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.
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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