Completed real right-hand side

In Mathematical, the completed real right indicates the unit $\ overline {\ mathbb {R}}$ made up of the real numbers with which one associates two noted elements $+ \ infty$ and $- \ infty$ (which is not regarded as real numbers) checking the following properties:

for any reality X , X < $+ \ infty$ ,
for any reality X , X > $- \ infty$

The addition and the multiplication defined on the whole of realities remain valid in the completed line, so that:

Addition: for any reality X ,

X + (

+ \ infty

) = (

+ \ infty

)

X + ( $- \ infty$ ) = ( $- \ infty$ )

( $+ \ infty$ ) + ( $+ \ infty$ ) = ( $+ \ infty$ )

( $- \ infty$ ) + ( $- \ infty$ ) = ( $- \ infty$ )

Multiplication: for any strictly positive reality X ( X > 0),

x \ times (+ \ infty)

= (

+ \ infty

)

$x \ times (- \ infty)$ = ( $- \ infty$ )

for any strictly negative reality X ( X < 0),

x \ times (+ \ infty)

= (

- \ infty

)

$x \ times (- \ infty)$ = ( $+ \ infty$ )

( $+ \ infty) \ times (+ \ infty$ ) = ( $+ \ infty$ )

( $+ \ infty) \ times (- \ infty$ ) = ( $- \ infty$ )

( $- \ infty) \ times (- \ infty$ ) = ( $+ \ infty$ )

on the other hand, the expressions

(

+ \ infty

) + (

- \ infty

$0 \ times (+ \ infty)$ and

$0 \ times (- \ infty)$ does not have any direction.

One of its notable characteristics is that any unit included in the completed real line admits a higher Borne and a lower Borne, including the Empty set (noted ∅, and who in the completed real line admits $+ \ infty$ as a LOWER limit, and $- \ infty$ as an UPPER limit).

This unit is very useful in analyzes, and particularly in certain theories of the integration.

See too

Compactifié d' Alexandroff

Random links: