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 The True Nature of Imaginary Numbers

The True Nature of Imaginary Numbers
 THE TRUE NATURE OF
IMAGINARY NUMBERS

"The metaphysical truth of the square root of −1 is elusive" (Gauss).

Imaginary numbers are amphibious between being and non-being" (Leibniz).


The Interpretation of Imaginary Unity i

The imaginary unit is not the square root of −1

It is usually stated that the solutions of the equation x2 = −1 are where i is an "imaginary" number, so called because there is no real number such that its square is −1.

However, i is not the square root of −1. It is not because if √(−1) were a solution of the equation x2 = − 1, we would reach a contradiction, if we want the property xr·y r = (x·y)r and, in particular, xr·xr = (x·x)r : Property (xr)s = (xs)r = xr·s, i.e., the exponents are not interchangeable. In effect: Another way to arrive at the same contradiction is the following. We have that i2 = −1. Multiplying both terms by −1, we have (−1)i2 = 1. Extracting the square root on both sides, we have: Therefore, the expression √(−1) makes no sense, just as it makes no sense to calculate the square of an apple or the square root of a car. That is why in physics it is written i and not √(−1), for example, in the Schrödinger equation and in the theory of relativity.


George Spencer-Brown's interpretation

George Spencer-Brown, in his book "The Laws of Form" [1972], interprets imaginary unity in another way. The expression x2 = −1 transforms it into the recursive expression x = −1/x, which produces the oscillating time sequence between two values: Asserts that, in such a recursive expression, the only possible value i of x is 1 or −1, so the above sequence becomes. or its inverse: According to this author: The above oscillating expression has the same structure as the famous liar's paradox.


Interpretation in MENTAL

The correct interpretation of the imaginary unit is very simple. The expression i2 = −1 is to be interpreted as a particular case of an imaginary expression, that is, a substitution expression in which the left-hand side is an algebraic expression. In this case, i is an entity whose square evaluates to −1. This interpretation allows us to define all kinds of imaginary numbers, as well as imaginary numbers of higher order. And, in general, imaginary expressions and higher-order imaginary expressions.

(i*i = −1) // definition of the imaginary unit i

(j*j = −i) // definition of an imaginary number of order 2

(k*k = −j) // definition of an imaginary number of order 3 ...

Generically, calling i/n to the imaginary unit of order n:

⟨( (i/n)*(i/n) = i/(n−1) )⟩
(i/1 = −1)
(i/2 = i) // imaginary unit

The fact of considering the number −1 as imaginary, links with the old belief that negative numbers were imaginary [see Addendum].

For centuries there has been speculation about the mysterious nature of imaginary numbers. The great Gauss (in 1825) claimed: "The true metaphysics of the square root of −1 is elusive". First it was given a geometrical interpretation (Wallis, Wessel, Argand and Gauss himself): the imaginary numbers as a dimension different from the real line. Other authors (Cauchy, Hamilton) gave a purely algebraic interpretation.

The interpretation as an imaginary expression (in the sense of MENTAL, i.e. as a substitution), also provides a unifying framework in which enter the imaginary numbers and the infinitesimal, the latter also defined by an imaginary expression: ε*ε = 0.

Just as from the expression ε2 = 0 it makes no sense to infer that ε = √(0) = 0, we cannot infer that i 2 = −1 be i = √(−1) because i2 = −1 is not an equation, but a substitution.


Complex numbers

Complex numbers have the form a + b*i, where a and b are real numbers. The real numbers are thus generalized, for when b is zero, we have the real numbers, and when a is zero, we have the imaginary numbers.

Complex numbers can be represented on a plane (the complex plane), the real numbers being on the horizontal axis and the imaginary numbers on the vertical axis. We can represent, if we wish, a complex number by its two coordinates (r1 and r2) and define, for example:

⟨( c(r1 r2) = (r1 + r2*i) )⟩


Some properties of complex numbers

Addenda

Negative numbers as imaginary numbers

There was a time when negative numbers were considered imaginary. For example, the equation x+3 = 0 has no natural number as a solution. Gerolamo Cardano, in the 16th century, called negative numbers "false", but in his Ars Magna (1545) he studied them exhaustively. Their introduction was extraordinarily slow, due to a certain refusal to consider them as numbers. Only negative quantities (associated, for example, with debts) were admitted. It was at the end of the 18th century when negative numbers were universally accepted.


Quaternions

Invented (or discovered) by Hamilton, quaternions are a generalization of complex numbers, with one real and three imaginary parts, with the following (purely algebraic) definition: (q = t + ui + vj + wk) (one quaternion)

(q' = t - ui - vj - wk) (conjugate of q)

(qq' = t2 + u2+ v2 + w2) (property similar to that of the complexes)

Quaternions have inverses too, like complex numbers.

In 1897, A.S. Hathaway formally extended Hamilton's ideas by moving from considering quaternions as 4 real numbers to the idea of 4 spatial dimensions.

In 1843, John Graves discovered that there is a type of double quaternion (octonions). They were rediscovered by Arthur Cayley in 1845.

There is no satisfactory generalization of octonions to higher dimensions.


History of imaginary numbers
Bibliography