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# Maths illusions watch

1. (Original post by nasht)
Because the person whom says this doesn't know proper arithmetic:

5+5+7 = 17 ?
2. (Original post by nasht)
Perhaps you mean that 5:00 (AM) + 5:00 + 7:00 = 17:00 = 5:00 PM.
3. (Original post by dacb1984)
Perhaps you mean that 5:00 (AM) + 5:00 + 7:00 = 17:00 = 5:00 PM.
Aha tis makes sense
4. (Original post by A. Nonny Mouse)
Anyone know the one about infinity hotel?

Infinity hotel has a room for every natural number (1,2,3,...). Each room can accomodate one person. One day, every room is occupied (this is alright, because the population of the mathiverse is infinite).

Another person arrives at the hotel. How does the manager fit him in?

An infinite number of people (one for every natural number) arrive. How does the manager fit them in?

Happy puzzling!

Samuel Borin
Recreational Mathematician
I suppose he could just get everybody to move over one room, in which case Room #1 would be free.
5. (Original post by dacb1984)
Perhaps you mean that 5:00 (AM) + 5:00 + 7:00 = 17:00 = 5:00 PM.
correct!!! lots of smart people here
6. (Original post by Gaz031)
Sorry but there isn't a difference.
There are numerous proofs without error, surely they show it to be true?
If you choose then for all , where is some fixed arbitrarily large integer and is arbitrarily small and can be made smaller by increasing N, which is the very definition of tending to as tends to infinity.

If you start at 1, and try to move away from 1 and toward 0.99999..., how far do you have to go to get to 0.99999... ? Any step you try to take will be too far, so you can't really move at all - which means that to move from 1 to 0.99999..., you have to stay at 1.
At risk of further damage to my mathematical ego, your proof that they're equivalent is very convincing. But would it not be equally valid to say that as they're both real numbers they both have decimal expansions, and since the decimal expansions are not equal then the numbers are not equal. Cantor's diagonal argument basically said "there exists a real number with different decimal expansions to all of these therefore there exists another number in the real set," and this is considered valid.
7. Proof that 1 = 0.999...

Consider two real numbers, m and n. If m ≠ n, then there should be an infinite number of real numbers, x, that satisfy either m < x < n or n < x < m. On the other hand, if m = n, then there can be no real number, x, that satisfies either of those two inequalities.

Suppose m = 0.999... and n = 1. What values can x take? Think about this. There is no real number, x, that lies between m = 0.999... and n = 1. Therefore, they must be the same number.
8. (Original post by Zuber)
here's a thread where you can post the best maths illusions. Please feel free to post illusions.

Here's mine. tell me what you think of it

a = b
times both sides by b
ab = b2
minus a2
ab – a2 = b2 – a2
factorise
a(b-a) = (b+a)(b-a)
divide by (b-a)
a = b+a
replace b by a
a = a + a
a = 2a
divide by a
1 = 2

If 1 = 2, the world would end
Are you allowed to 'replace' b by a

Maths probably isn't my strong point
9. (Original post by homoterror)
At risk of further damage to my mathematical ego, your proof that they're equivalent is very convincing. But would it not be equally valid to say that as they're both real numbers they both have decimal expansions, and since the decimal expansions are not equal then the numbers are not equal. Cantor's diagonal argument basically said "there exists a real number with different decimal expansions to all of these therefore there exists another number in the real set," and this is considered valid.
If you're not joking, Lex, then yes they are definitely equal.

Cantor's argument shows that the real are uncountable. The only reason that care is needed is because there are numbers with two decimal expansions - so in the proof one needs to be sure that the new number you've created, with a different decimal expansion, isn't the same number as one already on your list, but just a different expansion of it.
10. (Original post by RichE)
If you're not joking, Lex, then yes they are definitely equal.

