Precision Tail-off?

[Peter J. Holzer]

On 2023-02-17 10:27:08 +0000, Stephen Tucker wrote:
> This is a hugely controversial claim, I know, but I would consider this
> behaviour to be a serious deficiency in the IEEE standard.
> 
> Consider an integer N consisting of a finitely-long string of digits in
> base 10.
> 
> Consider the infinitely-precise cube root of N (yes I know that it could
> never be computed

However, computers exist to compute. Something which can never be
computed is outside of the realm of computing.

> unless N is the cube of an integer, but this is a mathematical
> argument, not a computational one), also in base 10. Let's call it
> RootN.
> 
> Now consider appending three zeroes to the right-hand end of N (let's call
> it NZZZ) and NZZZ's infinitely-precise cube root (RootNZZZ).
> 
> The *only *difference between RootN and RootNZZZ is that the decimal point
> in RootNZZZ is one place further to the right than the decimal point in
> RootN.

No. In mathematics there is no such thing as a decimal point. The only
difference is that RootNZZZ is RootN*10. But there is nothing special
about 10. You could multiply your original number by 512 and then the
new cube root would differ by a factor of 8 (which would show up as
shifted "binary point"[1] in binary but completely different digits in
decimal) or you could multiply by 1728 and then you would need base 12
to get the same digits with a shifted "duodecimal point".

        hp

[1] It's really unfortunate that the point which separates the integer
    and the fractional part of a number is called a "decimal point" in
    English. Makes it hard to talk about non-integer numbers in other
    bases.

-- 
   _  | Peter J. Holzer    | Story must make more sense than reality.
|_|_) |                    |
| |   | hjp at hjp.at         |    -- Charles Stross, "Creative writing
__/   | http://www.hjp.at/ |       challenge!"
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