Remember that the Babylonian method for finding square roots is divided in three steps:
Step 1:
- Make a reasonable guest. Since [tex] \sqrt{100} =10[/tex], our guest is that [tex] \sqrt{103} =10[/tex].
- Divide the original number by your guess. So, [tex] \frac{103}{10} =10.3[/tex].
- Find the average of those numbers. So,
[tex] \frac{10+10.3}{2} [/tex]
[tex] \frac{20.3}{2} [/tex]
[tex]10.15[/tex]
- Use that number as your next guess.
Step 2: Repeat step 1, but using the average, 10.15, as your next guess:
- [tex] \frac{103}{10.15} =10.14778325[/tex]
- [tex] \frac{10+10.14778325}{2} [/tex]
[tex] \frac{20.14778325}{2} [/tex]
[tex]10.07389163[/tex]
Step 3: repeat the process one last time, but using the average, 10.07389163, as your next guess:
- [tex] \frac{103}{10.07389163} =10.22444987[/tex]
- [tex] \frac{10+10.22444987}{2} [/tex]
[tex] \frac{20.22444987}{2} [/tex]
[tex]10.11222494[/tex]
We can conclude that [tex] \sqrt{103} [/tex], using the Babylonian method, is equal to 10.11222494
