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#19
Aug1114, 12:46 PM

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Between any powers of 2, are there 2^52 representable numbers? I just guessed on that one. Is the number of representable numbers between 2^3 and 2^4 2^52 as well?? Or how do you determine that
So you just express x as a power of 2, if its a whole number, then you do 2^x52, and that is eps(x). If yoy have something that is not expressed as a power of 2, then you just go back to the lower number that can be, so eps(5) = eps(4) So eps(4), 4 is 2^2, so eps(4) is 2^(252) = 2^50. So now, I can say with some confidence I understand F.1, F.3, and F.4. Now where I am stuck is F.2 and F.5. Edit: I might be able to justify F.5 now, let's give this a whirl So I know eps(2^52) is 1, because 2^(5252) is 0, hence 2^0 = 1. So, everything in between there also has an eps(x) = 1. That means you can have ##2^{52}, 1+2^{52}, 2+2^{52},3+2^{52} ....2^{53}##. So there are 2^52 numbers in between 2^52 and 2^53. So I guess what you should do to find the number of representable numbers is subtract the highest number from the lowerst number in the interval, then divide by eps(lower interval)?? ##2^{53}  2^{52} = 2^{52}/1 = 2^{52}## representable numbers So to create another example, the number of representable numbers between [4 8). eps(4) = 2^(252) = 2^50 spacing between. ##84 = 4##, so ##4/2^{50} = 4*2^{50}## representable numbers between 4 and 8 


#20
Aug1214, 11:59 AM

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There are 2^{52} representable numbers in the interval [2^{n},2^{n+1}) if 2^{n} has a normalized representation. For example, there are none in the interval [2^{1024},2^{1025}) because 2^{1024} is out of range.
What about the denormalized numbers? There is only one representable number in the interval [2^{1074},2^{1073}), 2^{1074} itself. The next largest representable number after 2^{1074} is 2^{1073}. There are two representable numbers in the interval [2^{1073},2^{1072}), three in [2^{1072},2^{1071}), and so on, until you get to the interval [2^{1022},2^{1021}), which contains 2^{52} representable numbers. Every power of 2 interval from that one to [2^{1023},2^{1024}) does contain 2^{52} representable numbers. 


#21
Aug1214, 09:19 PM

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#22
Aug1214, 09:20 PM

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Of course not. It means the spacing is the same for all representable numbers between 2^{n} and 2^{n+1}.



#23
Aug1214, 10:30 PM

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EDIT: nevermind, I get it now.
I still don't understand the thing about the smallest representable number. Someone said it was 1+2^53, that is very big to be a smallest representable number, how do you do the analysis to determine that? 


#24
Aug1214, 11:37 PM

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Your question F.2 asked for the smallest nonrepresentable integer. That obviously needs to be a biggish number.



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