11831709440945300741959396391796334941618289112058276067446850710278050710335270686353123138597455936105137955330600414620589343375360000000000000000000000

]]>Relentless is right. You have a 25% chance of getting the right answer, due to the fact that you have four choices.

]]>10X=0.9999×10 ⇒ 10X=9.999

⇒ 10X-X=9.999-0.9999

⇒ 9X=8.9991

⇒ X=8.9991÷ 9

⇒ X=0.9999 (which is the original value)

Therefore when we work with any finite recurring of 0.9 the result will be less then one. The problem only occurs when we let the recurring tend to infinity, but if we trace the behavour of the result as the recurring tend to ∞ we find:

X=0.99999 ⇒ X=8.99991÷9=0.99999

X=0.999999 ⇒ X=8.999991÷9=0.999999

... ... ...

∴ X=0.999... ⇒ X=8.999..÷9=0.999... (which is the correct answer).

What does equal "1" is not the number 0.9̅ instead it is the sum of the following infinite geometric series:

∑0.9/10ᵐ as m → ∞ which is:

S= 0.9+0.09+0.009...= 0.9/(1-1/10)=0.9/(9/10)=10/10=1

One fallacious argument which could lead to a similar paradox may goes as follow:

1=9/9=9×1/9=9×0.1̅=0.9̅ ⇒ 1=0.9̅ (these results follow since the set of real numbers is a field). However, the wrong hypothesis in this argument is "1/9=0.1̅" which should be "1/9≈0.1̅" since the number "0.1̅" gets closer and closer to the number "1/9" but never reaches it (this is because we can not reach infinity). So the correct angument is:

1=9/9=9×1/9≈9×0.1̅=0.9̅ ⇒ 1≈0.9̅

Lets take "1/64" as an example to illustrate this problem. We know that 1/64=0.015625 in this case we can not say 1/64=0.015 or 1/64=0.0156 or 1/64=0.1562 the correct propositions are 1/64≈0.015, 1/64≈0.0156 and 1/64≈0.01562 respectively. On the other hand the proposition "1/64=0.015625" is definitely true. But as we have mentioned it is impossible to reach an end to the number "0.1̅" therefore it allways approximates but never equals the number "1/9". The value which does in fact equal that number is the sum of the infinite series ∑0.1/10ᵐ as m → ∞. So the corrcet argument should be:

1=9/9=9×1/9=9×(∑0.1/10ᵐ as m → ∞)=9×(0.1/(1-1/10))=9×(0.1/(9/10))=9×10/10×1/9=1 ⇒ 1=1

Which is the expected and correct value.

Note: when we do math we should not apply the rules without reasoning, we should follow the logic since all math is logic.

Q.E.D

]]>Only if the threatener has information and motives such that he would probably benefit from carrying the threat out.

]]>3! + 0 + 0 + 4 + 3 = 13

]]>Lol, interesting how you specifically wanted to avoid what Relentless and I found.

I found the point that you and Relentless brought up intriguing and I have engaged with it at considerable length - see my entry at 2016-01-08 17:51:00. But there are two paradoxes here, both of them interesting and each highlighting different aspects of infinite series. If you leave the diary paradox in its original form, the statement that the gods required Achilles to 'eventually take all the pages' allows Achilles to postpone taking non-birthdays indefinitely. This is an interesting observation, but it obscures my original point that the subset of birthday pages is equinumerous with its complement (i.e. countably infinite).

I've been toying with the idea of starting a separate thread to highlight the 'permanent postponement paradox'...what do you think?

]]>The series continues: ...24, 19, 875, 5471, 20424...

]]>Basically, find what A and B are equal to, or in other words...

A =

B =

*edit* I fixed the mistake.

]]>Or any positive amount of anything for that matter. In principle, if there are infinitely many people with any amount of something and the ability to transfer that something in a chain (it need only be one-way), then they can all acquire a limitless amount of it simply by redistribution (bounded only by time, perhaps).]]>