math question
math question
is 0.9 reoccuring the same as 1?
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NO ROID NO!!! This question has brought many of forum to its knees!
http://www.blizzard.com/press/040401.shtml
The straight dope:
http://www.straightdope.com/columns/030711.html
wikipedia's proof:
http://en.wikipedia.org/wiki/Talk:Proof ... ._equals_1
last time the question was posed on a forum I am on, it reached 9 pages in two days, lol (one guy stopped it by talking nonsense):
http://www.linuxforums.org/forum/topic- ... hlight=999
typically people attribute the answer of .999~ = 1 to the fact to the CPU's inability to handle 'infinites'. but using some interesting calculus, most people have 'proven' that it does in fact = 1.
my vote is that .999~ will always approach one, but never reach it...
This is blizzards answer:Blizzard wrote:For seven and a half years , enthusiastic forum-goers have fervently debated the issue on the Battle.net® forums
http://www.blizzard.com/press/040401.shtml
The straight dope:
http://www.straightdope.com/columns/030711.html
wikipedia's proof:
http://en.wikipedia.org/wiki/Talk:Proof ... ._equals_1
last time the question was posed on a forum I am on, it reached 9 pages in two days, lol (one guy stopped it by talking nonsense):
http://www.linuxforums.org/forum/topic- ... hlight=999
typically people attribute the answer of .999~ = 1 to the fact to the CPU's inability to handle 'infinites'. but using some interesting calculus, most people have 'proven' that it does in fact = 1.
my vote is that .999~ will always approach one, but never reach it...
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.99999 (infinitely recurring) is, in fact, exactly 1.
Not "approaching 1", as some have claimed. A sequence of 9's getting longer and longer (that is, more and more 9's, getting closer to an infinite number of 9's) would approach 1, but if the sequence is actually infinite, it's not "approaching" any more, it's there.
Some might argue that you can't sum together an infinite series, or that you can't have an infinite number of 9's... that it's some kind of limit but never makes it. This is because they're arguing from intuition rather than from the definitions. Read the wikipedia comment by Monguin61 at 00:52, 9 December 2005 (UTC) for a more thorough explanation.
Not "approaching 1", as some have claimed. A sequence of 9's getting longer and longer (that is, more and more 9's, getting closer to an infinite number of 9's) would approach 1, but if the sequence is actually infinite, it's not "approaching" any more, it's there.
Some might argue that you can't sum together an infinite series, or that you can't have an infinite number of 9's... that it's some kind of limit but never makes it. This is because they're arguing from intuition rather than from the definitions. Read the wikipedia comment by Monguin61 at 00:52, 9 December 2005 (UTC) for a more thorough explanation.
Right, blizzard's solution depends on it being infinitely repeating, otherwise 9.99999..... (a finite amount of times) - .999999..... (The same finite number of nines) = 8.9999.....91, thus their solution becomes invalid. My question is, even with an infinite number of nines, is 9.999.... - .999.... = 9? I remember talking about degrees of infinity... like 1^infinity < 9^inifinity. Wouldn't it also be correct to say that .999999.... < 1 because of the degrees on finity stuff when looking at 9.999....-.999999?
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Nope. Degrees of infinity don't have anything to do with this.snoopy wrote:Wouldn't it also be correct to say that .999999.... < 1 because of the degrees on finity stuff when looking at 9.999....-.999999?
9.9999... - .999... would give you just 9. Same as 10-1. Again, this is provided you're using ... to mean an infinite number of 9's rather than just a large but finite number of 9's.
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Nobody credible. Well, except for people talking about programming -- in that case, they're right. You can't sum together an infinite series sequentially using finite time and resources, which is what you'd be trying on a computer.DCrazy wrote:Lothar: who says you can't sum together an infinite series?
You can't do that sum, but it's not because of what those people say.Does this mean that you can't do Sum[ i = 0 to infinity ] ( (-1)^i ) = 0?
Rather, it's because the sum is divergent. Take the sum from 0 to n and let n->infinity, and notice that you alternate between 1 and 0. There is no limit. (Those who work with divergent series would say the sum is 1/2 -- the average of 0 and 1 -- because it makes certain calculations convenient. Within their particular fields, they're entitled to make the definitions that way.)
Well, evidently, Wikipedia does. Check out that link fyrephile posted:DCrazy wrote:Lothar: who says you can't sum together an infinite series?
