V. Few words about complex numbers
The one who does not posses the knowledge of complex numbers may be surprised that something like that actually does exist. How the square of a number can be negative when the square of any real number must be non-negative. Well, this is the core of the issue that real numbers do not have the roots in negative numbers, although they are sometimes needed.
What actually do people need numbers for? A long time ago people managed to live without them. It can be assumed that, at first, small natural numbers were needed in order to count members of a group or animals. With time people learned how to count. By adding two natural numbers we will receive a natural number again, only the higher one. The upper limit of the used numbers raised when the number of members in a group or number of used items or livestock increased. After that, people learned how to subtract numbers. For a long time people subtracted lower numbers from the higher ones. It was considered impossible that quantity can be negative. Only when loans appeared people realised that negative numbers might be useful.
The above indicates that interpretation of numbers and calculations is not so strict and evident as we often think. It had taken hundreds of years before people learned how to freely use the negative numbers. When it comes to complex numbers it may take even longer…
Natural numbers together with their negative equivalents and the zero value constitute integer numbers. If only integer numbers are selected, someone can freely add numbers and subtract numbers and the result will be always an integer number. Integer numbers can be also multiplied. Multiplication of integer numbers appeared to be useful, for example, at estimating the area of rectangles. Only from time to time dividing something into parts was found necessary and for that the integer numbers were not enough. In such cases, fractions appeared to be necessary. This lead us to the rational numbers, that is, the numbers that can be written in the form of a quotient of two integer numbers (where the second one is other than zero).
A surprise was the postulate that irrational numbers do exist. It turned out that they are necessary, for example, for estimating the length of a diagonal of a square and generally, when computing roots of numbers. Rational numbers together with irrational ones constitute real numbers, which in geometrical interpretation create a straight line so-called a numerical axis. When at the straight line we indicate one point and mark it as zero, and another one is marked as one, then every real number can be unequivocally attributed to one point at the straight line and vice versa, every point at the straight line refers to one real number. There is only one problem. When comes to real numbers not all numbers have roots. Because the squared real number cannot give a negative result, accordingly, the square root of a negative number cannot exist. When for some reason we need to determine it, then we have to go beyond real numbers. Therefore, complex numbers were invented.
Complex numbers originate from real numbers, that is, to a real number that is the real part of a complex number we add an imaginary part. The imaginary part can be understood as an additional dimension in the direction of the imaginary unit defined by letter i. When we imagine the real numbers as a horizontal axis and the imaginary numbers as a vertical axis then the complex numbers will constitute a complex plane where the complex number a+bi will be defined by coordinates (a,b).
We see that all numbers mentioned so far in the geometrical interpretation are related to the Euclidean geometry. In other words, all computations conducted with the use of such numbers are based on a hidden assumption that space is Euclidean. It appears that for the computations related to cosmology, a mathematical model corresponding to the non-Euclidean geometries would be helpful…