Final answer:
The square root of 2 can be manually calculated using the long division method, by starting with an initial guess and iteratively improving that guess through averaging. Eventually, this results in an approximation of the square root of 2 to three decimal places, which is approximately 1.414.
Step-by-step explanation:
To calculate the square root of 2 manually with three decimal places, we can use the long division method also known as the Babylonian method or Heron's method:
- Start with an initial guess, say 1, because the actual square root has to be more than 1.
- Divide the number for which you want to find the square root (2) by the guess (1). This gives you 2/1 = 2.
- Average this result with the guess: (2 + 1) / 2 = 1.5. This is your new guess.
- Repeat this process using the new guess: Divide 2 by 1.5, which gives us 1.3333.
- Again, take the average with the previous guess: (1.5 + 1.3333) / 2 = 1.41665.
- Continue this iterative process until the result is accurate to three decimal places. After a few more iterations, you will find that the square root of 2 is approximately 1.414.
This method will give you a more and more accurate result with each iteration.