The nonsense of imaginary distance.

Definitions.

In the work of John B. Conway [1] on the page 1 it is stated the following «We define C, the complex numbers, to be the set of all ordered pairs (a,b) where a and b are real numbers and where addition and multiplication are defined by:

$(a,b)+(c,d)=(a+c,b+d)$

$(a,b)(c,d)=(ac-bd,bc+ad)$

…

If  we put $i=(0,1)$ then $(a,b)=a+bi$. From this point on we abandon the ordered pair notification for complex numbers.

...

When we write $z=a+ib(a,b\in R)$ we call a and b the real and imaginary parts of z and denote this by a=Re z, b=Im z.

...

If  $z=x+iy(x,y\in R)$ then we define $|z|=(x^2+y^2{)}^{1/2}$ to be the absolute value of z.»

In the same reference [1] on the page 3 it is stated that

«From the definition of complex numbers it is clear that each z in C can be identified by unic point (Re z, Im z) in the plane R². The addition of complex numbers is exactly the addition law of the vector space R². If z and w are in C then draw the strait lines from z and w to 0 (=(0,0)). These form two sides of parallelogram with 0, z and w as three vertices. The fourth vertex turns out to be z+w.

Note also that |z-w| is exactly the distance between z and w.»

Later on, in the same reference [1] on the page 4 the Author states the following

«Consider the point z=x+iy in the complex plane C. This point has polar coordinates $(r,\theta )$, $x=rcos\theta ,y=rsin\theta$. Clearly r=|z| and $\theta$ is the angle between the positive real axis and the line segment from 0 to z.».

r=|z| is the distance between point 0 and point z. This is the geometrical interpretation of the absolute value of number z.

Regarding the Minkowski space, in the work of L. D. Landau and E. M. Lifshitz [2] in section 2 on the page 4 it is stated the following

«$d{s}^2=c^{2}d{t}^2-d{x}^2-d{y}^2-d{z}^2$ (2.4)

The form of expression ... (2.4) permits us to regard the interval, from the formal point of view, as the distance between two points in the fictitious four-dimensional space (whose axis are labeled by x; y; z, and the product ct). But there is a basic difference between the rule for forming this quantity and the rule in ordinary geometry: In forming the square of the interval, the squares of the coordinate differences along with the different axes are summed, not with the same sign, but rather with varying signs».

Since the concept of strait line is the same in both Euclidean and Minkowski space, in two-dimensional Minkowski space the formula $|z|=(x^2-y^2{)}^{1/2}$ describes the distance between the points 0 and z on the strait line.

Axiom of the space contradicts the distance in Minkowski space.

In the Minkowski space, the distance r between the points (0,0) and (0,1) is described as $r=|i|=\sqrt{0^2-1^2}=i$ for complex number $i=(0,1)$. Equality of values r and i means that this is no difference between them. So, if Re i and Im i are two parts of complex number i then the distance r should have the same two parts. Are parts of the length of vector z are projections of vector on coordinate axes? No, for instance, if we put $z_1=(2,3)$ then $z_1=\sqrt{2^2-3^2}=0+\sqrt{5}i$, $0\ne 2,0\ne 3,\sqrt{5}\ne 2,\sqrt{5}\ne 3$ .

Assuming all stated abobe, the main question in this respect is where are two parts of the complex length of the vector?

The formula $|z|=r=x+iy$ indicates that there are two different segments between the beginning and end of the vector, which both determine the length of vector, however such a situation contradicts the axiom of the space «For every two points there exists no more than one line that contains them both», i.e. the situation is nonsense, therefore the imaginary distanse is impossible.

Conclusion.

In Euclidean space $|1|=1$ for number $1=1+i0$ and $|i|=1$ for number $i=0+i1$. So, number 1 is expressed by the vector with length 1 and number i is expressed by the vector with length 1 too. Hence, it is not possible to say whethe the numbers 1 and i are real or imaginary numbers due to the length of vectors. Length of the vector does not express the imaginarity of complex number. But there are only two properties of any vector, then if it`s not the length then this is the direction that expresses the imaginarity of complex number.

Explanations what is imaginarity in fact and why $i^2=-1$ is incorrect expression are in the blog at https://a-y-nechitaylo.blogspot.com/2024/05/the-paradox-of-minkowski-space.html .

Literature.

1. Functions of one complex variable / Conway, John B. (Graduate Texts in Mathematics; 11) (Springer-Verlag, New York, 1973) (https://psm73.wordpress.com/wp-content/uploads/2009/03/conway.pdf).

2. L. D. Landau and E. M. Lifshitz, The Classical Theory of Fields, Third Revised English Edition (Course of Theoretical Physics Volume 2) (Pergamon Press, New York, NY, 1971).