ROFL. That seriously pegged my WTF-meter. Never thought I'd see anyone talk that way about math, except maybe those LaRouche supporters.Wikipedia wrote:There is no proof that demonstrates 0.999... = 1 but there is solid proof that demonstrates 0.999... < 1. See the proof by induction if the sysops have not censored it yet. The number 0.999... is not a limit, it is an infinite sum. Even if you were to treat it as a limit, the limit is not equal to the infinite sum. Just try adding the following: 9/10 + 9/100 + 9/1000 + .... add as many terms as you like. You will notice that the terms keep getting closer to zero and the sum keeps getting closer to 1. However, no term ever becomes zero and no sum ever becomes 1. There is no such thing as an infinite sum that can be calculated. There is no definition that says the limit of an infinite sum is the infinite sum for if there were, it would be nonsense and thus untrue. This is a typical misconception.... Unlike this article by Ksmrq that is written in a trust me approach, I am challenging all thinkers and teachers to think for themselves. Examine the terminology, examine the definitions, think long and hard and if you find a flaw, discard everything and start all over again. This page contains proofs that debunk the myths stating the real number system collapses if 0.999... is not equal to 1. I am not asking you to believe me, I am asking you to think for yourselves. Please do not be fooled by arguments presented in group theory and the so-called definition of real numbers. Real numbers existed long before the concepts of groups and fields came into existence. You do not have to pass a course in real analysis or abstract algebra to figure this out. In fact, you do not need to know anything else besides high school math. Do not be intimidated by those who are able to write a lot of BS that is in the first place irrelevant and serves to confuse rather than enlighten. I have studied and passed all these courses and they are worthless. Finally, have a backbone and post your opinion.... When incorrect knowledge is propagated forcefully and the truth is rejected.... progress stops.
I won't add to the arguments given here that it equals one, except to note that it's a question of definitions. Under the definition of the real numbers I've seen, the decimal notation for the real numbers is defined to mean the infinite sum the of digits multiplied by place value, where the infinite sum is interpreted in the usual sense. Which means that when you write 0.999..., what you mean is the sum, from one to infinity, of 9 times 10^-i, where (lest the over-excitable friend from Wikipedia object), that infinite sum is defined in the usual epsilon-delta sense of a converging limit of partial sums and yada yada yada. It's exactly one.
By way of consolation, though, I'll add that there are other fields you can work in. One of my favorite fields is the hyperreals. It consists of the real numbers and an extra number--epsilon--which is smaller than every other real number but still bigger than zero. 2e, 5e, 1000e... they're all smaller than the smallest real number. In fact, there's a whole little infinitesimal real line of epsilons around each real number. And also many numbers bigger than any real number.
It's cool because you can do calculus there. Under the rules of arithmetic, you calculate (f(x + e) - f(x)) / e ... and you get the derivative you'd normally get on the real line. Only you're not taking a limit, you're just doing arithmetic. It's pretty cool.
So even though 0.999... is equal to one, if you wish it wasn't, and you'd like to see a space where numbers can be "infinitely close", and you understood any of what I just said, you should check out the hyperreals. 1 - e is the number you're wishing existed.
wow, i thought i was just asking a simple question.
i was just doing some simple algebra, and windows calculator gave me 0.99999 recurring as a result for something i did. Since i was previously asking Lothar if an infinitely small sphere/circle/line/plane would just be a POINT (0 dimentional), when i got this result on the calculator i absentmindedly wondered why it didn't just tell me 1.
i didn't expect to have stumbled on such a hotly debated mathematical topic, while simply re-teaching myself basic mathematics .
i guess it's cool i did though.
i was just doing some simple algebra, and windows calculator gave me 0.99999 recurring as a result for something i did. Since i was previously asking Lothar if an infinitely small sphere/circle/line/plane would just be a POINT (0 dimentional), when i got this result on the calculator i absentmindedly wondered why it didn't just tell me 1.
i didn't expect to have stumbled on such a hotly debated mathematical topic, while simply re-teaching myself basic mathematics .
i guess it's cool i did though.
thx that makes a lot of senseTricord wrote:Answer is yes. There's a simple proof using infinite substractions, but I can't remember it exactly.
If you calculate 1 - 0.999.... you basically have to put the 1 on infinity. And so you get zero.
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It is a basic question, and it shouldn't be hotly debated (and it isn't, except by people with Larouche-follower-like math skeelz.) As long as you're working within the real numbers, the answer is clearly defined and well known.roid wrote:wow, i thought i was just asking a simple question....
i didn't expect to have stumbled on such a hotly debated mathematical topic
If you're working in some system other than the real numbers, of course, all bets are off...
It depends. If you are an engineer 0.9 is identicle to 1.0 i nmost cases.
This reminds me of a joke I heard once.
A Mathematician, a Physicist, and an Engineer all go to the same job interview.
The mathematician gets called in first. He gets asked one question. ?What is one plus one?? Sensing a trap, he thinks about it for a bit, and then answers ?I am not sure but I am fairly certain it converges?.
