The six countable chiralkine numbers (a, ~~b~~, c, ~~a~~, b, ~~c~~) constituted in three pairs (a, ~~a~~, b, ~~b~~, and c, ~~c~~) provide an alternative to the five numbers conventionally used in computation (0 and the countable number pairs +1, -1 and +i, –i). The benefits of performing computation on the principle of an equation are sacrificed for the benefit of treating the two mutually exclusive sides of distinctions as mirror opposites: symmetrically.

Whereas conventional computation switches back and forth between a singular zero and pairs of positive and negative integers, treating the two sides of a relationship independently, computation using chiralkine number pairs proceeds in a cycle, coupling the sides of relationships and so ensuring that they are treated symmetrically.

One of the ways to understand chiralkine numbers deeply is to consider square roots, which arise in conventional mathematics, because it is based on equations. The square root of the product of two numbers a and b is equal to two numbers: a positive number c and a negative number c, independent of the order in which a and b are multiplied together.

√ a.b = √ b.a = ±c

However, reversing the operation by squaring the positive number c or the negative number c affords only one number: a.b.

(+c)^{2} = (-c)^{2} = a.b = b.a

So if a is +1 and b is -1, the square root of a.b is ±i.

Chiralkine numbers do not work with equations. The order in which the numbers are “multiplied” matters.

√ a.~~b~~ ≠ √ ~~b~~.a

a is turned by ~~b~~ into ~~c~~, while ~~b~~ is turned by a into c

There is a powerful analogy with spinors, which are constituted using equations, but are likened to the square root of geometry.

Conventional numbers (0, ±1, ±i) are founded by drawing distinctions using the mutually exclusive, mirror opposite logic gates XOR and XNOR, but privileging one over the other such that all operations are commutative (a.b = b.a). Chiralkine numbers (a, ~~b~~, c, ~~a~~, b, ~~c~~) are also founded by drawing distinctions using the mutually exclusive, mirror opposite logic gates XOR and XNOR, but without privileging one over the other such that all operations are non-commutative (a.~~b~~ ≠ ~~b~~.a).