Does 0.3333... = 1/3?
But, when we change the question and ask:
Does 0.9999... = 1?
In fact, last semester, in a College Algebra class that I was teaching, we took at vote. The post-it's from the votes are below.
The voting results were similar this semester with a different group of students - adults taking a math for elementary education class.
I began wondering: Why is this so challenging of a topic?
Why is it that people can believe 0.3333...= 1/3, but feel comfortable when I say 0.9999...= 1?
I think one reason it is a challenging topic is that we do not talk about it enough. We talk about how 1/2 = 0.5, 1/4 = 0.25, 1/3 = 0.3333...., 2/3 = 0.6666.... Yet, we spend little time or no time in elementary, middle, or high school talking about how 0.9999.... =1.
I think a second reasoning this topic is challenging is that our proofs used at the elementary level can be quite unconvincing.
One proof that students often present is: If 1/3 = 0.3333... and we multiply both sides of the equation by 3 we obtain, 3 * 1/3 = 3 * 0.3333... which is equivalent to 1 = 0.9999... Students will often support this type of proof with a circle graph where they show a circle divided into three sectors and each sector labeled 1/3 and 0.3333... Each of these 3 sectors add up to 1 and 0.9999.... Although I think this is great when a student creates this, often the response from other students is, "Ok, I see what is happening here. But, now I do not believe that 1/3 = 0.3333...." And, there is some truth embedded there. This proof is completely contingent upon the assumption that 1/3 does in fact equal 0.3333.... This proof is thus only convincing if 1/3 = 0.3333... is believed or proven first.
A second proof that students often find on the internet or in a book is an algebraic one. This proof I often call "mathemagic" because it is not a proof that is intuitive the the students and not one that I have seen the students invents themselves in class.