The Physicist gets called in next. He is asked the same question. ?What is one plus one?? Equally as cautious he ponders for a bit then replies ?I am not sure but I am fairly certain it is on the order of one?.
The Engineer gets called in last. He is asked the same question. ?What is one plus one?? He thinks for a second, and then answers ?what do you want it to be??
This reminds me of a joke I heard once.
A Mathematician, a Physicist, and an Engineer all go to the same job interview.
The mathematician gets called in first. He gets asked one question. ?What is one plus one?? Sensing a trap, he thinks about it for a bit, and then answers ?I am not sure but I am fairly certain it converges?.
The Physicist gets called in next. He is asked the same question. ?What is one plus one?? Equally as cautious he ponders for a bit then replies ?I am not sure but I am fairly certain it is on the order of one?.
The Engineer gets called in last. He is asked the same question. ?What is one plus one?? He thinks for a second, and then answers ?what do you want it to be??
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it can never = 1 as it will NEVER get past the decimal point to be a whole number. ANY WHOLE NUMBER. this is like saying 9 becomes 10 eventually. what, are you folks products of outcome based education? a whole number is a whole number by definition. a fraction is a fraction and is PART of a whole number 0.xxxxxxx can NEVER BE A WHOLE NUMBER. not even if christ resurrects, muhammed returns and every one attains nirvana (the state of being not the band) .
these are the same arguments used to claim that a bullet never really reaches it's target.
since .333... never reaches an end then 3 x .333 does not equal 1. it equals 3 times infinity which is infinity. it never reaches 1.
p.s. you can't represent 1/3 with a decimal number. so there isn't an equivalency since 1 divided by 3 is a repetitive decimal. ergo 1/3 <> .333...
since .333... never reaches an end then 3 x .333 does not equal 1. it equals 3 times infinity which is infinity. it never reaches 1.
p.s. you can't represent 1/3 with a decimal number. so there isn't an equivalency since 1 divided by 3 is a repetitive decimal. ergo 1/3 <> .333...
Here, I googled a page that will teach you how to convert repeating decimals to fractions. I learned this around the 7th grade. Give it a shot:
http://www.homeschoolmath.net/teaching/ ... umbers.php
http://www.homeschoolmath.net/teaching/ ... umbers.php
yeah it's one value and it can't be written down as it's infinite. as 1 cannot be divided by 3 the answer can only be an approximation.
i defy you to take one and divide by 3 the old fashioned way and reach an end to it. here's what happens.
3/1 = .33333333333333333333333333333333333333333333
3333333333333333333333333333333333333333333333333333
33333333333333333333333333333333333333333333333333333
ad infinitum. i don't care what home school new math mumbo jumbo you use. 1 cannot be divided by 3 evenly.
i defy you to take one and divide by 3 the old fashioned way and reach an end to it. here's what happens.
3/1 = .33333333333333333333333333333333333333333333
3333333333333333333333333333333333333333333333333333
33333333333333333333333333333333333333333333333333333
ad infinitum. i don't care what home school new math mumbo jumbo you use. 1 cannot be divided by 3 evenly.
yes it can, as a fraction.jakee308 wrote:yeah it's one value and it can't be written down as it's infinite. as 1 cannot be divided by 3 the answer can only be an approximation.
i defy you to take one and divide by 3 the old fashioned way and reach an end to it. here's what happens.
3/1 = .33333333333333333333333333333333333333333333
3333333333333333333333333333333333333333333333333333
33333333333333333333333333333333333333333333333333333
ad infinitum. i don't care what home school new math mumbo jumbo you use. 1 cannot be divided by 3 evenly.
if you can cut an apple in to thirds, you can divide 1 by 3.
and people have been dividing one into three alot longer than we've had zero.
wait... before zero? so like before they didnt have an apple in front of them...fliptw wrote:yes it can, as a fraction.jakee308 wrote:yeah it's one value and it can't be written down as it's infinite. as 1 cannot be divided by 3 the answer can only be an approximation.
i defy you to take one and divide by 3 the old fashioned way and reach an end to it. here's what happens.
3/1 = .33333333333333333333333333333333333333333333
3333333333333333333333333333333333333333333333333333
33333333333333333333333333333333333333333333333333333
ad infinitum. i don't care what home school new math mumbo jumbo you use. 1 cannot be divided by 3 evenly.
if you can cut an apple in to thirds, you can divide 1 by 3.
and people have been dividing one into three alot longer than we've had zero.
1/3 is a value. Just because you're having trouble expressing it in a particular base-ten number system using decimals without repeating decimals really isn't my problem.jakee308 wrote:yeah it's one value and it can't be written down as it's infinite. as 1 cannot be divided by 3 the answer can only be an approximation.
i defy you to take one and divide by 3 the old fashioned way and reach an end to it. here's what happens.
