Hi, I am in Honors Math, and have confronted everyone, including teachers, about repeating decimals. What interests me is the number 0.9... and 1. Everyone says that since there is no number between 0.9...(repeating) and 1, that 0.9... = 1. However, isn't a repeating number a representation of a number, and not a real number? Let's look at it this way. 0.9 is close to 1. 0.99 is closer. 0.99999999999999 is even closer. so, 0.9... is a representation of it's closeness to 1. it's an active number... I don't understand how 0.9... is equal to 1. Please help me prove that 0.9... does NOT = 1.

Hi Stan,

You are right when you say that a repeating decimal is a representation of a number, and not a real number. There are different ways to represent numbers, for example 1/2, 0.5 and 0.5000... are three ways to represent the same number. 1 and 0.999... are two ways to represent the number one. Here is one way to show this fact.