Likewise neither number system in isolation, but rather the integration of both constitutes dynamic mathematical understanding. 

To achieve this integration we need a means of translating 

(a) the (psychological) qualitative, in complementary (mathematical) quantitative terms and 

(b) the (mathematical) quantitative, in complementary (psychological) qualitative terms. 

These indirect translations lead in both cases to the emergence of fascinating circular number systems. 

Thus the linear (psychological) qualitative system, has an indirect reduced translation as a circular (mathematical) quantitative system. 

Likewise the linear (mathematical) quantitative system, has an indirect raised translation as a circular (psychological) qualitative system. 

It is through the integration of these linear and circular number systems, that the inherent complementarity of mathematics and psychology is best appreciated. 
 

Quantitative Circular Numbers 

As we have seen the first four natural numbers in the (psychological) qualitative system are 1¹, 1², 1³ and 1⁴ respectively. 

To translate these numbers in indirect (mathematical) quantitative terms, we must express each number in terms of an invariant 1st dimension. This simply implies taking the successive roots of each number. 

The one root of 1, which could be written 1¹ is of course 1 or +1. This number lies on the corresponding circle of unit radius. 

The two roots (i.e. square root) of 1 or 1^1/2 are +1 and -1. Thus expressed in one dimensional linear (quantitative) terms, the 2nd (qualitative) dimension has complementary aspects (i.e. two numbers of opposite sign). These two numbers again lie - as both ends of a diameter - on the corresponding circle of unit radius. 

The three roots of unity roots i.e 1^1/3 again lie on the corresponding circle of unit radius (where horizontal axis represents real, and vertical axis represents imaginary number systems respectively). 

The four roots of 1 or 1^1/4 are 1, -1, i and -i respectively. These four roots again lie on corresponding circle of unit radius (with four points representing the extremities of real and imaginary axes). 

Higher order roots will also lie on the same circle. If we obtained for example the 100 roots of unity (i.e. 1^1/100), these would lie as 100 equidistant points on the same circle). 

Therefore there is a coherent (mathematical) quantitative system which is circular corresponding to the (psychological) qualitative system which is linear. 
 

Qualitative Circular Numbers 

Let us take as an example the simple equation x = 1. In conventional practice when we square both sides we get x² = 1. However when we reverse this operation - by extracting the square root - we get x = +1 and x = -1. Thus, whereas x formerly had one solution, now it has two! 

This lack of reverse symmetry in operations highlights an important problem. 

When we start with x = 1 and square both sides, strictly speaking the result is x² = 1² . However in conventional terms, 1² is interpreted (solely) in reduced quantitative terms as 1(i.e. 1¹⁾. In other words the qualitative transformation involved in moving from the first to second dimension is thereby lost. 

If we are to interpret it correctly, when we move from 1 (i.e. 1¹ ) to 1², there is a psychological transformation involved. 1² represents a "higher" order unity than 1¹. This second dimension in fact corresponds to (circular) intuitive understanding based on the principle of the complementarity of opposites. This exactly matches the complementary mathematical transformation (i.e. the square root of unity). In like fashion this and all higher dimensional numbers can be referred to - in psychological terms - as circular.