Is it possible to find square root using only rational numbers and elementary arithmetic operators

Question

Suppose I have a number a

How can I find it's square root using only +, -, /, * and rational numbers?

If it is impossible how to prove it?

Answer 1

The set of rational numbers is closed under the elementary arithmetic operations (except for division by zero), i.e. if you have two rational numbers and take their sum, difference, product, or quotient, then the result is again a rational number (again, except for division by zero). Since $\sqrt{a}$ can fail to be a rational number (say, $\sqrt{2}$), there is no formula for $\sqrt{a}$ involving only elementary arithmetic operations.

Proof link

Answer 2

When you use an infinite ammount of operations, you can use:

$$ \!\ \sqrt{2} = 1 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \cfrac{1}{2 + \ddots}}}}. $$ from here...

Answer 3

If you allow infinite number of operations, then you can use some algorithm.

One easy example is root searching via Newton's method. Here we do the iteration $$x_{n+1} = \frac{a + x_n^2}{2x_n},$$ which eventually converges to $\sqrt{a}$ if $a$ and $x_0$ are positive.

See https://en.wikipedia.org/wiki/Methods_of_computing_square_roots for other methods.