Why 0.9999… equal to 1

Recently, I am trying to understand better about calculus and limit. And there is a famous debate or question about how could 0.9999… is exactly equal to 1, or in other words, 0.9999… and 1 is the same thing.

It is a bit counter intuition, but still understandable.

Solution #1: Analogy

1/9 = 0.1111…

2/9 = 0.2222…

3/9 = 0.3333…

…

8/9 = 0.8888…

follow the same pattern

9/9 = 0.9999… = 1

Solution #2: Algebra

let x = 0.9999…, then 10x = 9.9999…

10x -x= 9x = 9.9999…- 0.9999…=9

so, 9x = 9 , and it is easy to see that x = 1. so that we know 0.9999… is equal to 1.

Solution #3: Reasoning

For number A and B, if and only if that we can’t find another number between A and B, we say A equal to B. To understand 0.9999… is equal to 1. let’s try to find a number x that is between these two numbers.

Assume that there is a number x that meet 0.9999… <x <1.

in this case assume x = 0.999999. to make sure x> 0.9999… . x need more 9 in the end. However, no matter how many “9” x have in its number, it always less than 0.9999… . in other words. we can’t find a number x between 0.9999… and 1.

So we say 0.9999… equal to 1.