posted
Ok... so I think I'm the only one who could end up in an overwhelmingly painful argument about whether or not .99999999999........ is equal to 1.

Yesterday my little brother made a half hearted comment during a boring part of the superbowl about how it was stupid to say that .333333333.. was actually equal to 1/3 because 3*1/3 equalled 1, but 3*.33333333... only equalled .99999999..., thus, being the big sister I am, I felt it my duty to explain to him that in fact .9999999.. was equal to one.

It caused a HUGE stir... now, I wasn't hurt when my mom disagreed, she knows nothing about math, and believes in Sylvia Brown but not Jesus, nor was I surprised when my 12-year-old brother expressed his disbelief, and the 16-year-old brother argued that it couldn't be true. I proceeded to print a short proof of the fact from the net, and showed it to them. What realy stung however is that my father, who used to be a math teacher, refused to agree. He didn't disagree on any logical means, but disagreed by saying that anyone who actually believed such proofs was obviously wrong, but only thought they were right because they were intellectual snobs who thought they were better than the rest of the world.

Ugh!!!! it really hurt my feelings, I'm an intellectual SNOB because I believe in a mathematical REALITY... I swear, only MY family could get nasty over something like this.
Posts: 1321 | Registered: Jun 2006
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posted
Dear, while your feelings may be hurt, the little intrigue that .99999... is "equal" (whatever the heck *that* means, because it quite clearly isn't "equal" to 1 by any colloquial means) to 1 really isn't something to get that upset about.

Still, sorry about the familial tensions. Strength!

Yeah, and that mathematical reality thing is nice 'n all, but try living in it.
Posts: 3060 | Registered: Nov 2003
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posted
Actually there was a Hatrack thread long ago that went several pages of people arguing back and forth over that exact thing. Looks like it's gone, though -- it's referred to (by me at least once), but must have gotten removed long ago. Maybe someone could find it on the wayback machine, if they really wanted to.

My family argues over everything. There's been a standing disagreement over the past 20 or so years as to whether or not steam and water vapor are the same thing. It isn't nasty any more -- more an in-joke -- but yeah, we'll get nasty over pretty petty things.
Posts: 6213 | Registered: May 2001
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posted
Been there. Done that. It's likely your dad (and whole family) don't understand that it hurt you. You could tell them, but it would probably be more conducive to your own maturing to just be able to shrug it off and understand that they love you and don't want you to be upset.
Posts: 2827 | Registered: Jul 2005
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quote:Originally posted by Phanto: Dear, while your feelings may be hurt, the little intrigue that .99999... is "equal" (whatever the heck *that* means, because it quite clearly isn't "equal" to 1 by any colloquial means) to 1 really isn't something to get that upset about.

*Moves Phanto to short list for when I come to power*

Yes, it is equal in the colloquial sense of referring to "one apple". What other sense did you expect a number to have? One apple is nine-tenths of an apple, plus nine-hundredths of an apple, plus nine-thousandths of an apple, and so on.
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posted
King of Men, dear, then it just seems like I'm gonna have ta stop you from coming into power. Arguably, you may have won this point, though.
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posted
I cannot for the life of me imagine how someone could convince me 1 and .999999 are equal.

Then again, I'm the kid who won grade school arguments against kids who said "well you're stupid times infinity!" by saying "well you're stupid times infinity plus one!"
Posts: 4313 | Registered: Sep 2004
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posted
I can see how it would be frustrating, but I would try to pick your battles more selectively.

this is a fairly arcane case of mathematics which conflicts with general gut logic/understanding of numbers and really doesn't affect anything unless you're really getting into some complex math (as far as I can tell). As PM said, if a board of pretty intelligent people can argue back and forth for a few pages at least on the matter I'd argue that it's not completely clear cut (since both sides have reasonably concrete logical arguments for/against the proof)

Now if they're taking this kind of tone arguing against logic without any of their own on another more clear-cut and important issue, I'd try to keep explaining it to them.
Posts: 1038 | Registered: Feb 2006
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posted
.9999999999 might be equivalent to 1, but it's not equal to 1. Neither is 1/3 equal to .333333. That whole 1/3 vs .333333 thing always bugs me as well.

The fact that they are not equal is why fractions are still used. You just can't express all numbers in decimals.
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posted
Arguably, .333..... is the decimal representation of the fraction, 1/3. Although it is less elegant, the decimal version, are they not identical representations of the same number?
Posts: 3060 | Registered: Nov 2003
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posted
I think DDD's issue isn't that he didn't win, but that his father said bad things about his character.

