irrationals: * see Endnote 54

real numbers which are not rational numbers (which are numbers constructed from ratios of integers); when the ratio of lengths of two line segments is an irrational number, the line segments are also described as being incommensurable, meaning that they share no "measure" in common, that is, there is no length ("the measure"), no matter how short, that could be used to express the lengths of both of the two given segments as integer multiples of itself.

 

unidimensional:

having 1 dimension only

 

continuum:

the dimension of a continuum usually means its topological dimension, a 1 dimensional continuum is often called a curve.

 

“cut” (in Dedekind’s sense): * see Endnote 55

(aka Dedekind’s cut) introduced by Richard Dedekind in 1858-62, who used it to construct the irrational, real numbers. 

​

imaginary:

numbers – such as the square root of -1; the standard notation for this number is i; a non-real numbers (real numbers represent a quantity on a line).

​

complex numbers:

a number expressed in the form a + bi (or x + yi) where a and b are real numbers, and i is a solution to the equation x² = −1, which is called an imaginary number because there is no real number that satisfies this equation.  

 

linear continuum: * see Endnote 56

the real number line, whose points are the real numbers; the real line is the set R of all real numbers, viewed as a geometric space, namely the Euclidean space of dimension one.  It is a set of points in which the distances between all members of the set are defined.

 

aggregate:

a set or a collection of objects or elements classed together.

 

homogenous:

having a common property throughout (e.g. all numbers greater than 0).

​

real numbers:

a number that can be expressed as the quotient or fraction p/q of two integers, a numerator p and a non-zero denominator q.  eg ¾  or 2 (2/1)

 

number surfaces: * see Endnote 57

a surface is a geometrical shape that resembles a deformed plane (e.g. spheres); in differential geometry, a surface is a 2-dimensional manifold, may be an "abstract surface" not embedded in any Euclidean space (e.g. the Klein bottle is a surface, which cannot be represented in the 3-dimensional Euclidean space without introducing self-intersections).

​

x + yi:

notation form for a complex number

​

stereometric:

the measure of volumes.

 

Leucippus:

(5th cent. BC) the earliest Greek to develop the theory of atomism—the idea that everything is composed entirely of various imperishable, indivisible elements called atoms; often appears as the master to his pupil Democritus, a philosopher (who is also touted as the originator of the atomic theory); most likely born in Miletus.  We know nothing of his life.