Not ad infinitum

In Mathematical, in the under-field of the projective Geometry, the not ad infinitum is a not which can be added to a body.

Real numbers

When the point ad infinitum is added to the right-hand side of the real numbers, this one forms a Courbe closed, called the projective Droite real, $\ mathbb P^1 (\ mathbb {R})$ .

Let us consider two lines now parallel S in the projective Plan $\ mathbb P^2 (\ mathbb {R})$ . Since the two lines are parallel, they ad infinitum cross in a not . This one is on the Droite ad infinitum in $\ mathbb P^2 (\ mathbb {R})$ .

Moreover, each line is, in $\ mathbb P^2 (\ mathbb {R})$ , a projective Droite: each one with its clean not ad infinitum . Two lines of a Espace refines are parallel if and only if they have same the not ad infinitum .

Complex numbers

The point ad infinitum can also be added to the complex plan $\ mathbb C^1$ in order to obtain the projective line complexes, i.e. the Sphère of Riemann. One can conceive this by imagining the opposite process initially: that is to say a Sphere which one perforates. The resulting hole forms a edge which one stretches towards the infinite one. This forms the complex plan. By reversing this process, one transforms the complex plan into projective right-hand side complexes $\ mathbb P^1 (\ mathbb {C})$ : the hole is " déperforé" by ajouant the not ad infinitum , which is equivalent to each point on the edge which formed the edges of the hole.

See too

Right ad infinitum
Hyperplane of infinite the (plane of infinite)

Random links: