Why 0.9999999… is Equal To 1

One of the most debated topics between mathematicians is whether 0.999999999… (where the 9s go on forever) and 1 are the same. The reason for such a debate is due to the fact that whenever a number less than 9 is divided by 9 it becomes a decimal where the number keeps repeating forever. What I mean, for example is when we have the fraction 1/9 it equals 0.111111111….. and 2/9 = 0.2222222222…. and 3/9 (which simplifies to 1/3) = 0.33333333333 and 4/9 = 0.444444444444444… and 5/9 = 0.555555555555… and 6/9 (which simplifies to 2/3) is 0.6666666666… and 7/9 = 0.77777777777… and 8/9 = 0.8888888888… but when we reach the expression 9/9, it becomes confusing all of a sudden. 9/9 = 1 and not 0.9999999999…! This is what drives some people to say 0.999999999… and 1 are the same, and others are in a dilemma and not ready to accept the fact. This is what causes the huge debate between mathematicians. We will see another example where the same thing happens. Suppose we have a variable x which equals 0.99999…. and we multiply it by 10 to get 10x = 9.99999… and in this method our goal is to cancel out the repeating decimal and end up with a whole number so we subtract x which equals 0.999999… from 10x to get a result for s again. This method is actually used to find fractional equivalents of repeating decimals. The diagram below shows what happens when x is subtracted 10x.

x = 0.99999999…

     10x = 9.99999999….

      10x = 9.9999999…

     -x = 0.9999999..

      9x = 9

x = 1

The original value for x was 0.9999999…. but now we got the result of x = 1. This is not supposed to happen! This proves that 1 and 0.9999999… are in fact the same number, just written in a different way. Many of the readers still might not be convinced, so the reasoning for the claim that both the given numbers (0.9999999 and 1) are equal is the fact that if we decide to subtract 0.999999999… from 1 we will end up with none other than 0 because common sense tells us that 1 – 0.99999999.. has to be 0.0000000… where the zeroes repeat infinitely and at the end would be a 1. But, there cannot be a number at the end of an infinite series of numbers because an infinite sequence has no end! So it wouldn’t end up being an extremely minute infinitesimal, it would be 0.

That is all for today, thank you for reading!

Vishrut