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European Journal of Applied Sciences – Vol. 12, No. 2
Publication Date: April 25, 2024
DOI:10.14738/aivp.122.16273
Antonov, A. A. (2024). Proving Physical Reality and Explaining the Physical Essence of Imaginary Numbers. European Journal of
Applied Sciences, Vol - 12(2). 362-381.
Services for Science and Education – United Kingdom
Proving Physical Reality and Explaining the Physical Essence of
Imaginary Numbers1
Alexander Alexandrovich Antonov
Independent researcher, Kiev, Ukraine
ABSTRACT
In the article it is shown that the version of the special theory of relativity (STR),
stated in all textbooks of physics, is wrong as the relativistic formulas received in it
are wrong, they are incorrectly with use of wrong principle of non-exceeding of
speed of light are explained and from them wrong conclusions about physical
unreality of imaginary numbers and also about existence in the nature of our only
visible universe are made. This generally recognized version of STR is refuted
experimentally proved as a result of research of transient processes in linear
electric circuits by the general scientific principle of physical reality of imaginary
numbers discovered 500 years ago. It is explained that imaginary numbers in
astrophysics correspond to the world of invisible parallel universes in other
dimensions. Its cognition is the task of future science. However, the neighboring
universes can be seen on the starry sky in portals even now. The corrected
relativistic formulas are obtained and the corrected version of STR corresponding
to them is created.
Keywords: imaginary numbers, special theory of relativity, invisible universes, hidden
Multiverse, Hyperverse
INTRODUCTION
Imaginary numbers were discovered 500 years ago by Scipione Del Ferro, Niccolo Fontana
Tartaglia, Gerolamo Cardano, Lodovico Ferrari and Raphael Bombelli [1]. And perhaps even
earlier than them such a scientific discovery was made by Paolo Valmes [2], who was burned
alive at the stake for this by the verdict of the Spanish inquisitor Thomas de Torquemada. Even
Sir Isaac Newton 2 was forced to take into account the opinion of the Inquisition about
imaginary numbers, who therefore preferred not to use them in his works.
However, their physical significance remains unknown in science to this day. Indeed, everyone
knows what 7 seconds, 12 meters, or 19 grams are, but no one knows what 7i seconds, 12i
meters and 19i grams, where
i = −1
, are. We all know that 7, 12 and 19 are simply numbers
1 This is reprint of the article “Antonov A. A. Proving physical reality and explanation physical nature of imaginary
numbers”. Norwegian Journal of development of the International Science”. International Science. 123. 26-36.
https://doi.org/10.5281/zenodo.10451085
2
In the atmosphere of the omnipotence of the Inquisition and intolerance of dissent that existed at that time,
Newton's friend William Whiston was stripped of his professorship in 1710 for some of his careless statements and
expelled from Cambridge University.
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363
Antonov, A. A. (2024). Proving Physical Reality and Explaining the Physical Essence of Imaginary Numbers. European Journal of Applied Sciences, Vol
- 12(2). 362-381.
URL: http://dx.doi.org/10.14738/aivp.122.16273
having no physical significance outside of their context. However, this knowledge was not
enough to understand the STR.
THE PROBLEM OF UNDERSTANDING IMAGINARY NUMBERS
Works of famous mathematicians Abraham de Moivre, Leonhard Euler, Jean le Rond
d’Alembert, Caspar Wessel, Pierre-Simon de Laplace, Jean-Robert Argand, Johann Carl
Friedrich Gauss, Augustin Louis Cauchy, Karl Theodor Wilhelm Weierstrass, William Rowan
Hamilton, Pierre Alphonse Laurent, Georg Friedrich Bernhard Riemann, Oliver Heaviside, Jan
Mikusiński and others contributed to creation of a perfect theory of functions of a complex
variable. However, the theory neither proves physical reality of imaginary numbers nor
explains their physical significance3.
Imaginary numbers are now widely used in all exact sciences, including radio engineering,
electrical engineering, optics, mechanics, acoustics, etc. But in them also the physical reality of
imaginary numbers is not proved and their physical meaning is not explained4.
But in the generally accepted version of the special theory of relativity (STR) [3]-[5], which is
rightly considered one of the most outstanding theories created in the 20th century and is
therefore currently studied in all physics textbooks, it is even denied, since its creators were
unable to explain the relativistic formulas obtained therein.
0
2
1 ( )
m
m
v
c
=
−
(1)
2
0
1 ( ) v t t
c
= −
(2)
2
0
1 ( ) v l l
c
= −
(3)
Where,
• m0
is the rest mass of a moving physical body;
• m
is the relativistic mass of a moving physical body;
• 0 t
is the rest time of a moving physical body;
• t
is the relativistic time of a moving physical body;
• 0
l
is the rest length of a moving physical body;
• l
is the relativistic length of a moving physical body;
3 Naturally, about physical reality and physical essence of imaginary numbers, as well as real numbers, we can speak
only in relation to named numbers, equipped with indications on the used units of measurement of corresponding
parameters of physical objects and processes.
4 More precisely, in radio engineering and electrical engineering it is actually revealed in the process of their practical
use, but nothing is written about this in textbooks, so as not to refute physics.
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Services for Science and Education – United Kingdom 364
European Journal of Applied Sciences (EJAS) Vol. 12, Issue 2, April-2024
• v
is the velocity of a moving physical body;
• c
is the speed of light;
Fig. 1: Graphs of functions
mv( ) , t v( )
and
l v( )
corresponding to the existing and the
corrected versions of the STR in the subluminal
v c
and superluminal
v c
ranges
They could not explain physical significance of these formulas for the superluminal velocity
range, where, according to these formulas, mass, time, and distance were measured in
imaginary numbers (see Fig. 1a, b, c). However, since a theory that could not be explained even
by its creators would be useless to anyone, in the STR had to introduce a postulate
5, known as
the principle of light speed non-exceedance, the meaning of which is clear from its name.
In relation, for example, to the Lorentz-Einstein formula (1), it was explained as follows. The
postulate asserted that since the situation at
v c
never occurred anywhere in the early 20th
century, it did not need any explanation. Thus, imaginary numbers were unnecessary. i.e., non- existent. Moreover, they were even called imaginary
However, since the existing version of the STR was based solely on a postulate, that is, an
unproven assumption, there was no complete certainty that it was correct. Actually, it turned
out to be incorrect, since in 2008-2010 (i.e., even before publication of results of the
unsuccessful OPERA experiment6 conducted at the Large Hadron Collider in 2011), it was
experimentally proven [6]-[10] that imaginary numbers are physically real.
5
Since it has never been proven theoretically or confirmed experimentally by anyone.
6 Which was no longer needed