How is .9 repeating equal to 1? - Answers

We can show this in many ways. Here are 4 different ways.1.Let's say we do not know what .9999999999999 repeating for ever equals. I have no way to put the bar over the 9 here so I am going to write .9... to mean .9 repeating foreverNow let's call whatever it equals x. So we have x=.9...Since this is an equation, we can multiply both sides by 10 and we have10x=9.9...We are allowed to subtract the same thing from both sides of an equations.Since x=.9...we subtract x from the left side and .9... from the right side. We can do this because they are the same thing. That is how we defined x, in fact. Now we have10x-x=9.9...-(.9...)which tells usx=1This is because the 1-x-9x=1x and the 9.9...-(.9...)=12.Here is another way to think of it that is easier for some people.We accept and know that 1/3 is .333 repeating forever, or as we write it .3...Now 1/3 added to itself 3 times of 1/3 x3 is 3/3. Any number divided by itself is 1. This tells us .3...+.3...+.3...=.9... but we know that .3... is 1/3 and 1/3+1/3+1/3=3/3=1. We conclude the .9...=13.Here is a slightly more advanced but very pretty explanation that uses infinite geometric series.Find the exact value of a repeating decimal, asks the same question as "find the sum of an infinite geometric series":Think of .9... and 9/10+9/100+9/1000 etc.This is better written with exponents so .9...=9x10-1 +9x10-2 +9x10-3 +...This is an infinite geometric series and there is a convergence rule that tells us how to find the sum. Since the common ratio here is between -1 and 1 or |r| < 1, we know the series converges.In our case the common ratio, r=1/10. The first number in the series is a= 9/10.We know that the finite sum is given by the formula, S=a(1-rn)/(1-r). If -1