Until the middle of the 16th century, equations like x2 – 6x + 10 = 0 were simply considered “no solution”. This was because, according to Bhaskara's formula, when solving this equation, the result found would be:
Δ = (–6)2 – 4·1·10
Δ = 36 – 40
Δ = – 4
x = –(– 6) ± √– 4
2·1
x = 6 ± √– 4
2
The problem was found in √– 4, which has no solution within the set of real numbers, that is, no there is a real number that, multiplied by itself, yields √– 4, since 2·2 = 4 and (–2)(–2) = 4.
In 1572, Rafael Bombelli was busy solving the equation x3 – 15x – 4 = 0 using Cardano's formula. Through this formula, it is concluded that this equation does not have real roots, as it ends up being necessary to calculate √– 121. However, after a few attempts, it is possible to find that 43 – 15·4 – 4 = 0 and therefore that x = 4 is a root of this equation.
Considering the existence of real roots not expressed by Cardano's formula, Bombelli had the idea of supposing that √– 121 would result in √(– 11·11) = 11·√– 1 and this could be an “unreal” root for the equation studied. Thus, √– 121 would be part of a new type of number that makes up the other unfound roots of this equation. So the equation x
3 – 15x – 4 = 0, which has three roots, would have x = 4 as the real root and two other roots belonging to this new type of number.In the late 18th century, Gauss named these numbers as complex numbers. At that time, complex numbers were already taking the form a + bi, with i = √– 1. Furthermore, The and B they were already considered points of a Cartesian plane, known as the Argand-Gauss plane. Thus, the complex number Z = a + bi had as its geometric representation a point P (a, b) of the Cartesian plane.
Therefore, the expression “complex numbers” started to be used in reference to the numerical set whose representatives are: Z = a + bi, with i = √– 1 and with The and B belonging to the set of real numbers. This representation is called the algebraic form of complex number Z.
Since complex numbers are formed by two real numbers and one of them is multiplied by √– 1, these real numbers have been given a special name. Considering the complex number Z = a + bi, a is the "real part of Z" and b is the "imaginary part of Z". Mathematically, we can write, respectively: Re (Z) = a and Im (Z) = b.
The idea of modulus of a complex number is crystallized analogously to the idea of modulus of a real number. Considering the point P(a, b) as a geometric representation of the complex number Z = a + bi, the distance between the point P and the point (0,0) is given by:
|Z| = √(The2 + b2)
A second way to represent complex numbers is through the Polar or trigonometric form. This form uses the modulus of a complex number in its constitution. The complex number Z, algebraically Z = a + bi, can be represented with the polar form by:
Z = |Z|·(cosθ + icosθ)
It is interesting to note that the Cartesian plane is defined by two orthogonal lines, known as the x and y axes. We know that real numbers can be represented by a line, on which all rational numbers are placed. The remaining spaces are filled with the irrational numbers. Whereas the real numbers are all on the line known as X axis from the Cartesian plane, all other points belonging to that plane would be the difference between complex numbers and real numbers. Thus, the set of real numbers is contained in the set of complex numbers.
By Luiz Paulo Moreira
Graduated in Mathematics
Source: Brazil School - https://brasilescola.uol.com.br/o-que-e/matematica/o-que-sao-numeros-complexos.htm
