math question

roid · Post by **roid** » Mon Dec 12, 2005 7:13 am

is 0.9 reoccuring the same as 1?

Darkside Heartless · Post by **Darkside Heartless** » Mon Dec 12, 2005 7:20 am

In a single word, no.
0.9 recurring comes infinitly close to 1, but it is never quite there.

DCrazy · Post by **DCrazy** » Mon Dec 12, 2005 7:28 am

I wouldn't be too sure, but here's my take on it:

lim_x->infinty (1 - 10^-x) = 1

Notice that for larger and larger x, (1 - 10^-x) approaches 1. Writing 1 as a lim_x->infinity (1) means that 0.999... does indeed equal 1.

Tricord · Post by **Tricord** » Mon Dec 12, 2005 10:24 am

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.

fliptw · Post by **fliptw** » Mon Dec 12, 2005 11:30 am

consdier:

1 divided by 3 is 0.333...

0.999... divided by 3 is also 0.333...

ergo 0.999 = 1.

De Rigueur · Post by **De Rigueur** » Mon Dec 12, 2005 12:26 pm

.999... = x
9.999... = 10x
9 = 9x
1 = x

This is how it was explained to me.

fyrephlie · Post by **fyrephlie** » Mon Dec 12, 2005 12:37 pm

NO ROID NO!!! This question has brought many of forum to its knees!

Blizzard wrote:For seven and a half years , enthusiastic forum-goers have fervently debated the issue on the Battle.net® forums

This is blizzards answer:
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...

fyrephlie · Post by **fyrephlie** » Mon Dec 12, 2005 4:39 pm

De Rigueur wrote:.999... = x
9.999... = 10x
9 = 9x
1 = x

This is how it was explained to me.

9x = 9 on lazy calculators....

Lothar · Post by **Lothar** » Mon Dec 12, 2005 7:15 pm

.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.

snoopy · Post by **snoopy** » Mon Dec 12, 2005 7:42 pm

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?

Lothar · Post by **Lothar** » Mon Dec 12, 2005 7:49 pm

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?

Nope. Degrees of infinity don't have anything to do with this.

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.

DCrazy · Post by **DCrazy** » Mon Dec 12, 2005 8:03 pm

Lothar: who says you can't sum together an infinite series? Does this mean that you can't do Sum[ i = 0 to infinity ] ( (-1)^i ) = 0?

Lothar · Post by **Lothar** » Mon Dec 12, 2005 8:30 pm

DCrazy wrote:Lothar: who says you can't sum together an infinite series?

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.

Does this mean that you can't do Sum[ i = 0 to infinity ] ( (-1)^i ) = 0?

You can't do that sum, but it's not because of what those people say.

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.)

Drakona · Post by **Drakona** » Tue Dec 13, 2005 12:55 am

DCrazy wrote:Lothar: who says you can't sum together an infinite series?

Well, evidently, Wikipedia does. Check out that link fyrephile posted:

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.

ROFL. That seriously pegged my WTF-meter. Never thought I'd see anyone talk that way about math, except maybe those LaRouche supporters.

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.

roid · Post by **roid** » Tue Dec 13, 2005 1:11 am

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.

Tricord 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.

thx that makes a lot of sense

Lothar · Post by **Lothar** » Tue Dec 13, 2005 1:35 am

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

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.

If you're working in some system other than the real numbers, of course, all bets are off...

Mobius · Post by **Mobius** » Tue Dec 13, 2005 12:43 pm

Hmmm. Tell me again why this is interesting?

fliptw · Post by **fliptw** » Tue Dec 13, 2005 1:22 pm

Lothar wrote:[dyk]that Mobi needs to STFU?[/dyk]

Dedman · Post by **Dedman** » Tue Dec 13, 2005 1:23 pm

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??

Vindicator · Post by **Vindicator** » Tue Dec 13, 2005 5:22 pm

My super-easy proof:

1/3 -> 0.3333333...
2/3 -> 0.6666666...
3/3 -> 0.9999999...

