What is the best way to do X^2= 618

Question

What would be the best and easiest way to do X^2 = 618 (Without a calculator) Wouldn't it just be considered irrational (Solving for x)

Answer 1

You want to find the positive zero of function $$f(x)=x^2-618$$ Notice that $f(25)=7$ and the $f''(x) > ~~ \forall x$. Then, by Darboux theorem, Newton method will converge without overshooting the solution at any time.

So, starting with $x_0=25$, perform the iterations $$x_{n+1}= x_n-\frac{x_n^2-618}{2 x_n}$$ This would generate the sequence $$\left\{25,\frac{1243}{50},\frac{3090049}{124300}\right\}$$ Notice that $$\frac{1243}{50}=24.86 \qquad \text{while} \qquad \sqrt{618}=24.8596\cdots$$

I do not see the problem to get the first iterate by hand.

Sooner or later, you will learn about methods of higher order. Fo example, Halley method would give for its first iteration $$\frac{61975}{2493}=24.8596069\qquad \text{while} \qquad \sqrt{618}=24.8596058$$ I made it by hand.