting rS[sub]n[/sub] from S[sub]n[/sub] leads to S[sub]n[/sub] - rS[sub]n[/sub] = a - ar[sup]n[/sup]
This means S[sub]n[/sub](1-r) = a(1 - r[sup]n[/sup])
And S[sub]n[/sub] = a(1 - r[sup]n[/sup])/(1-r)
Now, to find the sum to infinity, n must be equal to infinity. If |r| > 1, r[sup]infinity[/sup] is infinite. If |r| < 1, r[sup]infinity[/sup] is equal to zero. (If |r| = 1, we end up with 0/0, and I don't wanna go there (it's not 1)).
Thus, S[sub]infinity[/sub] = a(1 - r[sup]infinity[/sup])/(1-r) = a(1-0)/(1-r) = a/(1-r) when |r| < 1
Satisfied now?
Infinity doesn't scare us at all, which is why we do math with it on a regular basis.SomethingAmazing said:
And it actually equals both .999... and 1, but that's semantics.
1/3 = .333...
.333... * 3 = .999...
1/3 * 3 = 3/3
3/3 = 1
.999... = 1
I'm not saying that there arent other good ideas backing it up to be "true." I'm just saying that this is the same logic that explains how the holocaust started. Someone of authority said something was right, and the people followed.Rabid Toilet said:
and dont tell me it has nothing to do with this, because it has everything to do with the reasoning behind as to why some believe its right and others dont.
Also:
I'm glad that we can make fun of others in the thread.Jaime_Wolf said:
Unfortunately, we've used that proof at least twice, to no avail. Good effort though!Redingold said:
And yet someone in authority said that 2 + 2 = 4, and the people followed.emeraldrafael said:
Why does 2 + 2 = 4? Because they say so.
Using the very laws of mathematics that were invented so long ago by those people, .99... and 1 are the same number.
Oh, well that's completely different. I thought you were referring to a situation like this one.IMakeIce said:
You're right that .3333 does not equal 1/3.SomethingAmazing said:
However, .333... (note the repeating, there's an infinite number of threes) exactly equals 1/3.
I like the fact that most people don't seem to know that this is true only because it is defined to be so within the most commonly used numbering system. You can define systems where 0.(9) =/= 1 you know. You just have to define numbers such as 0.000...0001 (an `infinitesimal'). Of course, then many of the things we take for granted (like three thirds making up a whole) don't work any more.
that's not the point. in the problem you are looking at:x^2=x can equal 1 and 0emeraldrafael said:
1^2=1
0^2=0
in this situation, 0 and 1 are not interchangeable which I think is what you are trying to say. What the person who brought up this problem was trying to say is that 2 things can represent the same thing: 1 can be x and 0 can be x
with the topic's problem, the question is: is .999 = 1? yes or no?
people are using 2 different approaches.
1: using methods to show that using the same logic with the same numbers you can end up with 1 and .999...
-3(1/3) = 3(.333...)
2: showing that the difference between .999... and 1 is so small that it practically does not exist.
-the guy with the limits and such. I didn't read it.
with these 2 approaches 1 and .999... are so similar that they can just be the same thing.
numbers are a human creation to conceptualize quantities. numbers are not Absolutes. Numbers are tools.
to serve its function, .999... and 1 both represent the same idea.
assume you are you. if you lose 1 strand of hair, are you still you?
technically not the exact same you, but the difference is so small that it does not really matter.
Nah.BlacklightVirus said:
And actually, my information was slightly flawed. They banned it after years of it being discussed (it would die for a few months and reappear, in exactly the same fashion...several - multitudes of threads much like this one), which always devolved into hurling insults. They finally opened the can of worms with the april fools joke because so many people were made aware of it and flocked in droves to discuss it in the general forum. After that, it was not to be repeated because of the backlash it caused.
Considering how you tore into the one poster, I wouldnt go calling any kettles black.Jaime_Wolf said:
Also, no, I'm just stating. The reasons ideas become fact is cause the idea carries in the human mind. If i were to kill everyone who thinks .999...! = 1 (not a threat) and the idea died with them, then the truth would be .999...! doesnt equal 1 because no one would beleive it. Again, I'm going to have to wait till my math professor gets back to me, I just thought it was interesting that thats why students who carry the ideas after the teacher is dead accept the idea.
I think it has more to do with the fact that people who believe they're right have actually taken higher level math courses and know what they're talking about, whereas those who disagree think infinitesimals exist when they don't.emeraldrafael said:
Also, did you just bring up the holocaust in the middle of a math discussion? -facepalm-