Re: Definition of real number ℝ --infinitesimal--

On 4/1/2024 3:59 PM, André G. Isaak wrote:

On 2024-04-01 14:30, olcott wrote:

On 4/1/2024 2:37 PM, Fred. Zwarts wrote:

Op 01.apr.2024 om 20:54 schreef olcott:

 

Since PI is represented by a single geometric point on the number line
then 0.999... would be correctly represented by the geometric point
immediately to the left of 1.0 on the number line or the RHS of this
interval [0,0, 1.0).

>
In the real number system it is incorrect to talk about a number immediately next to another number. So, this is not about real numbers.
>

>
PI is a real number.
If there is no real number that represents 0.999...
that does not provide a reason to say 0.999... = 1.0.

 I'm a bit unclear why you keep bringing pi into this. pi isn't a repeating decimal, unlike 0.999... which is.
 But if you want to talk about pi, that also can be construed as the limit of an infinite series:
 π/4 = 1 - 1/3 + 1/5 - 1/7 + 1/9 - 1/11...
 For any *finite* number of terms, the above series never quite reaches pi, but the LIMIT of this series is exactly equal to pi, not to some

To says that 0.999... = 1.0 means that after the never ending
sequence ends (a contradiction) then we reach exactly 1.0.

value one 'geometric point' (which has a length of exactly zero) away from that limit And for this series your peculiar notion that it is a geometric point away is particularly absurd since it isn't clear whether you'd want it to be one 'geometric point' greater or less than this limit since the series doesn't converge on its limit from a single direction.
 Similarly, the value of the series 9/10 + 99/100 + 999/1000... is exactly equal to the LIMIT of that series. That's what the notation 0.999... means, by definition.
 André
 

--
Copyright 2024 Olcott "Talent hits a target no one else can hit; Genius
hits a target no one else can see." Arthur Schopenhauer