Theory: R is not a number line...

Thread Starter

Jennifer Solomon

Joined Mar 20, 2017
15
The real number line (set) ℝ is not a set, it is an infinite 2D plane, where the x axis is the whole natural integers, and the y axis is the fractions of those integers. The x and y here are not to be confused with an actual Cartesian plane, but analogous to it.

ℕ represents the counting numbers, and every other real number we call a number is really a fractionated concatenation involving concatenations of elements of ℕ. For example, in base 10, 4.25 is a numeric expression composed of two things: a whole number and an algorithmic fraction of 1. In this case, 4 plus 25/100’s as its mantissa. 4 is a fraction of infinity, and .25 is a fraction of the infinitesimal.

Seeing ℝ as a plane and not as a line solves the conceptual problem of infinite, non-terminating irrationals that are in a dynamic state of unresolved finitude co-existing in the same “space.”

It is not proper to see pi, for example, as a finite point on a number line. The 3 portion is on the x axis, and the mantissa portion on the y axis (.141592...) which may be truncated to “dial in“ a user-defined amount of exactitude depending on the application. NASA uses more digits to specify pi than someone working around their house. They are both pi.

Pi is composed of 3 and an infinitesimal additional quantity, an unresolved portion of infinity. 3 is a rational, or “knowable” portion of infinity, and .14592... is an irrational or “unknowable” portion of infinity. Both are part of the ℝ plane.

In order to give spatial dimension to functions, we graph them on a spatial plane. y=x shows us x infinitely varies over a real plane and determines the state of y, another ℝ plane. Both x and y could be considered “instantiations” of ℝ set themselves, which we call “variables”: as x infinitely varies over ℝ, it renders y as another incarnation of the continuum of the ℝ plane intersecting it. There’s no such thing as a true 2D object, and thus a Euclidean line is really a 3D object with infinitesimal, non-terminating length and width, since if a point has “no measurable dimension,” one must specify minimally length and width when speaking of its spatiality.
 
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ci139

Joined Jul 11, 2016
1,696
i think i know that the Zero is mostly NaN ... perhaps nonlinear within the \(\left\{{{\mathbb{R}}^-,{\mathbb{R}}^0,{\mathbb{R}}^+}\right\}\) ... and the "linearity" of \(\mathbb{Z}\) may be valid near Zero only ... (not explaining)
 

ci139

Joined Jul 11, 2016
1,696
(( actually i don't get your integer-?quanted? X-axes -- but everything is cool as long you get it yourself -- even if it "does not describe" too well ))
...
also does the y has only positive <1 "offsets"
or
the "whole" reals (with their integer part)
,
is X-axes defined only at integer points
or
the integer value is a "staircase" (along the X-axis)
 
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Thread Starter

Jennifer Solomon

Joined Mar 20, 2017
15
(( actually i don't get your integer-?quanted? X-axes -- but everything is cool as long you get it yourself -- even if it "does not describe" too well ))
...
also does the y has only positive <1 "offsets"
or
the "whole" reals (with their integer part)
,
is X-axes defined only at integer points
or
the integer value is a "staircase" (along the X-axis)
You’re correct... perhaps the description is not ideal quite yet.

I say x and y only for depicting 2 dimensions to the number—“horizontal vs. vertical.“ In practice, in the Cartesian plane, we see it as a line: 2.645 is somewhere on that line between 2 and 3, but conceptually the number has 2 separate informational dimensions concatenated together.

So in the case of pi on a Cartesian plane, you’d graph it on the x or y axis, but its true nature would be a kind of superpositional hidden axis for the non-terminating points of the mantissa. The axes on the Cartesian plane can be scaled to essentially infinite density.

I believe the whole numbers are the only “actual” numbers (as mathematician L. Kronecker said), and then everything else is a compounded arithmetic expression using fractions or decimal points to “append” an additional amount to the actual integer. The appendage is not a number but a fraction of 1, which in the irrationals is a non-terminating “pursuit“ of the infinitesimal.

What is pi? It’s 3 “plus” some undefined amount appended. 3 is the natural portion, and the mantissa is some dynamic numeric “flux” that is appended. We can truncate that appended “flux” for as much accuracy we need, like a “dimmer switch.“ This concept would be base agnostic. It makes no sense to see each possible incarnation of pi on that actual axis. So its ”dimmer switch” is a shadow “potentiometer“-esque axis since irrationals cannot be all crammed into the same line.
 
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ci139

Joined Jul 11, 2016
1,696
irrationals cannot be all crammed into the same line.
you talk like that or your teacher ... or is it trivial today ... i learned math decades ago in a different language (Estonian) ((maybe i just misinterpret things))
_____________
(in my timeline) the π comes from Egypt and was a composite of ? 1.7???... and 2.5???... just pehaps 1.7? + 2.5? - 1 . . . the each next decimal in 1.7? was iterated from it's previous more significant one keeping the chk-sum digit in mind for each iteration - was some "mind algorithm" of finding π . . . was long ago - don't quite remember
it's possible the similar iteration for the 2.5? was done
(( "1" states for "God" ... frequently ? so "X minus God equals Pi ???" // the above might be not correct ... running random chk-s ))
((( ??? π+1=X -- Closed-time + God = X . . . inside of the creator + outside of the creator = X ??? X = E -- is an Universal set - Wikipedia /!\ = perhaps the above is non.Err /!\ )))
(((( the \(\frac{10}9\) for "1" checks up better 1.111.... -1.7? = 1.5? // 1.7?≈√3 would \(\rightarrow\) 2.5206529571320270560463081528847 likely not !!! ... i guess ))))
____________
it's possible the method was not the only --or-- the universal one - though . . .
 
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Thread Starter

Jennifer Solomon

Joined Mar 20, 2017
15
(( actually i don't get your integer-?quanted? X-axes -- but everything is cool as long you get it yourself -- even if it "does not describe" too well ))
...
also does the y has only positive <1 "offsets"
or
the "whole" reals (with their integer part)
,
is X-axes defined only at integer points
or
the integer value is a "staircase" (along the X-axis)
To more directly address this...

When we work with variables over the reals, we are discretizing them to make them rational, and thus knowable (”sane”). There is obviously no such thing as a real “line” or “circle” composed of infinite points in observable nature, only in our mind’s definitions.

Thusly, graphing a line or any other geometric figure is using truncated portions of the ℝ continuum that our mind interpolates to indissociably “real.”

I would therefore see ℝ as composed of countably infinite integers for the horizontal component and then the vertical component would be uncountably infinite infinitesimal portions, rendering ℝ as a kind of plane.

But since we discretize the continuum for calculations, we only see the rational “linear“ dimension.
 

ci139

Joined Jul 11, 2016
1,696
Thanks for sharing . . . i hovever must decode your 1-st and other post to get into this (am old , witty and tired - so donno when - but i will work out what you are on about)

there was someone http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/contactron.html you might drop a question (if he's not retired) . . . although i donno him in person nor wheather he gets/gives any credit to your theory ... such must not discourge you as other people busy by their own life etc.

other wise your posting here would be a backup for your intellectual property rights (if such ever comes an issue)
 

Thread Starter

Jennifer Solomon

Joined Mar 20, 2017
15
Thanks for sharing . . . i hovever must decode your 1-st and other post to get into this (am old , witty and tired - so donno when - but i will work out what you are on about)

there was someone http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/contactron.html you might drop a question (if he's not retired) . . . although i donno him in person nor wheather he gets/gives any credit to your theory ... such must not discourge you as other people busy by their own life etc.

other wise your posting here would be a backup for your intellectual property rights (if such ever comes an issue)
Thanks for the referral...
 
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