3/1 = .33333333333333333333333333333333333333333333
3333333333333333333333333333333333333333333333333333
33333333333333333333333333333333333333333333333333333
ad infinitum. i don't care what home school new math mumbo jumbo you use. 1 cannot be divided by 3 evenly.
Math is a pretty well defined logical system. If you think that the already defined rules are wrong and ought to conform to the way that you think that the universe should be, then you should invent your own math and then write a book about it or something. But as far as the math that everyone else in the world uses is concerned, .999... = 1.
no that's another place your wrong. what they say is the .999... is the equivalent of 1. it can never be equal to 1 or it would be 1. why are you making personal comments. they seem condescending and arrogant and assume that only you have the final truth but as i just wrote YOU are mistating the conclusion. .999... is the equivalent of 1. however although they use an equals sign it can never be 1. it is self defined as not being 1. don't ascribe desires that i haven't expressed and don't make sly little shots about how you learned whatever in 7th grade. what grade are you in now? by your desire to put me down over a question that millions of people have difficulty understanding i'd say about 8th grade. that .999... = 1 is of little use in the real world. oh and when you grow up, if you become an architect or civil engineer, please post a warning to others about how you use equivalencies in your calculations. concrete = real = hard and that's a fact jack.
jakee, numerous mathematical proofs have been given that .999... does in fact equal 1, some of which involve math no more advanced than what is taught at a 7th grade level, and all of which conform to the universal laws of mathematics established over the past 4 or 5 millennia. Your inability to accept them is based on a preconceived notion that .999... does not equal 1, not on any fault in the proofs. Stop redirecting the conversation to hide this fact.
To further illustrate how little of a grasp you have on the concept:
And in case you're wondering, Jeff250 and I are both CS majors in college. Lothar and Drakona are both grad students who majored in Math, and I know at least Lothar is a math teacher.
To further illustrate how little of a grasp you have on the concept:
Physical measurements cannot be made with infinite precision, so this entire debate is irrelevant in the "concrete" world. This is pure math, not architecture or civil engineering.jakee wrote:if you become an architect or civil engineer, please post a warning to others about how you use equivalencies in your calculations.
And in case you're wondering, Jeff250 and I are both CS majors in college. Lothar and Drakona are both grad students who majored in Math, and I know at least Lothar is a math teacher.
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4/4 is equal to 1. It's just another way to write it. .999... is also another way to write 1. So is e^i*2*pi.jakee308 wrote:what they say is the .999... is the equivalent of 1. it can never be equal to 1 or it would be 1.
BY DEFINITION, a repeated decimal is the infinite sum of the digits multiplied by place value, and the infinite sum is defined as the limit of partial sums, where the limit is defined using standard epsilon-delta arguments (or, in this case, epsilon-M arguments.) I'm going to invoke completeness, for the sake of completeness (bonus points if you understood that, and double bonus points if you laughed.)
So, .999... (where the ... means there are an infinite number of 9's in sequence) is defined as the sum from i=1 to infinity of 9*1/10^i. This, in turn, is defined as the limit as n->infinity of the sum from i=1 to n of 9*1/10^i, provided the limit exists. We can quite directly demonstrate that the limit is, in fact, 1 -- first, notice that the sequence of partial sums is monotone as n increases. Second, given any epsilon>0, I can choose M large enough that the sum from i=1 to M of (the stuff) is between 1-epsilon and 1 (simply choose M=1/log(epsilon) and you're done.) Third, note that 1 is an upper bound for the sequence of partial sums. By the last 2 statements, 1 is clearly the supremum (least upper bound) of the sequence of partial sums. Because the real numbers are complete, we know any bounded monotone increasing sequence *must* converge to its supremum. Therefore, .999... = 1. QED
You might want to revise this statement.YOU are mistating the conclusion.
Math doesn't always work the way you expect or want it to. It works according to definitions, and sometimes those definitions lead to results that are surprising. In this case, the definitions show that .999... is in fact exactly equal to 1. (If you don't like that, learn enough about math that you can work in a space other than the real numbers.)
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Your information is out of date.DCrazy wrote:Lothar and Drakona are both grad students who majored in Math, and I know at least Lothar is a math teacher.
Neither of us are grad students any more. I have my masters in applied math, and she has her masters in math.
I'm not a math teacher any more, either. I work at a museum where I teach about airplanes, and especially about the principles of flight. I'm doing it until my brain has sufficiently recovered from grad school. My wife now works for an airplane manufacturer on military aircraft -- specifically, she writes software requirements for a program that simulates this unmanned aircraft. Her group's software is used to test the software the military uses to control the planes from the ground.
All that is to say... we're not math grad students any more, but we definitely know what we're talking about.