is ∞ a real number?


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Question Number 205577 by BaliramKumar last updated on 25/Mar/24

$is \infty a real number ?$
Commented by mr W last updated on 25/Mar/24

$i t i s o n l y a s y m b o l . i t m a y h a v e$ $d i f f e r e n t m e a n i n g s d e p e n d i n g o n$ $i n w h i c h c o n t e x t i t i s u s e d .$ $g e n e r a l l y p e o p l e u s e i t t o$ $d e s c r i b e s o m e t h i n g w i t h o u t a n y$ $l i m i t (b o u n d a r y) .$
Commented by BaliramKumar last updated on 25/Mar/24

Commented by mr W last updated on 25/Mar/24

$h e r e a c o n t e x t i s g i v e n, t h e t o p i c i s$ $a b o u t n u m b e r s . w h e n t a l k i n g a b o u t$ $n u m b e r s, \infty r e f e r s t o$ $- \infty < x < + \infty$ $b u t s t i l l w e c a n n o t s a y \infty i s a n u m b e r$ $o r \infty i s a r e a l n u m b e r .$
Commented by BaliramKumar last updated on 25/Mar/24

$i think Q is wrong$ $e is also irrational$
Commented by mr W last updated on 25/Mar/24

$y e s, q u e s t i o n i s n o t a g o o d o n e . t h e$ $a n s w e r g i v e n i s a l s o w r o n g .$
	Answered by TheHoneyCat last updated on 31/Mar/24
	$In standard math (Russel′s set theory, Bourbaki′s construction... ect...) R (the set of real numbers) is defined as the completion of Q (i.e. the limits of bounded increasing sequences in Q). So in all the usual meanings of “∞” ∞∉R This includes: formal topological closure of R R^� =R⊎{+∞,−∞} The projective plane RP^1 where ∞:=[0:1] The sur−real extension ∞=ℵ_0 (or any “infinite” number) But of course, there are numerous areas of math where this is not explicitely stated and we don′t pay attention wether R refers to real numbers or some extentions...$
	$In standard math ({Russel}^{'} s set theory,$ ${Bourbaki}^{'} s construction . . . ect . . .)$ $R (the set of real numbers) is defined as the$ $completion of Q (i . e . the limits of$ $bounded increasing sequences in Q) .$ $So in all the usual meanings of ‘ ‘ \infty^{″}$ $\infty \notin R$ $This includes : formal topological closure of R$ $\bar{R} = R ⨄ {+ \infty, - \infty}$ $The projective plane {RP}^{1} where \infty := [0 : 1]$ $The sur - real extension \infty = ℵ_{0} (or any$ $‘ ‘ {infinite}^{″} number)$ $But of course, there are numerous areas of$ $math where this is not explicitely stated$ $and we {don}^{'} t pay attention wether R refers$ $to real numbers or some extentions . . .$
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