Question

3. Use the Babylonian method to approximate square root 19 to the nearest hundredth

Answer 1

The Babylonian method for computing a square root involves refining an approximation to the root. If our first guess is \(x_0\), and we are computing the square root of \(x\), then we take the average of \(x_0\) and \(x/x_0\) and make that \(x_1\), our next guess. Repeat as many times as necessary to get the precision you want. We want the nearest hundredth, so as soon as our next guess is identical to the current guess through the hundredths place, we're done.

Answer 2

Reasonable starting point might be \(x_0 = \sqrt{16} = 4\). You'll get to your answer in 2 or 3 steps.

Answer 3

I kind of understand, but what do I do with the 4? do I come up with another guess and find the average?

Answer 4

Okay, if \(x_0 = 4\) is our first guess, our next guess is \[x_1 = \frac{1}{2}(x_0+\frac{19}{x_0}) = \frac{1}{2}(4+\frac{19}{4}) =4.375\]

Answer 5

Now we plug that in and turn the crank:
\[x_2 = \frac{1}{2}(x_1 + \frac{19}{x_1}) = \frac{1}{2}(4.375 + \frac{19}{4.375}) \approx 4.35893\]

Now find \(x_3\) by the same procedure...

Answer 6

oh, ok I understand, thank you so much for the help.