0.999...

0.999... is a recurring decimal with periodic '9'

representing an infinite string approaching 1.0

The number is assumed to be equal to 1, by using proofs

that ignore existence of infinitesimal numbers (

the actual difference is 1-0.999...=0.(...)1 or 1/10^n where n is equal to number of '9's in in 0.999... at any place).

Example of invalid math proofs that posit exact equality between 1 and 0.999...

Proofs with fractions(e.g. 1/3):

1/n = 0.xxx....

(1/n)*n=1= ((0.xxx...)*n)=0.999....

the proof assumes the fraction 1/n is exactly

equal to a recurring decimal form 0.xxx...,

which is in fact the process of division always leaves a remainder at every step, with such remainders resulting at 1/10^n difference at every n'th place of such proof:

1/3=0.333

(1/3)*3=1 != 0.333*3=0.999

Proofs by multiplication are similarly flawed:

x=0.111(n places)

10x=1.11(n-1 places)

9x=1.11-0.111=0.999 (with number of places n-1 <n)

Proofs by geometric series:

Proofs use the formula that discards size of series as opposed to finite formula, that can be extended to any

n places, showing a clear difference(simplified to a/1-r from a/(1-) -(a^(r*(n+1))/(1-r) the latter being the infinitesimal number discarded in the proof)

https://en.wikipedia.org/wiki/Geometric_progression#Infinite_geometric_series

The 'Archimedean property' employed in the definition

of real number set is the axiomatic flaw that lies behind

most other complex proofs, which assume a priori that

infinitesimals cannot exist and use to prove that

1-0.999...=1/10^n =0 which is circular reasoning. https://en.wikipedia.org/wiki/0.999...#Proofs_from_the_construction_of_the_real_numbers