which is the smallest even number...

[xilman]

Originally Posted by Dr Sardonicus

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.

[a1call]

Originally Posted by enzocreti

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.

[Dr Sardonicus]

Originally Posted by xilman

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.

[Uncwilly]

2₁₀ = the following in binary

[retina]

Originally Posted by Dr Sardonicus

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.

[a1call]

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).

[axn]

Originally Posted by retina

... <repeat ∞ times> ...

...is not an algorithm

[retina]

Originally Posted by a1call

1.11...₂ is not a defined mathematical quantity so it can not be equal to 2.

Are you sure about that?

[retina]

Oh, WP has an article about it.

https://en.wikipedia.org/wiki/0.999...

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.

[Uncwilly]

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:

Originally Posted by rudy235

How many months of the year have 28 days? Answer 12 . (all months have 28 days) Same thing here.

[kriesel]

Originally Posted by xilman

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.