Let $x=0.99999 \ldots$.
(a) Do you think that $x<1$ or $x=1$ ?
(b) Sum a geometric series to find the value of $x$.
(c) How many decimal representations does the number 1 have?
(d) Which numbers have more than one decimal representation?
To check: Whether the value of x is less than 1 or equal to 1 if x=0.99999....
The value of x is x=1.
Multiply both sides by 10.
Substitute x for 0.99999... in the above equation,
Divide both sides of the equation by 9.
Therefore, the value of x is equal to 1. That is, x=1.
Note: The value of x is not less than one (x<1) because x consists of an infinite number of 9s.
To find: The value of x.
The value of x=0.99999....
The sum of the geometric series ∑n=1∞arn−1 (or) a+ar+ar2+⋯ is a1−r, where a is the first term of the series and r is the common ratio of the series.
Consider, x=0.99999... and it can be written as,
Clearly, it is geometric series with the first term of the series is a=910 and the common ratio of the series is
Use the Result stated above, the sum of the series is
Therefore, the value of x is x=1.
To describe: The decimal representations of the number 1.
The number 1 has two decimal representations.
The decimal representations of the number 1 are 0.9999… and 1.0000….
To describe: The numbers have more than one decimal representation.
All terminating decimals other than zero have more than one decimal representation.
1. The number 2 can be written as 1.9999⋯ and 2.0000⋯.
2. The number 5.245 can be written as 5.245000⋯ and 5.244999⋯.