Does anyone understand why 1 is greater in value than 0.9…. ? ; the three dots represent the digit 9 repeating indefinitely.

Take two numbers and expand them to a function: ⅓ to 3/9, 33/99, 333/999, 3…/9…; etc. ; and expand the 0.3… to 3/10, 33/100, 333/1000, 3…/10…; etc. A rational person recognizes the difference between these numbers. Do the same for 1 and 0.9… : 1 to 10/10, 100/100, 1000/1000, 10…/10…; etc.; compared to 9/10, 99/100, 999/1000, 9…/10…; etc. ; Do you understand the difference between these numbers ?

The series 9/10 + 9/100 + 9/1000 +…. ; evaluated: 9/10 + 9/100 = 99/100; less than 100/100; a value of 1; adding 9/1000 to 99/100 is 999/1000; a value less than 1000/1000; a value of 1 and this will continue forever; the value will always be less than one.

For those in in elementary school:

1/3 is 1 divided by 3; or:

3 divided into 1 using the division process and confirmation (long division):

1 = 10/10;

3 divided into 10/10 = 3 + a remainder;

confirmed: 3 times 3/10 resulting in 9/10; not 10/10;

the remainder is 1/10 written as a 1 so you can add a 0 to divide 3 into 10;

and for all eternity, 3 times 3/10 will always be 9/10; never 10/10;

therefore always a remainder. That remainder needs to be added to 0.3… repeating to equal ⅓. Proving that ⅓ is not equal to 0.3… ; therefore a value less than 1/3 times 3 is less than 1 and 0.9… is not equal to 1.