# Theory: R is not a number line...

#### Jennifer Solomon

Joined Mar 20, 2017
112
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.

Last edited:

#### 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)

#### BobTPH

Joined Jun 5, 2013
3,689
Nope, I am not going to bite. Have fun.

Bob

#### atferrari

Joined Jan 6, 2004
4,433
Nope, I am not going to bite. Have fun.
Bob
Don't worry there is always candidates for that.

#### Jennifer Solomon

Joined Mar 20, 2017
112
Nope, I am not going to bite. Have fun.

Bob
Why not? What’s the big deal? Tell me why you think it’s wrong...

#### Jennifer Solomon

Joined Mar 20, 2017
112
Don't worry there is always candidates for that.
New theories are always spurned, aren’t they now...

#### 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)

Last edited:

#### Jennifer Solomon

Joined Mar 20, 2017
112
(( 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.

Last edited:

#### 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 . . .

Last edited:

#### Jennifer Solomon

Joined Mar 20, 2017
112
(( 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)

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)