Mathematicians distinguish between real numbers, which have real square roots, and imaginary numbers, which do not.
When a number is squared, it is multiplied by itself. The logic from which this is constructed is that same as that from which zero is constructed.
+1 squared is +1 (+ x + is +) and -1 squared is also +1 (- x – is +). Once a number has been squared, knowledge of its prior state is forgotten. When the reverse operation is applied (taking the square root), then there are two roots: +1 and -1.
Taking the square root of -1 presents a conflict with the rules for multiplying signs. Mathematicians get around this by defining the square roots of -1 as +i and -i. +i squared is -1 and -i squared is also -1 . They then apply the same sign multiplication rules to +i and -i as they do for +1 and -1. However, these so-called imaginary numbers are actually conforming to sign multiplication rules mirror opposite to those of the real numbers: (+ x + is -) and (- x – is -).
Applying the sign multiplication rules for real numbers to the imaginary numbers privileges the real numbers over the imaginary numbers. Signed units that obey one set of multiplication rules are privileged over signed units that obey the mirror opposite multiplication rules. This creates a conundrum that becomes clear when the definition of the square root of minus one is cast as the equation: i = -1/i, because it sets up an oscillation: if i on the left is positive, (the same) i on the right must be negative, but if i on the left is positive, (the same) i on the right must be negative.
Multiplying a number by itself to afford its square, like drawing a distinction between a number and itself to afford zero, enables meanings to be assigned to the products of the operations, but also creates a logical problem. In the real numbers, + means the same and – means different. In the imaginary numbers, + means different and – means the same. The first is operating in Boolean XOR and the second in Boolean XNOR. The use of the terms “real” and “imaginary” is deceptive. It implies that one kind of number is found in the real physical world, while the other is abstract. What is actually happening is perceptual switching between XOR and XNOR. Representing “complex” numbers on an Argand diagram, on a complex number line projecting from one absolute origin, messes with your head! There is no one, absolute zero.
In the new mathematics, this will not stand. When two numbers operate on one another, even if those numbers are the same, the result must always be two numbers. When a number mutually negates with itself, it affords two zeroes, one from the perspective of each side. When a number is multiplied by itself, it affords two numbers, one from the perspective of each side. The interactions between the numbers are non-commutative: one acts in Boolean XOR and the other in Boolean XNOR.