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Old 2019-11-06, 22:16   #12
xilman
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Quote:
Originally Posted by Dr Sardonicus View Post
The question specified "converted in base 2."

Please explain the algorithm by which you convert 210 into base two, so as to get 1.111... rather than 10.
oh dear.

It's the same as 1.999999999... in decimal.
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Old 2019-11-07, 00:51   #13
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Quote:
Originally Posted by enzocreti View Post
which is the smallest even number N (in base 10) such that converted in base 2 contains five 1's?
Well, since the OP does not specify that "10" is a decimal notation it is quite obvious to me that 10 is equal to the number which is normally represented as 19 in decimal notation. So the correct answer is obviously 2613660 (when written in decimal notation.

https://www.wolframalpha.com/input/?..._19+to+decimal

ETA: Oops, got the 1's in the wrong base.
But at least I learned something: An odd number of 1's in an odd base will never evaluate to an even number, regardless of any 0's present anywhere.

Last fiddled with by a1call on 2019-11-07 at 01:18
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Old 2019-11-07, 01:37   #14
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Quote:
Originally Posted by xilman View Post
oh dear.

It's the same as 1.999999999... in decimal.
Maybe, but retina specified 210, not "1.999999999... in decimal."

You have to invoke limits to say they're "the same."

In any case, it doesn't answer my question -- what's the conversion algorithm?

It can reasonably be argued that using .999999999... in decimal, or .1111111... in binary to express an integer isn't algorithmic because the decimal or binary expansions do not terminate, so they can not express an integer exactly using a finite number of arithmetic operations with integers.
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Old 2019-11-07, 02:14   #15
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210 = the following in binary 10^{1^{1^{1^1}}}
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Old 2019-11-07, 03:42   #16
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Quote:
Originally Posted by Dr Sardonicus View Post
The question specified "converted in base 2."

Please explain the algorithm by which you convert 210 into base two, so as to get 1.111... rather than 10.
Okay, sure. Here is my method.

I start with 12 and convert to base 10 to get 110. Too small.
Then I try 1.12 and convert to base 10 to get 1.510. Too small.
Then I try 1.112 and convert to base 10 to get 1.7510. Too small.
Then I try 1.1112 and convert to base 10 to get 1.87510. Too small.
Then I try 1.11112 and convert to base 10 to get 1.937510. Too small.
... <repeat ∞ times> ...
Then I try 1.1111....2 and convert to base 10 to get 210. Yay. Found it.
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Old 2019-11-07, 04:06   #17
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Decimal 2 is a defined mathematical quantity.
1.11...2 is not a defined mathematical quantity so it can not be equal to 2.
This is despite the fact that the limit of the sum
1+1/2+1/4+... converges to 2 as the number of addends approaches Infinity (an undefined quantity).

Last fiddled with by a1call on 2019-11-07 at 04:07
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Old 2019-11-07, 04:21   #18
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Quote:
Originally Posted by retina View Post
... <repeat ∞ times> ...
...is not an algorithm
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Old 2019-11-07, 04:35   #19
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Quote:
Originally Posted by a1call View Post
1.11...2 is not a defined mathematical quantity so it can not be equal to 2.
Are you sure about that?
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Old 2019-11-07, 04:41   #20
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Oh, WP has an article about it.

https://en.wikipedia.org/wiki/0.999...
Quote:
This number is equal to 1. In other words, "0.999..." and "1" represent the same number. There are many ways of showing this equality, from intuitive arguments to mathematically rigorous proofs.
WP said it, so it must be true.
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Old 2019-11-07, 22:25   #21
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Mod note:
Timekeeping posts split off to this thread:
https://www.mersenneforum.org/showthread.php?t=24923

This the post that started it and refers to the prior posts in this thread:
Quote:
Originally Posted by rudy235 View Post
How many months of the year have 28 days? Answer 12 . (all months have 28 days) Same thing here.

Last fiddled with by Uncwilly on 2019-11-07 at 22:25
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Old 2020-07-05, 03:54   #22
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Quote:
Originally Posted by xilman View Post
Reminds me of the programmer who, before he went shopping, was told by his wife: "Buy a loaf of bread and, if they have any eggs, get a dozen." He came back with 12 loaves of bread and no eggs.
His programmer wife could have asked, why he came home one loaf short.
Buy a loaf (1), + if (they have any eggs) get 12 (loaves). 1+12=13.

Last fiddled with by kriesel on 2020-07-05 at 03:54
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