# Real and Imaginary Numbers

In conventional mathematics a distinction is drawn between two kinds of countable numbers: real and imaginary. Each kind of number comes in two mirror opposite forms, identified as positive and negative.

The positive and negative forms of their own kind sum to zero. This results from treating the difference between two numbers that are of the same kind and sign as nothing, and hence uncountable.

Rules have been developed for how the mirror opposite forms multiply together. These rules are the same as those for the truth tables for the logic gates XOR and XNOR.

An imaginary number is defined as the square root of -1. It is the number that when multiplied by itself outputs -1. Hence the square of the imaginary number is the same as the negative real number -1. The sum of the square of the imaginary number, -1, and the square of the real number, 1, is nothing, which is uncountable.

Chiralkine counting is a dual system, like the real and imaginary number system. However, it does not use equations. Hence it is not possible to reduce it to a singular nothing by squaring numbers of each kind (comparing each with itself) and summing them together.

There is a set of imaginary numbers invented by the mathematician, Hamilton, which are known as the quaternions. These have many useful applications, for example in controlling flight and in computer gaming. They form three mirror opposite pairs, generally represented by the symbols i and -i, j and -j and k and -k. Each is itself a square root of -1. However, these numbers conform to a set of multiplication rules where the order in which the multiplication is performed affects the sign of the product.

The numbers in chiralkine counting, a, b and c, each of two kinds, behave like the quaternions i, j and k, each of two kinds, in that they are ordered like i, j and k, but they are symmetric.

The counting cycle in chiralkine counting is non-commutative, like quaternion multiplication.

However, when the two are compared, it is clear that chiralkine counting is symmetric. The asymmetry of quaternions is due to the use of equations to relate the kinds of numbers together.

Quaternions consist of four ordered objects, three imaginary numbers i, j and k and a real number 1. Hence they correspond with four ordered objects a, b, c, d where a, b and c are imaginary and d is real. However, in chiralkine counting, being more symmetric than the quaternions, the distinction between absolute imaginary and absolute real is lost.