Let’s look at some properties of the square root of two. Well, firstly, we know it as an exact length that we can measure on a number line. We can do this with the help of a compass. I’ve drawn our right-angle triangle on a number line, and then placed my compass along the length of the hypotenuse, which is the square root of two, and then found that exact same measure on the number line. Because the square root of two is an exact length, it is said to be a real number.
A real number is any number that can be located as an exact position on a number line in the negative or positive direction. We know the square root of two is an exact length, but what happens when you try to find the value of it on your calculator? When I did, I got the square root of 2 equals 1.414213562, because my calculator only goes to 9 decimal places. When I looked up the square root of 2 on my computer’s calculator, I got an answer to 31 decimal places.
What can you see is happening with the decimal answer to the square root of two? It doesn’t appear to have an end to it, does it? No matter what the accuracy of my calculator, there is no end to the decimal. So we say that the answer to the square root of two is a nonterminating decimal, or infinite decimal.