dear math teacher,
i am algebra II and am in the 9th grade. today we were talking about rational and irrational numbers. When we were talking about square roots my friend and i were talking and we thought of something. if you have a square with sides of length one then the diagonal of the square is the square root of 2. Now the square root of two is never supposed to end. But the diagonal of the square ends so therefore doesn't the square root of 2 end. our math teacher did not really answer our question because it was not in the lesson plan and not to many people would see where we were coming from. the answer is really bugging me and i would like to have your input.

a wondering student

Hi

It is great that you and your friend talk about math problems. Many times we learn more by discussing problems with our friends then from our teachers. If you continue this approach to mathematics you will do better and better and have a ball!
   You are trying to represent or describe the number "the square root of 2" in two different ways. One way is to express it as a decimal, the other as "the length of the diagonal of a square with sides of length one". Both of these are valid ways to represent the square root of two. The confusion comes from your statement "the square root of two is never supposed to end". The correct statement is that if you represent the square root of two in decimal form then the non-zero digits after the decimal never end.
   You have seen something like this before. If you have a line that is 1 unit long, divide it into 3 equal parts and write the length of one of the parts in decimal form then the non-zero digits after the decimal never end. In this case you can write the length as 1/3 or 0.3333... with the digit 3 continuing forever. In the case of the square root of two there is the additional fact that the decimal part has no repeating pattern as it does for 1/3. This is what makes the square root of two irrational.

Cheers,
Jack and Penny