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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.

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I cannot for the life of me imagine how someone could convince me 1 and .999999 are equal.

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.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.

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.99999999999........ is equal to 1

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code:

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 9    9      9
-- + --- + ---- + ... 
10   100   1000

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code:

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  9      9      9  
---- + ---- + ---- + ...
10^1   10^2   10^3  

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They are not equal.

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Can you demonstrate their inequality, please?

They are equal. Things that look different can be equal.

2 + 2 = 4. I can clearly see that they are different.

But they're still equal.
 

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Posted by Mucus (Member # 9735) on :
 
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.
 

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Posted by rivka (Member # 4859) on :
 
DDDaysh is a she.

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?
 

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Posted by Papa Moose (Member # 1992) on :
 
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.
 

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Posted by rivka (Member # 4859) on :
 
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.
 

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Posted by Jhai (Member # 5633) on :
 
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 by Papa Moose (Member # 1992) on :
 
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.
 

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Posted by GaalDornick (Member # 8880) on :
 
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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Posted by Dagonee (Member # 5818) on :
 

quote:

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The numbers (conceptual) they represent are the same, and (thus) the expressions are equivalent, but they aren't technically equal.

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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.

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Would you explain the difference between mathematical equality and equivalence?
 

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Posted by Papa Moose (Member # 1992) on :
 
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.
 

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Posted by Papa Moose (Member # 1992) on :
 
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.
 

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Posted by rivka (Member # 4859) on :
 
It's like a state -- path is irrelevant.

.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 ).
 

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Posted by mr_porteiro_head (Member # 4644) on :
 
Wow. Your mom's a big nerd.

:wub:
 

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Posted by rivka (Member # 4859) on :
 
My mom has a Ph.D. in mathematics. She was in fact one of the first women to attend Princeton's grad school in the math department.

Of COURSE she's a big nerd!
 

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Posted by Dead_Horse (Member # 3027) on :
 
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...
 

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Posted by erosomniac (Member # 6834) on :
 
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.
 

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Posted by Papa Moose (Member # 1992) on :
 
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?
 

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Posted by mr_porteiro_head (Member # 4644) on :
 
I really like the algabraic proof from the wikipedia link.
 

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Posted by rivka (Member # 4859) on :
 

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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?

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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).
 

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Posted by Papa Moose (Member # 1992) on :
 
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.

--Pop
 

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Posted by rivka (Member # 4859) on :
 
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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Posted by Verily the Younger (Member # 6705) on :
 

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I might note that there are two extremely easy to understand proofs on Wikipedia.

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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?
 

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Posted by Papa Moose (Member # 1992) on :
 
<Sends rivka a Menger sponge (zero volume and infinite surface area) to clean up her computer screen and heads.>
 

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Posted by rivka (Member # 4859) on :
 
And this is supposed to help my poor head?
 

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Posted by anti_maven (Member # 9789) on :
 
OK, so if I may summarise:

0.9999999 is NOT EQUAL to 1

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
 

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Posted by Katarain (Member # 6659) on :
 
Can .888... be equal to .9, then?

How about .777... = .8?

and .666... = .7?

And so on?
 

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Posted by Xavier (Member # 405) on :
 

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Can .888... be equal to .9, then?

How about .777... = .8?

and .666... = .7?

And so on?

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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.

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

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

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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....

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*blink*

Daddy?



Seriously, I think my father used awfully similar examples and phrasing last night.
 

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Posted by katharina (Member # 827) on :
 
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.
 

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Posted by Will B (Member # 7931) on :
 
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."
 

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Posted by katharina (Member # 827) on :
 
Oh, I agree. I think he handled it badly.

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.
 

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Posted by Mucus (Member # 9735) on :
 
anti_maven: While .999...=1 may be not generally well known or understood, perhaps even unintuitive, I find your assertion that it is an "exotic theory" troubling.
To borrow an example from physics, the rules of thermodynamics and their full consequences are not easily understood by most people. However, they are fundamental and knowledge of them is in so many areas of modern industrial technology, such that calling them "exotic" is a doing them a disservice.

Similarly, .999...=1 is not easily understood and like thermodynamics, we do not need to fully understand it in order to make use of theories and technology that make use of it. However, it is far from "exotic."
IIRC, understanding it is essential for the concept of limits, which is a pretty basic building block for many of the proofs in calculus. In a related field, statistics, this also leads to consequences such as that implied by Papa Moose's area question. Unintuitively, if you have a continuous probability function, the probability at a single point is actually zero. In plain English, your probability of dying at 75 is zero. Your probability at dying at 75.0000001 is also zero. But your probability of dying between 75 and 75.0000001 is non-zero.

Failure to grasp it can lead to weird consequences such as Zeno's paradox. If you truly believe that .999.. != 1 then consider this (from MathWorld):

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Imagine the great Greek hero Achilles starting a race with a turtle. Achilles is a fast runner, running 10 metres per second, while the turtle is slow and runs at one metre per second. Therefore Achilles agrees to give the turtle some advantage and the turtle starts 10 metres in front of Achilles. The ancient Greek philosopher Zeno found the following “paradox”.