QED, bitches.

fyrephlie · Post by **fyrephlie** » Tue Dec 13, 2005 5:26 pm

Vindicator wrote:My super-easy proof:

1/3 -> 0.3333333...
2/3 -> 0.6666666...
3/3 -> 0.9999999...

QED, bitches.

lol ... new math is great!

DCrazy · Post by **DCrazy** » Tue Dec 13, 2005 5:29 pm

/me loves Fyrephlie's new avatar! I rue the day when Worms went 3-D...

Darktalyn1 · Post by **Darktalyn1** » Tue Dec 13, 2005 6:38 pm

The answer is yes. There are multiple proofs that prove this, but the one I usually remember and cite is the one posted earlier,

1/9 = 0.1111 repeating
2/9 = 0.2222 repeating

etc

9/9 = 0.9999 repeating = 1

jakee308 · Post by **jakee308** » Tue Dec 13, 2005 6:59 pm

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)

.

Jeff250 · Post by **Jeff250** » Tue Dec 13, 2005 7:33 pm

I wasn't already familiar with the subject, but the arguments presented here seem fairly cogent.
By definition:
.333... = 1/3
.666... = 2/3
.999... = 3/3
We're talking about symbols that represent a value, not an expression needed solving or infinite sum. By definition, ".999..." represents 1.

jakee308 · Post by **jakee308** » Tue Dec 13, 2005 7:38 pm

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...

Jeff250 · Post by **Jeff250** » Tue Dec 13, 2005 7:41 pm

No, you're making the mistake of trying to solve it or treat it as an infinite sum. It isn't. It represents one value, not an always increasing value.

Jeff250 · Post by **Jeff250** » Tue Dec 13, 2005 7:44 pm

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

jakee308 · Post by **jakee308** » Tue Dec 13, 2005 7:48 pm

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.

fliptw · Post by **fliptw** » Tue Dec 13, 2005 8:12 pm

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.

yes it can, as a fraction.

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.

fyrephlie · Post by **fyrephlie** » Tue Dec 13, 2005 8:14 pm

fliptw wrote:
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.
yes it can, as a fraction.

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...

Jeff250 · Post by **Jeff250** » Tue Dec 13, 2005 8:24 pm

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.

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.

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.

DCrazy · Post by **DCrazy** » Tue Dec 13, 2005 8:42 pm

"Home school new math mumbo-jumbo"? That's a new term for "way over my head".

Decimal points have absolutely nothing to do with it. They're a convenience for writing things down. Were we working in a base-9 number system, 1/3 would be equal to ...0001.0000... quite easily.

jakee308 · Post by **jakee308** » Tue Dec 13, 2005 8:49 pm

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.

DCrazy · Post by **DCrazy** » Tue Dec 13, 2005 9:28 pm

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:

jakee wrote:if you become an architect or civil engineer, please post a warning to others about how you use equivalencies in your calculations.

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.

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.

Lothar · Post by **Lothar** » Tue Dec 13, 2005 9:33 pm

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.

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.

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 are mistating the conclusion.

You might want to revise this statement.

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.)

Lothar · Post by **Lothar** » Tue Dec 13, 2005 9:41 pm

DCrazy wrote:Lothar and Drakona are both grad students who majored in Math, and I know at least Lothar is a math teacher.

Your information is out of date.

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.

Top Gun · Post by **Top Gun** » Tue Dec 13, 2005 9:41 pm

I'll have to admit that the equivalence seems to be completely counter-intuitive, but after learning special relativity, it's become clear to me that science and intuitiveness are often in complete opposition.

Jeff250 · Post by **Jeff250** » Tue Dec 13, 2005 9:46 pm

DCrazy wrote: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.

I'm not a CS major either.

DCrazy · Post by **DCrazy** » Tue Dec 13, 2005 11:55 pm

/me misremembered Jeff's Facebook profile. Lo_Wang = CS major.

Lothar wrote:I'm going to invoke completeness, for the sake of completeness (bonus points if you understood that, and double bonus points if you laughed.)

Ooh, shiny bonus points.