0.9 newspaper -lrb- 0,999 … , where the dígito 9 is repeated indefinitely -rrb- also writing as 0 , \ bar9 , 0 , \ dot9 or 0 , -lrb- 9 -rrb- , is a real number equal to 1 . that is , the notaciones ' 0,999 ... ' and ' 1 ' represent the same number real . although apparently contradictory , that equality is valid and can provide demonstrations with different degrees of rigour and has its origin in that representation decimalization of a real number is not necessarily only . the fact that there are spans real as 0,999 ... is not limited to decimalisation . the same phenomenon in all bases whole and the mathematical have also quantified modes of write 1 in basis . even this is an isolated phenomenon to 1 : all the decimal point terminals various zero of a real number have a twin with infinite new , for example 28,3287 and 28,3286999 … . arguments not formal then debated here three arguments non-formal to illustrate this and a demonstration mathematical that proves it . 0.333 ... in 160 ; = 1 ⁄ 3 3 × 0.333 ... in 160 ; = 3 × 1 ⁄ 3 0.999 ... in 160 ; = 1 multiplication of 1/3 given that 1/3 = 0,333 … causing 3:3 × -lrb- 1/3 -rrb- = 3 × 0,333 … = 0,999 … but then 0,999 ... must inevitably be 1 , since -lrb- 1/3 -rrb- × 3 = 1 . x - 160 ; = 0,999 … 10x in 160 ; = 9,999 … 10x − x - 160 ; = 9,999 … − 0,999 … 9x in 160 ; = 9 x - 160 ; = 1 with x = 0,999 … denoting by x the number 0,999 … : x = 0,999 … is multiplied by 10 the two figures : 10x = 9,999 … undermining both phrases : 10 x - x = 9,999 … - 0,999 … it concludes that 9x = 9 , is to say , x = 1 . 0 , xxx ... in 160 ; = x ⁄ 9 0,999 ... in 160 ; = 9 ⁄ 9 0,999 in 160 ; = 1 with mathematical formula if x is a number whole between 0 and 8 , it ascertains formula 0 , xxx \ ldots = \ frac -lcb- x -rcb- -lcb- 9 -rcb- taking the numerical value of x as 9 we should obtain that : 0,999 \ ldots = \ frac -lcb- 9 -rcb- -lcb- 9 -rcb- = 1 there is no real number between 0,999 … and 1 from the fact : if two numbers real are different , then there is at least a third between the two , different . if 0,999 … and 1 were different , would be possible to find between them , for example , the average arithmetic of the two . however , it is impossible to arrange any number between 0,999 … and 1 , and therefore must be equal . demonstration formal the next exhibition is a test to detail , but because 0,999 ... is a sum with a number ad infinitum of terms , his rigorous evaluation requires necessary form a concept of calculation infinitesimal : limit . by definition of representation decimalization : that is a number debt -lrb- of first term to = 9/10 and reason k = 1/10 -rrb- : in this way , we need calculate the value of the number debt from the factorización needs for x \ ne 1 and in particular , when / x / < 1 , the term x ^ -lcb- n + 1 -rcb- tends to zero , situation in which the number debt can assess : going back to 0,999 … are that the number debt with reason 1/10 is assessed : and therefore as i wanted to demonstrate . a demonstration similar -lrb- in fact that 10 amounts to 9,999 ... -rrb- is already emerging in the year 1770 in the book elements of algebra -lrb- vollständige anleitung zur algebra -rrb- of leonhard euler . generalisation the proof that 0,9999 \ ldots in basis 10 is exactly 1 , can be generalise for any basis does not necessarily 10 . on the basis n + 1 the number 0 , nnnnnnn \ ldots is exactly 1 . can verify that \ sunt _ -lcb- k = 1 -rcb- ^ -lcb- m -rcb- to ^ -lcb- k -rcb- = \ frac -lcb- to ^ -lcb- m + 1 -rcb- - -rcb- -lcb- a-1 -rcb- then \ begin -lcb- array -rcb- -lcb- rcl -rcb- 0 , nnn \ ldots & = & n \ left -lrb- n + 1 \ right -rrb- ^ -lcb- -1 -rcb- + n \ left -lrb- n + 1 \ right -rrb- ^ -lcb- -2 -rcb- + n \ left -lrb- n + 1 \ right -rrb- ^ -lcb- -3 -rcb- + \ cdots \ \ & = & \ displaystyle \ lim _ -lcb- m \ rightarrow \ infty -rcb- \ sunt _ -lcb- k = 1 -rcb- ^ -lcb- m -rcb- n \ left -lrb- m + 1 \ right -rrb- ^ -lcb- -k -rcb- \ \ & = & \ displaystyle \ lim _ -lcb- m \ rightarrow \ infty -rcb- n \ sunt _ -lcb- k = 1 -rcb- ^ -lcb- m -rcb- \ left -lrb- \ frac -lcb- 1 -rcb- -lcb- m + 1 -rcb- \ right -rrb- ^ -lcb- k -rcb- \ \ & = & \ displaystyle \ lim _ -lcb- m \ rightarrow \ infty -rcb- n \ frac -lcb- \ left -lrb- \ frac -lcb- 1 -rcb- -lcb- n + 1 -rcb- \ right -rrb- ^ -lcb- m + 1 -rcb- - \ left -lrb- \ frac -lcb- 1 -rcb- -lcb- n + 1 -rcb- \ right -rrb- -rcb- -lcb- \ left -lrb- \ frac -lcb- 1 -rcb- -lcb- n + 1 -rcb- \ right -rrb- -1 -rcb- \ \ & = & \ displaystyle \ lim _ -lcb- m \ rightarrow \ infty -rcb- 1 - \ frac -lcb- 1 -rcb- -lcb- \ left -lrb- n + 1 \ right -rrb- ^ -lcb- m -rcb- -rcb- \ \ & = & 1 \ end -lcb- array -rcb- it is to say that in binario 0,111 \ ldots = 1 , in octal 0,777 \ ldots = 1 , and in decimalisation 0,999 \ ldots = 1 , etc . see also one number decimalization newspaper references links external numbers