If Achilles wants to get in front of the turtle he first has to run to where the turtle started. But in that time the turtle has bridged some distance, which Achilles now has to run in order to take up. But in this time again the turtle has gone for some distance and Achilles is still in behind of the turtle. This process continues forever and apparently Achilles cannot pass the turtle.

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Originally posted by DDDaysh:
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.

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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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Originally posted by mr_porteiro_head:
I really like the algabraic proof from the wikipedia link.

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Originally posted by Xavier:
However:

.9 = .8999...

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Xavier
 

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Posted by Verily the Younger (Member # 6705) on :
 
I don't understand why math people still love Zeno's paradox so much. Okay, so it's unintuitive. It's also wrong.

The only thing Zeno's paradox proves is that it's possible to demonstrate mathematically things that are completely bogus in the real universe. Every day of our lives we see examples of things catching up to other things in motion, despite the fact that Zeno "proved" this is impossible. Every time a car passes you on the road, they prove Zeno wrong.

If Zeno had been right, then no one running away from anyone or anything would ever get caught, unless they were stupid enough to stop moving. Races would never be run, because whoever first got the lead position would be guaranteed to keep it. If someone shot at you, all you'd have to do to survive is start walking away; it doesn't matter how slowly you move, because as long as you keep moving, the bullet can never reach you.

It would be perverse to deny a lifetime's worth of empirical evidence and say that all the things we see don't really happen because you can mathematically prove they're impossible. Where the theories fail to match up with reality, it is the theory, not reality, that is in error.

So, given that everyone is trying to prove that 0.999...=1, that they really are equal in reality as well as mathematical theory, then I must ask how Zeno's paradox even relates.
 

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Posted by fugu13 (Member # 2859) on :
 
Verily: you misunderstand. Zeno's paradox only arises due to a misunderstanding of mathematics. When mathematics is properly applied, the paradox is resolved.

Specifically, even if you can break a finite distance into infinitely many parts, the sum of the infinitely many pieces of time to travel those infinitely many parts can still be finite, meaning that people do catch up to other people and run into walls.

Zeno though that because there were an infinity of parts of the journey there would be an infinity of time required, which is not the case. Properly applying mathematics shows that there is no paradox, its only a misunderstanding of mathematics that causes confusion.

The reason Zeno's paradox is brought up is because the mathematical issue at its heart is a limit, and .999999 . . . is just another way of expressing a very similar limit (you could even make it the same limit by doing Zeno's paradox with 9/10 instead of 1/2).
 

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Posted by Mucus (Member # 9735) on :
 
fugu13: Bravo, I was going to type out the explanation, but you jumped in and posted a much simpler one.

Verily: To elaborate, the whole point is that Zeno's paradox is used to challenge your initial intuition. If you follow the initial intuition that .999... != 1 then there are weird consequences including the unintuitive Zeno's paradox.
Your own belief that Zeno's paradox is wrong should nag and poke at your belief that .999.. != 1 until you realise that your intuition in that case is wrong.

(I'm using "your" in the generic sense)
 

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Posted by kmbboots (Member # 8576) on :
 
It isn't "intuitive" because our intuition of infinity is insufficient.
 

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Posted by mr_porteiro_head (Member # 4644) on :
 

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Originally posted by Lisa:

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Originally posted by mr_porteiro_head:
I really like the algabraic proof from the wikipedia link.

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The fraction proof is the one I usually use. It's simple and straightforward. Kind of obvious, actually.

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Originally posted by Will B:
Either that last proof is wrong, or I missed something. How can you numerically substitute the x in 99x to get 99, without assuming x=1?

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Originally posted by fugu13:
m_p_h: which counterintuitive solution? There's a correct one and an incorrect one, and the incorrect one is significantly more popular.

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Originally posted by mr_porteiro_head:

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Originally posted by Lisa:

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Originally posted by mr_porteiro_head:
I really like the algabraic proof from the wikipedia link.

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The fraction proof is the one I usually use. It's simple and straightforward. Kind of obvious, actually.

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I agree, but that can also be a weakness for some people.

It's so simple and obvious that some people will say "There's got to be a trick. You're missing something".

I actually had a conversation the other day with someone about the Monty Hall problem and when I proved to them the counterintuitive solution, they said exactly that.

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A friend of mine in college once defined a natural-born mathematician and someone to whom e to the i(pi) is obviously -1.

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Originally posted by Mucus:
kmboots: Yes.

There's actually an interesting lecture by Richard Dawkins on this subject here. For those that are religious and worried about his reputation as an aggressive atheist, rest assured that this lecture contains no religious (or anti-religious) content.

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So for your logic to be correct, somehow which door you pick has to influence which door the prize was placed behind.

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Let me make it simpler. If you had picked the other (unopened) door first, you'd be arguing that you were more likely to win if you switched to your current door. Both arguments cannot be correct, since the event of you picking a door is independent of the location of the prize.

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