Re: Newbye quetion: Why a double can store more number than a long ?

From: jose luis fernandez diaz (jose_luis_fdez_diaz_news_at_yahoo.es)
Date: 05/04/04 

Date: 4 May 2004 10:27:01 -0700

Ok. I understand.

Regards,
Jose Luis.

Karl Heinz Buchegger <kbuchegg@gascad.at> wrote in message news:<409779C7.65D57E09@gascad.at>...
> jose luis fernandez diaz wrote:
> >
> > Hi,
> >
> > I have made the program below to try to understand how a floating
> > point number is made by my computer (HP-UX cronos B.11.11 U 9000/800):
> >
>
> Your program won't help you in understanding what's the deal with
> floating point numbers and why they have a greater range then eg. long.
>
> Let us simplify the whole thing in a more familiar environment.
> Assume you are my 'computer'. But I will allow you to calulate
> with 5 digits only (and no sign for simplicity)
>
> If you use those 5 digits for 'long' only , you can use those 5
> digits to count from 00000 to 99999, so your range is 100000 numbers.
> But you can do different also. You can divide the available 5 digits
> into a mantissa and an exponent. Say you use 2 digits for the exponent
> and the remaining 3 digits for the mantissa (you are familiar with
> sientific notation, are you?). So eg. a number 100 can be represented
> in various ways:
>
> number scientific not. our notation
> **************************************
> 100 100 * 10^0 100 00
> 100 10 * 10^1 010 01
> 100 1 * 10^2 001 02
>
> Note: in the column 'our notation', no representation uses more then
> 5 digits which fits perfectly to what I allowed to you :-)
>
> Now lets see some other numbers.
> Say you need to represent 1000 in 'out notation'. How can you do that?
> Well you can eg. do 001 03. That would be 1 * 10^3. Or you could do
> 010 02 (10 * 10^2), or 100 01 (100 * 10^1). Fine. But now try to represent
> 1001 in 'out notation'. You can't. That's because the only way to get there
> would be to increment the mantissa. Let's see whats happening then:
>
> (1000) 001 03 -> (~1001) 002 03 -> 2 * 10^3 -> 2000
> (1000) 010 02 -> (~1001) 011 02 -> 11 * 10^2 -> 1100
> (1000) 100 01 -> (~1001) 101 01 -> 101 * 10^1 -> 1010
>
> You see, using only 5 digits, it is impossible to get at the number 1001. You can
> get 1000 and you can get 1010, but no numbers inbetween. It follows from the fact
> that you reserve some of the available 5 digits to represent an exponent, which
> drops the 'range' of numbers representable with the remaining 3 digits. But you
> get the freedome to move the comma around by fiddeling with the exponent. Using
> those 5 digit 'scientific notation' it is easy to represent
>
> 1000 -> 100 01
> 10000 -> 100 02
> 100000 -> 100 03
> 1000000 -> 100 04
> 10000000 -> 100 05
>
> Right now we are way 'over' the upper limit for 5-digit-long, which was
> 99999. But we bought this by reducing the granularity. We cannot represent
> all numbers in 10 millions, just some of them.
>
> 100 05 -> 10000000
> 101 05 -> 10100000
>
> All numbers between 10000000 and 10100000 are not representable by our 5 digit
> scheme but need to be approximated by either 10000000 or 10100000.
>
> Back to your computer. It doesn't use only 5 digits, but much more. Also the
> exponent is not taken to base 10, but to base 2. This reduces the effect you
> just saw. But the principle is still the same. You buy the greater range
> by reducing the granularity.

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