How does a calculator compute the square root of a number?

The purpose of this unit is to give an example of the algorithm that is executed inside a TI:83/84 calculator when he user punches the square root button.
Obviously the square root of every number is not stored in the calculator; it must be computed.

(The example below uses a TI-83/84 calculator.)

Let’s start with an example. Let the number be 35467. It has to be stored somewhere, so let’s store it in N:
35467→N
Next an initial guess has to be made. It may be a stupid guess such as 1, or 35000, but better guesses give the answer faster. (Inside the calculator, the guess is always
computed in some way. For example, if the number is N, the guess could be N/2.) So let’s guess 200 and store it in X:
200→X
Now, let’s form the second guess, N/X.
Notice that X * N/X = N. So, if one of the two guesses, X or N/X, is too small, the other one is too big. Thus when we take their average, (X + N/X)/2, as our next guess, we improve both of them.
We can repeat this process by storing this average back in X,
(X + N/X)→X. Repeat ENTER.

We can implement this method on the TI-84 calculator as follows:

And then:

Next

We continue to press ENTER, and we get this sequence of approximations:

When we get two approximations that are equal we stop.
We cannot get better accuracy on this calculator.
We then check whether the answer is correct by tracking the square root of N:

We are done!
Now its your turn!

1. Find the square root of 1789.34 using the technique above. Show your work.
Your first guess__________________
The sequence of approximations on the TI-84:

Now check that your answer is correct: (1789.34) =
2. Now find the square root of a number of your choice, using the same technique:
Number:
Your first guess:
Sequence of approximations:

Check: