Does 1 = 0.999... ?

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Does 1 = 0.999... ?

Postby indigoboy83 » Sat Aug 22, 2015 5:43 pm

Does 1 = 0.999... ?

Most if not all mathematicians say that 1 = 0.999...

They are an infinitesimal off.

I demonstrate this below, without the literal introduction of "an infinitesimal".

So, the following statements are correct:

Statement A: x < 1
Statement B: x <= 0.999...

Statement A = Statement B

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Another way of putting this is to create an algorithm to determine whether one number is less than another number. This is done on my website.
(https://sites.google.com/site/indigoboy83/home)
indigoboy83
 
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Re: Does 1 = 0.999... ?

Postby Mikko » Sun Aug 23, 2015 4:29 am

indigoboy83 wrote:Does 1 = 0.999... ?

Most if not all mathematicians say that 1 = 0.999...

All of them. Those who say otherwise do not understand enough math to be called mathematicians. In particular, not what the string "0.999..." means.
They are an infinitesimal off.

No, they are not. In the standard space of real numbers there are no infinitesimals. Even in non-standard spaces that do contain infintesimals these two expressions are exacly equal.
I demonstrate this below, without the literal introduction of "an infinitesimal".

So, the following statements are correct:

Statement A: x < 1
Statement B: x <= 0.999...

Statement A = Statement B

That does not demonstrate anything.
Another way of putting this is to create an algorithm to determine whether one number is less than another number. This is done on my website.
(https://sites.google.com/site/indigoboy83/home)

It is perfectly possible (and trivially simle) to create an algorithm to determine whatever one wants as long as the determinations may be incorrect. Besides, you cannot put an infinite string to an algorithm, as the definition of an algorithm only allows a finite amount of processing.

Note that this forum is for discussion of those parts and aspects and uses of calculus that are relevant to physics. This question is not. A physical quantity can be compared to another only approximately. Often only two significant digits are known or important, and always only a finite number of digits. If two quantities are closer than whatever tolerance is considered sufficient, they are approximately equal and it is either impossible or unimportant to determine which is bigger.
Mikko
 
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