(Or did he say things that you can *interpret* as bad about your character, DDD?)

Either way, although it's sad that family disrespect us sometimes, it *does* happen. What can you do? Respect yourself, I think. If your self-respect is weak, you can distance yourself from them or at least from such conversations. If it's stronger, you can think of yourself as teaching by example: I will show you respect, I will show you how I think people ought to be treated, *without any hint of demand that you agree*. If that's possible.

The way I dealt with parental disrespect (after I lived on my own) is that I stopped tolerating it. (By "not tolerating it" I mean it's clear that I won't participate in such a conversation.) As a result, my mother doesn't even try any more, and my father only tries if he's trying to manipulate me into something. Even that doesn't work, so he doesn't do it much.

I'm not sure this is the best model. I had a really awful home situation; there wasn't much to leave behind that I wanted to keep. I don't think refusing to be disrespected hurt anything, though.
Posts: 1877 | Registered: Apr 2005
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1-(3*0.33...) == ? It's that "..." that gets us where we're going. If there's no upper bound on the number of 3's in that repeating decimal, there's no upper bound in the number of 0's right of the . in the result. We have a simpler way of writing 0.000..., which is simply 0.

Put more formally, with calculus, lim as n->infinity (sum, i ranging 1 to n, of 9 x 10^(-i)) = 1.

[ February 05, 2007, 09:10 PM: Message edited by: Will B ]
Posts: 1877 | Registered: Apr 2005
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quote:I cannot for the life of me imagine how someone could convince me 1 and .999999 are equal.

quote:.9999999999 might be equivalent to 1, but it's not equal to 1. Neither is 1/3 equal to .333333. That whole 1/3 vs .333333 thing always bugs me as well.

.99999 is NOT equal to 1.

Neither is .99999999999999999999999999999999999999999.

Nor .999999999999999999999999999999999999999999999999999999999999999999999

Please note that no one on this thread has asserted that any of these three is equal to 1.

From the OP:

quote:.99999999999........ is equal to 1

Now, she included too many periods in the "..."

But that ... is very important because it represents this:

posted
They are not equal. But for all intents and purposes, they are close enough. Anyone who seriously thinks a person who says "1 and .99999... are not the same" is stupid for thinking that is just pathetic. Not the same. You can clearly see that they are different. But they are close enough. There really isn't ever going to be a situation where the difference between them is going to matter. So you are all right. YAY!
Posts: 1287 | Registered: Apr 2006
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posted
I might note that there are two extremely easy to understand proofs on Wikipedia. here

If you think this stuff hurts your brain, wait till you encounter NP=P or some of the weird non-deterministic Turing machine proofs.
Posts: 7593 | Registered: Sep 2006
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And yes, family tends to argue about stupid things. That is because it's easier than arguing about the real issues.

Anyway, we had this discussion (the one about the equivalence of .9999... and 1) a couple months ago, neh?
Posts: 32919 | Registered: Mar 2003
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posted
We had a much longer one once, rivka -- before your time, I believe. Much longer, and with more expletives and hurt feelings.

Well, that might be a slight exaggeration -- I can't remember for sure. But it was definitely longer.
Posts: 6213 | Registered: May 2001
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posted
I know. Someone mentioned it (in chat?) at the time of the one in October.

I was trying to short-circuit THIS argument, not claim that was the thread you meant. Sorry if I was unclear.
Posts: 32919 | Registered: Mar 2003
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posted
They're equal. Just because you can't understand that they're equal, and aren't willing to look at, think about, and understand the proofs, doesn't make them not equal.

This is why I don't discuss any sort of my "learned" knowledge with any of my immediate family. I'll be entering graduate school in economics next fall, but I don't dare talk about the economy or things of that nature with my parents, since they will simply discount that knowledge. *shrug* Sometimes there's not a lot you can do to get your parents to accept you growing up or having knowledge that they can't understand.
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posted
Saying they are equal is imprecise. The numbers (conceptual) they represent are the same, and (thus) the expressions are equivalent, but they aren't technically equal. In real life, however, the difference between "equal" and "equivalent" is often negligible.
Posts: 6213 | Registered: May 2001
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posted
I had a very similar argument with my family once. I posted about it here . Hatrack is a very good place to post rants because the people here are so understanding. Anyways, so yeah I understand why it frustrated you so much. Just try to see it from their point of view. The first time you heard that .99999 is equal to 1, did you immediately believe them? Didn't you think that's crazy at first until you had time to really think about it? Even if you didn't, I'm sure you can see why other people would doubt it upon first hearing of it. It just seems to go against logic.

