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New Proof that 1 = .999999999 Repeating
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In both questions, we're trying to measure a quantity.... a quantity which cannot be measured using our number system which was only meant to measure finite things.
The decimal system isn't cut out to do that stuff. It becomes a limitation of our numbering system.
In our numbering system, it's able to measure things in wholes, halfs, and tenths pretty efficiently. But things like thirds don't calculate so easily, the number will continue forever using the decimal system.
Also, simple things like circles are not measured easily by our system. A simple relationship such as the relationship between a circle's diameter and its circumference are not easy, round numbers. It should be, and we gave it a simple name, Pi, to symbolize this simple relationship, but using the decimal system that number doesn't work out. Do the calculation and it spews out an endless barrage of non-repeating numbers.
MrDudeMan
Lifer
thats the thing...its a different way to say the same thing, so in theory and reality, it is exactly 1. ill be back in a little while...im going to go try to find the examples presented in a slideshow from a while back that clearly ends this thread and all discussion that disagrees with, mathematically and logically, .999.... = 1.
MrDudeMan
Lifer
thats about the quality of reply i expected. there seems to be a clear division in the type of people in this debate. people with no prior mathematical knowledge or insanely large pride complexes have an issue with .999... = 1, which basically makes them troll in the thread, while the exact opposite types of people are on the other side of the fence. interesting observation IMO.
MrDudeMan
Lifer
Originally posted by: nakedfrog
:roll:
I don't think you read my post well enough. I've bolded the really important part of it for you. Here's afor your quality reply. Oh, and saying that there's people that agree with you, and people that are wrong isn't really an interesting observation.
i agree that it doesnt matter in the long run.
saying there are people that agree with you and then others that are wrong doesnt typically work, but in this situation it does because there is clearly a true answer to the problem. its a problem of mathematics, and those of us that claim to know better than mathematics are simply ignorant to facts and logic (ironically).
http://mathforum.org/library/drmath/view/53387.html
Given that a geometric series has a sum of a/(1-r) where a is the first term and r is the common ratio between terms, 0.9... has a = 0.9 and r = 1/10 (each successive .9 is ten times less than the previous), (9/10)/[1-(9/10)] = 1.
Given that a geometric series has a sum of a/(1-r) where a is the first term and r is the common ratio between terms, 0.9... has a = 0.9 and r = 1/10 (each successive .9 is ten times less than the previous), (9/10)/[1-(9/10)] = 1.
Classic exmaple of a guy who knows just enough to be dangerous. You can tell he doesn't know what he's talking about when he uses language like "essentially the same quantity". Give me a break.
edit: read what Dr. Math says instead
In the base-10 (decimal) system, each successive placeholder is 1/10th the value of the previous placeholder. Starting with the first place on the left side of the decimal point, you input its value. If you have 10, then the previous placeholder gets a "1".
9 falls short of 10
99 falls short of 100
999 falls short of 1000
9999 falls short of 10,000
In each of these cases, had the number been 1 larger, the placeholders would not have been filled with 9's... they'd have an extra placeholder added on the left, which would be filled with a 1, with the remaining digits becoming zeros.
Likewise,
0.9 falls short of 1.0
0.99 falls short of 1.00
0.999 falls short of 1.000
0.9999 falls short of 1.0000
In each of these cases, as you increase the amount of decimal places, the amount that it falls short by decreases by 1/10th. But the fact remains that it must have fallen short... otherwise you wouldn't be putting a "9" in the successive placeholder, you'd be putting a "1" in the previous placeholder.
The fact that you have repeating 9's tells you that you've fallen short. Since the number of 9's is infinite, you have missed by an infinitely small margin, but nevertheless, you have missed.
9 falls short of 10
99 falls short of 100
999 falls short of 1000
9999 falls short of 10,000
In each of these cases, had the number been 1 larger, the placeholders would not have been filled with 9's... they'd have an extra placeholder added on the left, which would be filled with a 1, with the remaining digits becoming zeros.
Likewise,
0.9 falls short of 1.0
0.99 falls short of 1.00
0.999 falls short of 1.000
0.9999 falls short of 1.0000
In each of these cases, as you increase the amount of decimal places, the amount that it falls short by decreases by 1/10th. But the fact remains that it must have fallen short... otherwise you wouldn't be putting a "9" in the successive placeholder, you'd be putting a "1" in the previous placeholder.
The fact that you have repeating 9's tells you that you've fallen short. Since the number of 9's is infinite, you have missed by an infinitely small margin, but nevertheless, you have missed.
You're completely correct up until that last sentence. What do you mean by fallen short? You mean that one value is less than an other. The is not true of the 9 decimal series. Besides, you can't compare infinity to normal, everyday numbers.
Have you taken calculus yet? Back in the day, my reasoning was identical to yours, until I took calculus and realized I was wrong. If you haven't gotten there, trust us and you'll see 😉 If you still won't let it go, I recommend that Dr. Math link I put in my edit.
Have you taken calculus yet? Back in the day, my reasoning was identical to yours, until I took calculus and realized I was wrong. If you haven't gotten there, trust us and you'll see 😉 If you still won't let it go, I recommend that Dr. Math link I put in my edit.
I'm saying that a given number will always be smaller than a number that has a larger value in the previous (leftward) placeholder.
Some fields in math must go an an assumption. Otherwise they won't work. They may be right 99.999R% of the time but they won't be right 100% of the time. Take Newtonian physics for example. Nobody would have thought that as you increase in speed the laws of physics can begin to follow other principles. But Einstein formulated that as you approach the speed of light, you gain mass.
Rock Hydra
Diamond Member
Infinity is a variable, therefore it cannot be assigned a numeric value.
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