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2=1.999...infinity! What are your thoughts?
- Thread starter sjjordan
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monke said:7 years trying to solve anything would be brutal, I would quit after a week.
Welcome to my life in grad school...
only problem with the
1/9 =.11bar and then going up to 9/9=.9bar is that it is not a mathatical proof. (those can use only letters. x/x=1 and that is always true.
Now do the math math by hand on 8/9 and you do get the .8 bar because of how it works out. You always have that remainder of 8 for every spot. But when you go up to 9/9 it a one. Now some where in that enteter .9bar one of the numbers was able to go up one and that cause the enter thing to go to 1.
Now for pratical reasons I could care less 1.9999....= 2 to me. heck I really for what I do I really only use 4-5 sig figures any way and just round the rest. so for examppel 1.9996=2 for all care and in the end my answer is still going to be close enough and will work because I well the knowns in the problems are really only with in maybe 5% of what they should be any way.
1/9 =.11bar and then going up to 9/9=.9bar is that it is not a mathatical proof. (those can use only letters. x/x=1 and that is always true.
Now do the math math by hand on 8/9 and you do get the .8 bar because of how it works out. You always have that remainder of 8 for every spot. But when you go up to 9/9 it a one. Now some where in that enteter .9bar one of the numbers was able to go up one and that cause the enter thing to go to 1.
Now for pratical reasons I could care less 1.9999....= 2 to me. heck I really for what I do I really only use 4-5 sig figures any way and just round the rest. so for examppel 1.9996=2 for all care and in the end my answer is still going to be close enough and will work because I well the knowns in the problems are really only with in maybe 5% of what they should be any way.
It's worth pointing out that Iavhé asked for evidence that a proof was wrong. Several of us posted other ways to prove that 1=.9bar. While that technically proved that Iavhé's proof is invalid, it didn't show why, which is more enlightening. Luckily, atszyman already gave the specific answer.
Somewhere in my collection of math stuff on one of my old Macs I have a trigonometry "proof" of a false claim. It's hard to spot the error, which is why I like it. I think it is in a ClarisWorks document. If I run across it, I'll post it for forum members to ponder.
Somewhere in my collection of math stuff on one of my old Macs I have a trigonometry "proof" of a false claim. It's hard to spot the error, which is why I like it. I think it is in a ClarisWorks document. If I run across it, I'll post it for forum members to ponder.
I think the concept of infinity is a bit of a hard concept to grasp in this thread. when you have .999999|9 there is always another 9 after the previous one. One way to illustrate this is to read up on Hilbert's Hotel. It's a fun logical exercise that illustrates some of the issues with an infinite sequence of 9s.
Like the fact that shifting one of the 9's over to be to the left of the decimal point does not result in a zero at the end of the sequence, but instead puts another 9 there.
By this method the set of all real integers has the same number of elements as the set of only the odd valued real integers...
Of course I would know almost nothing about this if we hadn't covered it only a month ago in my last grad school class.
Like the fact that shifting one of the 9's over to be to the left of the decimal point does not result in a zero at the end of the sequence, but instead puts another 9 there.
Hilbert's Hotel said:
By this method the set of all real integers has the same number of elements as the set of only the odd valued real integers...
Of course I would know almost nothing about this if we hadn't covered it only a month ago in my last grad school class.
I've stayed at Hilbert's Hotel. It's a nice place, but rather expensive: 1 cent for the first minute, 2 cents for the next half minute, 4 cents for the next quarter minute, etc.
This reviewer actually got a photo of it!
This reviewer actually got a photo of it!
mangoduck said:
3. math is NOT the language of pure and universal logic.
It's A language of a particular class of universal logic at most.
There have been numerous attempts to reconstruct math by formally creating a logical system in which some or all of math would be a part of, therefore, abstractify math to the level of simple logic and algorithms. This is logicism, which created "Principia Mathematica" by Bertrand Russell and Alfred Whitehead, which recreated three volumes of math into simpler logic including set theory, number theory, and arithmetic, at which point a fundamental contradiction was realized (see also Godel's incompleteness theorem), and then logicism was abandoned. Which was too bad because this would unite mathematics with logic and analytical philosophy and give philosophy its much needed backing and solid proofs.
So now the general understanding of math is that it is an abstraction layer above simple logic and algorithmic language like cellular automata and your base-4 genetics. By the fact that the two systems can not interconvert perfectly, you can never arrive to the fact that Math itself can become the single, monolithic description of universal logic and being perfect.
This thread made my day yesterday... I was really bored at work and mad at my boss and decided enough was enough for the day: therefore, I had the chance to spend most of my day reading through this post and the other links found in this thread.
I obviously don't have a definitive answer on the issue but tend to agree with the math reasoning, assuming we take math as a number of rules and assumptions which lead to apparently contradictory results in particular situations.
I believe, under a mathematical reasoning, 0.9| definitely equals one, even if it seems to defeat the logic we have grown to associate to the representation of numbers in a decimal form.
I obviously don't have a definitive answer on the issue but tend to agree with the math reasoning, assuming we take math as a number of rules and assumptions which lead to apparently contradictory results in particular situations.
I believe, under a mathematical reasoning, 0.9| definitely equals one, even if it seems to defeat the logic we have grown to associate to the representation of numbers in a decimal form.
sjjordan said:
Write down a clear definition what an "infinite decimal fraction" is and what it means. It is pointless to go any further before you have done that.
this thread is rather impressive really - considering the fact that it sprung from errors in a calculator - where I believe I feel more at home than most people on this board - on account of building one, so kudos all around, this was very entertaining
I'll go a non-geeky way.