What I'm trying to say is try to understand why they disagreed with you and think about how you can improve your argument next time rather than just placing the blame on them and thinking the whole argument was their fault.
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quote:Saying they are equal is imprecise. The numbers (conceptual) they represent are the same, and (thus) the expressions are equivalent, but they aren't technically equal.

Would you explain the difference between mathematical equality and equivalence?
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posted
Under that usage, yes. But that's imprecise usage of the term "equal." It doesn't really matter in most math you deal with (anything algebraic, pretty much), but when dealing with set theory and some other abstract concepts it can make a difference.
Posts: 6213 | Registered: May 2001
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posted
Well, to give a really bad analogy, think of them as different paths up a mountain. In algebra, the only thing that matters is the destination. In analysis and topology, the path itself matters. So .999... could be one path up the hill, and 1.0 could be another path, and they're equal in that they arrive at the same point, but the paths aren't identical.

As I said, a really bad analogy.
Posts: 6213 | Registered: May 2001
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.999... = 1.0 for the simple reason that there exists no smallest fraction.

I just called my parents. They both agree that they are always and absolutely equal, regardless of set theory and other abstractions (assuming, my mother points out, that everything stays in base 10 ).
Posts: 32919 | Registered: Mar 2003
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posted
My second grade teacher insisted that there was no such things as negative numbers, even when I showed her how they worked and everything. Now I wonder if she was just trying not to confuse the other second graders or if she really thought that.

She also insisted that I print big with a pencil on primary paper. I must have been such a trial to her...
Posts: 1379 | Registered: Feb 2002
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posted
Thanks for the explanations, everyone--especially that Wikipedia link. I now at least understand, technically, why the two are equal, even if I'll never really accept .99999... as equivilant to 1.
Posts: 4313 | Registered: Sep 2004
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Ask her to calculate the area under the graph where y=1 for the interval (0,1) and y=0 everywhere else. Then calculate the area under the graph where y=1 on the interval [0,1] and y=0 everywhere else. Then calculate the area under the graph where y=1 for all irrational x in the interval (0,1) and y=0 everywhere else. Are the areas under those graphs equal, equivalent, both, or neither?
Posts: 6213 | Registered: May 2001
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quote:Originally posted by Papa Moose: I love you and your mom.

Ask her to calculate the area under the graph where y=1 for the interval (0,1) and y=0 everywhere else. Then calculate the area under the graph where y=1 on the interval [0,1] and y=0 everywhere else. Then calculate the area under the graph where y=1 for all irrational x in the interval (0,1) and y=0 everywhere else. Are the areas under those graphs equal, equivalent, both, or neither?

Ok. After 15 minutes on the phone with both parents (simultaneously) and another 15 consulting Wikipedia, I can tell you that

a) My dad refuses to believe that there is any difference between equal and equivalent, and my mother says it is a question of definitions (or was it the other way around?); b) The first two sets are equal, because the difference (boundary lines) are infinitely thin (as all lines are, since they only have two dimensions), and therefore make no difference to the area, which is 1 in both cases; c) The third one, it depends what method you use to sum. If you use the more typical Riemann integral (which is apparently also called finding the Jordan content), the area is not defined. However, using the Lebesgue integral, the area is equal to 1, just like the first two; d) My head hurts, and I haven't even finished trying to understand the Banach-Tarski paradox, which my dad said I should as a follow-up to this discussion; e) My parents say none of this changes the fact that .999...=1, and that = can be read as equal or equivalent, whichever you like.

Oh, and last but not least, my father hopes this makes me contented (and he explained that).
Posts: 32919 | Registered: Mar 2003
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posted
Well, I'll continue to disagree with your father, and agree with your mother that it's a question of definitions. I believe the definitions I use to be more correct both mathematically and semantically, but I'm almost certain to be in the minority there. I can deal with that.

posted
My head just exploded. Fortunately, the pieces are immeasurable, and we will be able to construct two perfectly good heads from them. Once we scrape them off the computer screen . . .
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quote:I might note that there are two extremely easy to understand proofs on Wikipedia.

Says you.

