Question

Using the Babylonian method find √600 ???????????? √235. how?

Answer 1

Answer:

With the Babylonian method $\sqrt{600} \approx 24.494897$ and $\sqrt{235} \approx 15.329709$.

Step-by-step explanation:

• To find the square root of 600, do the following:

1. Make an initial guess:  Because $\sqrt{576}=24$ and $\sqrt{625} =25$ you can start with $x_{0}=24$ as your initial guess.

2. Apply the formula:

$x_{1}=\frac{(x_{0}+\frac{x}{x_{0}})}{2}$ where $x=600$

$x_{1}=\frac{(24+\frac{600}{24})}{2}\\x_{1}=24.5$

The number $x_{1}$ is a better approximation to $\sqrt{600}$

3. Iterate until convergence:

For this apply the formula:

$x_{n+1}=\frac{(x_{n}+\frac{x}{x_{n}})}{2}$

Convergence is achieved when the digits of $x_{n+1}$ and $x_{n}$ agree to as many decimal places as you desire.

$x_{2}=\frac{(x_{1}+\frac{x}{x_{1}})}{2}\\x_{2}=\frac{(24.5+\frac{600}{24.5})}{2}\\x_{2}=24.494897$

$x_{3}=\frac{(x_{2}+\frac{x}{x_{2}})}{2}\\x_{3}=\frac{(24.494897+\frac{600}{24.494897})}{2}\\x_{3}=24.494897$

Because $x_{2}$ and $x_{3}$ agree to six decimal places. we can say that an estimate for $\sqrt{600} \approx 24.494897$.

You can compare with the value that WolframAlpha gives you which is $\sqrt{600} \approx 24.49489742$ you can see that it agrees to six decimal places.

• To find the square root of 235, do the following:

1. Make an initial guess:  Because $\sqrt{225}=15$ and $\sqrt{256} =16$ you can start with $x_{0}=15$ as your initial guess.

2. Apply the formula:

$x_{1}=\frac{(x_{0}+\frac{x}{x_{0}})}{2}$ where $x=235$

$x_{1}=\frac{(15+\frac{235}{15})}{2}\\x_{1}=\frac{46}{3}$

3. Iterate until convergence:

Apply the formula:

$x_{n+1}=\frac{(x_{n}+\frac{x}{x_{n}})}{2}$

$x_{2}=\frac{(x_{1}+\frac{235}{x_{1}})}{2}\\x_{2}=\frac{(\frac{46}{3}+\frac{235}{\frac{46}{3}})}{2} \\x_{2}=15.329710$

$x_{3}=\frac{(x_{2}+\frac{235}{x_{2}})}{2}\\x_{3}=\frac{(15.329710+\frac{235}{15.329710})}{2} \\x_{3}=15.329709$

$x_{4}=\frac{(x_{3}+\frac{235}{x_{3}})}{2}\\x_{4}=\frac{(15.329709+\frac{235}{15.329709})}{2} \\x_{3}=15.329709$

Because $x_{3}$ and $x_{4}$ agree to six decimal places. We can say that an estimate for $\sqrt{235} \approx 15.329709$.

WolframAlpha gives you $\sqrt{235} \approx 15.329709716$