I have 2 apples on my desk. I can't have 1.999... apples on my desk. It isn't physically possible to cut some away leaving 1.999.... exactly. But it would be physically possible to cut my apple to any size if I had a thin enough knife, etc. Therefore 1.999... doesn't exist and must just be poncey talk for 2. But it's a theoretical number, you say. Well, numbers are only symbols representing matter on God's earth. What about infinity, you ask. It does exist. It's the size of the Universe. Or maybe it has no meaning. Everything is finite.
I'm already looking forward to the "Does 2.99...=3" thread.
I have 2 apples on my desk. I can't have 1.999... apples on my desk. It isn't physically possible to cut some away leaving 1.999.... exactly. But it would be physically possible to cut my apple to any size if I had a thin enough knife, etc. Therefore 1.999... doesn't exist and must just be poncey talk for 2. But it's a theoretical number, you say. Well, numbers are only symbols representing matter on God's earth. What about infinity, you ask. It does exist. It's the size of the Universe. Or maybe it has no meaning. Everything is finite.
I'm already looking forward to the "Does 2.99...=3" thread.
was googling something about infinity and this came up and i couldn't resist enlightening anyone interested, as a teacher with a math degree.
2 is in fact equal to 1.9999999...
i won't get into a rigorous proof, however, here's a quick sketch
1.9999999... is actually 1 + 9/10 + 9/100 + 9/1000 + ...
this is called a geometric series, that is, an infinite sum of the form ar^n
it can be shown that the geometric series converges if and only if |r|<1, and it converges exactly to ar/(1-r)
now we can think of the terms after the 1 as the terms to the right of the decimal place, and those terms form a geometric series with a=9 and r=(1/10), so if our series converges to x, we can essentially say 2=1.x whatever x is
now since i already have said that geometric series converge to a/(1-r) you can plug in our a and r to get:
((9)(1/10))/(1-(1/10)=(9/10)/(9/10)=1
adding that previous 1 term in we have 1+1=2
so there you have it, a rough sketch of a rigorous proof to show that 2 = 1.9999999999999
there are many other interesting facts that arise on the subject of infinite sums, this being one of the less noteworthy ones, i'd urge anyone with an interest in math to skim through a calculus textbook or something on real analysis for more information concerning these topics
2 is in fact equal to 1.9999999...
i won't get into a rigorous proof, however, here's a quick sketch
1.9999999... is actually 1 + 9/10 + 9/100 + 9/1000 + ...
this is called a geometric series, that is, an infinite sum of the form ar^n
it can be shown that the geometric series converges if and only if |r|<1, and it converges exactly to ar/(1-r)
now we can think of the terms after the 1 as the terms to the right of the decimal place, and those terms form a geometric series with a=9 and r=(1/10), so if our series converges to x, we can essentially say 2=1.x whatever x is
now since i already have said that geometric series converge to a/(1-r) you can plug in our a and r to get:
((9)(1/10))/(1-(1/10)=(9/10)/(9/10)=1
adding that previous 1 term in we have 1+1=2
so there you have it, a rough sketch of a rigorous proof to show that 2 = 1.9999999999999
there are many other interesting facts that arise on the subject of infinite sums, this being one of the less noteworthy ones, i'd urge anyone with an interest in math to skim through a calculus textbook or something on real analysis for more information concerning these topics
My turn, if Im repeating someone I guess it's just they way they expressed it was different from my way of thinking (EDIT: as I now realise, this thread is actually 4 pages instead of just the first one i read)
EDIT: it made sense in my head when I started...but now I'm not so sure. At (*) because it only holds true when you actually take the limit?
Code:
x = 1 - 1/n lim n -> inf
10x = 10 - 10/n lim n -> inf
Now:
10/n lim n -> inf = 1/n lim n -> inf (*)
Hence:
10x = 10 - 1/n lim n -> inf
Therefore:
10x - x = (10 - 1/n) - (1 - 1/n) lim n -> inf
9x = 9
x = 1Holy cow this is an old thread.
That's like saying that π (pi) doesn't have any meaning. The use of numbers is purely theoretical until you apply it to something. If you have a cubic metre/foot of sand, and you took a single grain out, then you would have yourself 0.9999999...(arbitary) cubic units of sand. Using an apple is difficult because of what you define what a single unit of an apple is.
That's like saying that π (pi) doesn't have any meaning. The use of numbers is purely theoretical until you apply it to something. If you have a cubic metre/foot of sand, and you took a single grain out, then you would have yourself 0.9999999...(arbitary) cubic units of sand. Using an apple is difficult because of what you define what a single unit of an apple is.
Look at it on a graph think about it visually zooming in infinitely .99999999999999999infinite does not equal 1 it's a vertical line on a graph that is .infinite01 away from 1 almost equaling 1 but not 1 the proper symbol for this relationship would be the exact use of the equal sign as a pair of wavy lines like this "≈" this making 1≈.9999999999999999999999999........... the truest statement. Though stating 1≠.999999999999999............ is a rougher guess just as is stating 1=.9999999999999999..........
the problem I've seen here is incorrect usage of the mathematical symbols.
the problem I've seen here is incorrect usage of the mathematical symbols.
I think you need to review your definition of "infinite."
By any mathematical definition, 0.(9) = 1, period. There is no ambiguity, no wiggle room, no grey area. 0.999~ =1.
In my first university calculus class, the professor wrote on the board:
0.999... infinity
1
And asked us if they were different numbers. Per her argument, they are not, since there's no number in between 0.999... and 1. Ergo, they're the same. Or, more technically,
0.999... infinity
1
And asked us if they were different numbers. Per her argument, they are not, since there's no number in between 0.999... and 1. Ergo, they're the same. Or, more technically,