Egad, I hate math. Nothing makes me feel sad quite like trying to read about mathematical concepts. I went to a special elementary school for advanced children! I've been reading since age two! So why does math always make me feel so stupid?
Posts: 1814 | Registered: Jul 2004
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posted
<Sends rivka a Menger sponge (zero volume and infinite surface area) to clean up her computer screen and heads.>
Posts: 6213 | Registered: May 2001
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0.99999... is, to all intents and purposes EQUIVALENT to one.

I can live with that.

By the way. I am an engineer and use maths a lot, but my head pops like microwave popcorn when folks start doing wierd things with exotic theories. After reading the exchange between rivka and Papa Moose, I need a lie-down
Posts: 892 | Registered: Oct 2006
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posted
AHHHHH! So much misinformation! (You knew I'd have to step into this discussion at some point)

Let me clarify by saying everything Rivka('s parents) have said so far is 100% correct.

It is NOT the case that ".99999..... and 1 are equivalent for all intents an purposes", it IS the case that .999999....=1, in EXACTLY the same way 2+2 = 4, 3*5 = 15. It's the same meaning of "equals" in all 3 cases. It doesn't matter what field of mathematics (set theory, analysis, measure theory, etc) you're working in, .9999...=1 period.

In fact, all of you are used to using multiple representations for the same number. For instance, no one would argue that 1/2 and .5 are different numbers, yet 1/2 = .5 in exactly the same was as 1 = .999999999....

This is a general flaw of representing numbers as decimals (or in any integer base). In fact, ANY time you have repeating 9's, you can remove them and bump the preceding digit up 1.

For instance, 3238.3248329999999999.... = 3238.324833,

.599999... = .6 an so on.

And Papa Moose, you may disagree with the definitions of equality vs equivalence, and I make no argument about their semantic value, but mathematically, these notions coincide (at least in this case ;-) )

quote:Originally posted by Mathematician: Let me clarify by saying everything Rivka('s parents) have said so far is 100% correct.

Good to know their years at Princeton (and elsewhere) were not wasted.

quote:Originally posted by Mathematician: It is NOT the case that ".99999..... and 1 are equivalent for all intents an purposes", it IS the case that .999999....=1, in EXACTLY the same way 2+2 = 4, 3*5 = 15. It's the same meaning of "equals" in all 3 cases. It doesn't matter what field of mathematics (set theory, analysis, measure theory, etc) you're working in, .9999...=1 period.

In fact, all of you are used to using multiple representations for the same number. For instance, no one would argue that 1/2 and .5 are different numbers, yet 1/2 = .5 in exactly the same was as 1 = .999999999....

*blink*

Daddy?

Seriously, I think my father used awfully similar examples and phrasing last night.
Posts: 32919 | Registered: Mar 2003
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posted
I think the hard part here is that you were stating something that was correct, and your father chose to make the other members of the family feel better rather than 1) aknowledge you were right, or 2) worry about your feelings.

My parents would do the same thing. I HATED it - I hate being disrespected, and I still don't think it is okay. This probably won't help, but I wonder...I wonder if your dad saw the other members of the family as the underdogs. It's clear that you know what you're talking about, so, in a sense, you "won" the argument. Your dad tried to spin the end by making it so your other family members didn't "lose". Maybe, in his way, he was thinking that everyone gets a little bit of something positive. You get to be right, and they get to feel better about not knowing it.

Of course, that means everyone gets something negative, too - you get disrespected, and they get to be bested by you. I'm sorry.

My advice (not that you asked): Let the proof go. You were right, but it's a Phyrric victory. Prove yourself academically in other settings, because in isn't working to do it in your family. I assure you that your family knows you are smart. I think you'll be happier if you prove your brains at school and here and let your family be the arena of support and love - from you to them. That's hard, but I think you'll be happier.
Posts: 26077 | Registered: Mar 2000
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posted
Thing is, a wise father doesn't make part of the family feel better by picking someone to be the loser. It would be possible to say, "Regardless of who's right about this little math problem, you're my family and you're all worthy of respect."
Posts: 1877 | Registered: Apr 2005
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Unfortunately, we can't change how other people handle things - we can only change how we handle things. While it would be a blast to discuss mathematical proofs at the table, it looks like that isn't possible. I have some of the same thing in my family - I don't discuss religion with one brother, and I don't discuss school or money with another. They don't discuss their disapproval of my life with me. On the one hand, this is not a perfect solution - I wish there weren't things that we don't talk about. On the other, I'd rather have my brother than another venue to discuss religion, so there it is. Maybe when life changes a little we can revisit the topic.
Posts: 26077 | Registered: Mar 2000
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