does the long division method gives the exact value of square root of any non perfect square natural number?

Question

does long division method or the other manual algorithms that are used to calculate square root of any number , gives the exact value of square root of any number OR they just approximate it as accurately as they can ? Please guide me.

Answer 1

Any numeric method will give an approximation to the square root. As the decimal expansion goes on forever, you can't write down an exact value. Generally you can go as long as you want to get a value that is as accurate as you want, but you can't get the exact value. There is no way to write the number that is $\sqrt 2$ exactly unless you just write $\sqrt 2$ or some equivalent expression. Any of the numeric methods will give you something like $1.414213$, which is not exact.