International High IQ Society

I'd like to think 149.9999999... is more exact though, hehe.Also, in the question it says "...If there are an infinite amount of flaps, what would be the area when they are ALL unfolded" (emphasis added)I think it could be worded differently--

Actually what "I am" doing in the picture is repremanding my recently graduated helper monkey for being a silly amature.

Well, it would be, if you could write it out. But since you'll never be able to write anything but an arbitrarily close approximation in the way you're doing it now, why don't you write the exact number: 150? Right. All the flaps. In other words, "given the sequence 100 + 100(1/4 + 1/8 + 1/16...)" what is the limit of this sequence?" 150.

Your proof is good (and correct as I'm sure you know) but I tried to use that one once and someone accused me of using a "mathematical trick". They didn't believe it. A shorter proof is that between any two real numbers there is a rational number, what number is between 149.999999... and 150?

bob_gray98 responds to me:

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A shorter proof is that between any two real numbers there is a rational number, what number is between 149.999999... and 150?

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Well, no, that isn't proof. That's punting. That puts the onus on the person you're arguing with to prove the opposite rather than you stepping up to do your side of the deal.Now, if you were to show that there is no number between 149.999... and 150 (no need to restrict it to rational numbers...between any two non-idential numbers, there are an infinite number of rational and irrational numbers), then you would have proven it.There is a reason to point out that these two are the same thing, however. It is used in the common proof that the size of the reals is larger than the size of the rationals:From previous work, we know that the size of the rationals is the same as the size of the integers. Thus, if we can come up with a 1-1 correspondance of a set with the integers, the two are the same size.Suppose we have a list of all the numbers between 0 and 1 (and thus, a 1-1 correspondance with the set of integers). Let's write them out in their infinite decimal expansion. Thus, we have:a = 0.a₁a₂a₃a₄...
b = 0.b₁b₂b₃b₄...
c = 0.c₁c₂c₃c₄...
.
.
.Then let's create a new number, p where:p = 0.a₁b₂c₃d₄...But, let's alter it so that if p_i = 2, then p_i = 3 and if p₁ <> 2,then p_i = 2.Whew!Now, what we end up with is a number that is different from every single number in the list...specifically at the ith position.

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Rrhain
WWJD? JWRTFM!

Is that Cantor's diagonal proof?I always remember it as an example of a proof so elegant and simple that it blows my mind. The problem is, I have such a bad head for math I can never remember how it goes, or what it proves.

crashfrog responds to me:

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Is that Cantor's diagonal proof?

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Yep.

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Rrhain
WWJD? JWRTFM!

Rrhain responds to me:

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Well, no, that isn't proof. That's punting. That puts the onus on the person you're arguing with to prove the opposite rather than you stepping up to do your side of the deal.

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I shouldn't have used the word proof, it is not appropriate in this situation. However, the situation I was in when I came up with this explanation was Thanksgiving with the family and, as I said, they weren't having any of my "mathematical tricks". I think that the explanation that there needs to be a rational number between these two, while not a formal proof, is probably sufficient to convince anyone of the truth of 149.99999... = 150. Anyway, I wouldn't consider it a punt but it certainly isn't a proof.Also, when I was writing this short explanation I couldn’t recall a theorem that said that there was necessarily an irrational number between any two reals, only a rational so I decided to err on the side of caution. Of course my memory is notoriously bad.

bob_gray98 writes:

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Also, when I was writing this short explanation I couldn’t recall a theorem that said that there was necessarily an irrational number between any two reals, only a rational so I decided to err on the side of caution. Of course my memory is notoriously bad.

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'Sno biggie. I was merely being formal.One of the weird aspects of the numbers is that despite the fact that there are more irrational numbers than rationals, the rationals are "dense" in the reals. That is, between any two real numbers, even two irrational numbers, you will find a rational number.And, as you suspected, the irrationals are dense in the reals, too. Between any two reals, even two rationals, there is an irrational.

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Rrhain
WWJD? JWRTFM!