Cantor's argument shows that the real are uncountable. The only reason that care is needed is because there are numbers with two decimal expansions - so in the proof one needs to be sure that the new number you've created, with a different decimal expansion, isn't the same number as one already on your list, but just a different expansion of it.
Oh. Then I misunderstand his proof. I always had it go "the new number formed differs from the r'th number by the r'th decimal place and therefore must be a different number from any in the hypothetical list of elements of [0,1]." Is there some step where you show that the new number formed is none of the other numbers with a different expansion?
11. (Original post by homoterror)
Oh. Then I misunderstand his proof. I always had it go "the new number formed differs from the r'th number by the r'th decimal place and therefore must be a different number from any in the hypothetical list of elements of [0,1]." Is there some step where you show that the new number formed is none of the other numbers with a different expansion?
that's right except if you're not careful then you might end up with the new number being 1.0000000000... and 0.999999999999... being on your list.

How do you feel Cantor's proof though has anything to do with what is just summing a GP?

Are you happy that

0.3333333333... = 1/3

for example and to just times that by 3?
12. (Original post by RichE)
that's right except if you're not careful then you might end up with the new number being 1.0000000000... and 0.999999999999... being on your list.

How do you feel Cantor's proof though has anything to do with what is just summing a GP?

Are you happy that

0.3333333333... = 1/3

for example and to just times that by 3?
I do acknowledge the sum of the GP is entirely valid as a proof. I'm not arguing that the numbers aren't the same. I just want to bust the counterproof. If somebody says I've found a counterproof to fermat's last theorem you'd have to analyse it yeah? =P

So when you're going through Cantor's proof, how can you tell that you're not describing a number which is already on your list with a different decimal expansion?
13. Who dared revive this thread?
14. (Original post by homoterror)
I do acknowledge the sum of the GP is entirely valid as a proof. I'm not arguing that the numbers aren't the same. I just want to bust the counterproof. If somebody says I've found a counterproof to fermat's last theorem you'd have to analyse it yeah? =P

So when you're going through Cantor's proof, how can you tell that you're not describing a number which is already on your list with a different decimal expansion?
Because one way to create the new number is to have its nth digit be a 6 unless the nth number has nth digit 6 in which case change it to a 7.

That way a number having the same decimal expansion doesn't arise as it only occurs for numbers that end all 0 or 9 in their expansions.
15. (Original post by RichE)
Because one way to create the new number is to have its nth digit be a 6 unless the nth number has nth digit 6 in which case change it to a 7.
But this uses the logic "it's different in every decimal place than any other number." Something which can be said about 0.9999... and 1.0000...
16. (Original post by homoterror)
But this uses the logic "it's different in every decimal place than any other number." Something which can be said about 0.9999... and 1.0000...
No, it doesn't, not quite as simply as you've phrased it.

Almost all real numbers have one decimal expansion, some don't. If done properly you'll end up producing a number that, say, has only 6s and 7s in it. And that number has only one decimal expansion and we are sure that number isn't on the list.

But some care needs to be taken to make sure a different expansion of a number on the list isn't generated.
17. (Original post by dvs)
-1 = -1
1/-1 = -1/1
sqrt(1/-1) = sqrt(-1/1)
1/i = i/1
1 = i^2
1 = -1

Isn't it just wrong because of our assumption that line 4 is right (or follows from line 3), which indeed is false as 5 isn't true per defintion? Sort of going round and round though.
18. This is a nice quiz a friend told me:

Assume you have some sort of Chinese candle (those that smell nice). Now you have two of those candles and I would like you to measure 45 minutes with those candles. Both candles are straight and if you lit an end, they burn down in an hour. You *could* measure the length of the candles, however the density isn't uniform ... ie our candles sometimes decide to burn down slower and sometimes faster. The total time for one candle stays 1 hour though. So, how would you do it?
19. (Original post by xyro)
This is a nice quiz a friend told me:

Assume you have some sort of Chinese candle (those that smell nice). Now you have two of those candles and I would like you to measure 45 minutes with those candles. Both candles are straight and if you lit an end, they burn down in an hour. You *could* measure the length of the candles, however the density isn't uniform ... ie our candles sometimes decide to burn down slower and sometimes faster. The total time for one candle stays 1 hour though. So, how would you do it?
1.Light candle A at both ends.
2.Light candle B at one end.
3.When candle A has finished light the other end of candle B.
20. Yes, correct.

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