**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

*i*is on the boundary of the set C = {z ε*C*/ | z | <1}*i*is 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.