# Answer to Question #1350 in Abstract Algebra for saim

Question #1350

Show that [0,1] is uncountable?

Expert's answer

To prove that some interval is countable, we have to associate every number in the interval with other countable set, that is natural numbers. Any number

same number as 0.15000.

Let's denote any decimal number as shown below and associate it with natural numbers:

[1] -> 0.a

[2] -> 0.b

[3] -> 0.c

.......

[n] -> ///

etc. where a

In such case we cannot say that the interval (0,1) is countable, so [0,1] is not countable too.

*x*in the interval (0, 1) can be expressed as a unique, never-ending decimal, but this is not quite true: 0.1499999... is thesame number as 0.15000.

Let's denote any decimal number as shown below and associate it with natural numbers:

[1] -> 0.a

_{1}a_{2}a_{3}....[2] -> 0.b

_{1}b_{2}b_{3}....[3] -> 0.c

_{1}c_{2}c_{3}..........

[n] -> ///

etc. where a

_{1}..., b_{1}..., c_{1}... - are digits and construct number: 0.X_{1}X_{2}X_{3}... using the following rules: X_{1}is not equal to 0,9 and a_{1}, X_{2}is not equal to 0, 9 and b_{2}, X_{3}is not equal to 0, 9 and c_{3}and so on. Thus we obtain the new number which is not in our associated set.In such case we cannot say that the interval (0,1) is countable, so [0,1] is not countable too.

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