Imaginary unit

Imaginary unit is used to construct complex numbers, an algebraic field where it is possible to solve all algebraic equations of rational coefficients [1] . The impossibility of solving the equation x 2 + 6x + 10 = 0 → (x + 3) 2 = -1, would suppose to know the square root of -1. This requires defining a new mathematical entity:

The imaginary unit i . And it is agreed that i 2 + 1 = 0. From the above it follows that the square roots of -1 are i, -i.

Summary

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  • 1 Other possibilities
  • 2 Finite group with i
  • 3 Topological aspect
  • 4 Geometry
  • 5 References
  • 6 See also

Other possibilities

Other exits fit. The concept of “generalized complex number” can be formulated.

However, given the equation x 2 + 2x +2 = 0, the existence of a “non-real number” ε is proposed , such that ε 2 + 2ε +2 = 0, constructing a new set of «flat numbers», of the form a + bε, where a and b are real numbers.

[2] Cardano, when solving cubic equations, had to resort to new mathematical objects and thus arise the so-called “complex numbers” of the form x + yi where x, y are real numbers, i is the non-real solution of x 2 + 1 = 0.

Finite group with i

The roots of the fourth-degree algebraic equation x 4 -1 = 0 determine the set H = {1, -1, i, -i}, which is a cyclic multiplicative group of order 4, with two generators i , -i . These last two roots of the equation are called the primitive roots of unity. The elements of the set H represent the vertices of a square, located on the unit circumference of the center at the origin of coordinates and radius equal to 1.

Topological aspect

  1. iis on the boundary of the set C = {z ε C / | z | <1}
  2. iis an accumulation point of the set D = {z ε C / | z | ≥ 1}

Geometry

as vector or point of the plane i = (0, 1).

When multiplying the real complex number (a, 0) by i is ai = (0, a), geometrically it is a counterclockwise rotation of 90º; and the product a (-i) = (0, -a) is a counterclockwise rotation of 270º.

Multiplying the complex number z = (a, b) by i is (a, b) i = (-b, a) is a counterclockwise rotation of z by 90º.

Multiplying the complex number z = (a, b) (-i) = = (a, b) × (-1) × i = (b, -a) is a counterclockwise rotation of 180º, followed by a counterclockwise rotation of 90º.

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