Is every number in the real number system unique? Can real numbers represent every point in a continuous line?

Question

Is there a real number that is less than the number one (1) and greater than the number represented by a decimal followed by an infinite number of nines (.999999....)?

if (.99999...) is a transfinite number (as sugggested by Curt Monash) there is no final number. Than for every number in the real number system with a decimal followed by a series of nines is it not true that it is always possible to insert a '0' after the Nth digit even if 'N' goes to infinity? Then for every number including (.9999...) if it is a real number the choice is available to not select a '0' after the Nth digit thus selecting a number 'X' which is >1and <(.99999...)?

Answer 1

.999 ... is not a transcendental number. It's algebraic. In fact, it's rational. In fact, it equals 1.

If you don't know the difference between "transcendental" and "transfinite", you probably shouldn't try to speak on the subject with authority.

The other answers are all good.

Answer 2

Can real numbers represent every point in a number line?
YES

Is there a real number that is less than the number one (1) and greater than the number represented by a decimal followed by an infinite number of nines (.999999....)?
NO

Answer 3

To answer your second question first, yes, real numbers can and do represent every point on a continuous line. In set theory, this is referred to as being "complete."

As far as every number in the real number system being unique, there is no real consensus on the matter; it is being actively researched even today. The trouble stems from a flaw in the formal definition of real numbers: the set of axioms that define them does not allow for a proof, or disproof, of uniqueness. The example given above (where x = 0.999... and so forth) is persuasive, but may not be valid because 0.999... is a trancedental number.

Answer 4

i myself think of WWE's finished interest is on the adult men, they're in simple terms sidetracking the Divas. the adult men are frequently the the spotlight & interest bringers for the business enterprise, So WWE has in simple terms gotten caught in the shuffle with stressful approximately how nicely can they carry out in the adult men's branch, fairly than the girls human beings's. yet I consider you!, I ignored those previous days while the divas have been given fairly feuds and storylines. each little thing now's no longer something yet prepare fights and contact suits, and while is the final time we actually considered a diva promo ?, no longer those days, they only situation we can get out of the divas today is a Interview. WWE had some thing going with the McCool/ Maria deal, yet while they might have chosen another Diva as nicely Maria, to insurrection against McCool, that feud might have been greater constructive. yet they only dropped it after the final 2 cat fights. while i think of of great diva feuds, i think of of Trish and Mickie, Lita and Trish, Victoria and Lita, Hell even Victoria and Christy Hemme replaced into some what greater good of today. i myself do no longer understand the place WWE's mind's eye went in the divas branch, yet they confident choose it returned. I assure if WWE had greater constructive diva feuds and suits, no person could be complaining with regard to the divas branch like they're today. The Wrestling has gotten much greater constructive, yet while your being provided as no longer something substantial, anybody is going to talk approximately how unimportant you're!, we could particularly desire this form in WWE on no longer giving divas popularity like adult men, wears out speedy.

Answer 5

Yes aand this includes irrational numbers such as pi and sqrt(2). Since .999 goes on infinitely you would never come to the end and be able to put another number there.

Answer 6

0.99999... is equal to 1, so there is no real number greater than 0.99999... but less than one. (believe me!)

Hope that helps!