Some have difficulty understanding why .999... = 1 . The main reason seems to be related to the use of .999... by mathematicians to represent 2 different ideas . These 2 ideas are one finite and the other infinite , so the difference is the difference between the finite and the infinite . Sometimes .999... is used to mean a finite quantity close to 1 . I.e. , .999... suggests writing , or equivalently adding , 9s to some point , perhaps far to the right , and then stopping . E.g. , perhaps .99999999999999999999 . It is hard to conceive of any ordinary situation , apparatus or construction where any practical difference between .99999999999999999999 and 1 would arise . Usually , mathematicians write .999... with a different meaning . The construction for this meaning is the same as for the first meaning , except that the last step is skipped , and there is no stopping . No matter how far to the right we look , we always find a 9 there . There are an infinite number of 9s . If you don't see the difference between these 2 constructions , then you have not yet "bought" into the concept of the infinite , or infinity . The product of .99999999999999999999 and 10 is 9.9999999999999999999 . The product of .9999999999999999999 and 10 is 9.999999999999999999 . Note that , just as .99999999999999999999 is not = .9999999999999999999 , neither is 9.9999999999999999999 = 9.999999999999999999 . Mathematicians like to think very carefully about the exact meaning of the following calculation , but you can see it's a powerful argument . The product of .999... and 10 is 9.999... . After multiplying by 10 , the infinite number of 9s is still there to the right of the decimal point . Every distant decimal place still has the same digit , namely , 9 . The product of x and 10 is 10 * x = 10x = 9x + x . Let x = .999... . Then , 10x = 9.999... . Then , 9x + .999... = 9.999... . Then , 9x + .999... - .999... = 9.999... - .999... . Then , 9x = 9 . Then , x = 1 . Thus , .999... = 1 . Walter Nissen 2008-04-11