Exercises: Square Roots

The method of nested intervals is a way to approximately compute square roots.

As a first step, you have to find an interval in which the value of $\sqrt{7}$ is definitely included. For this, it is useful to know the nearest smaller and larger square numbers.

Step 1

You already know the values of:

$\sqrt{4}$ and $\sqrt{9}$

This gives you the following estimates:

$\begin{array}{ccccc}{2}^2& <& 7& <& 3^2\\ 4& <& 7& <& 9\end{array}$

Taking the square root, this implies:

So the value of $\sqrt{7}$ is contained in the interval:

The next steps always follow the same algorithm:

First you find the mean value of the old interval boundaries. This mean value will become one of the new interval boundaries. The other new interval boundary will be taken from the old interval.

In order to decide whether the new value is the upper or lower interval boundary, you proceed as follows: Take the square of the mean value and compare it with 7.

• If its square is greater than 7, then the mean value becomes the new upper boundary.

• If its square is greater than 7, then the mean value becomes the new lower boundary.

Step 2

Calculate the mean value of the interval boundaries of $I_0$.

${\displaystyle \frac{2+3}{2}}=2.5$

Calculate the square of $2.5$ and compare it with $7$.

${2.5}^2=6.25$

$6.25<7<9$

${2.5}^2<7<3^2$

Now define the new interval.

Step 3

$I_1=(2.5,3)$

Calculate the mean value of the interval boundaries of $I_1$.

Calculate the square of this mean value and compare it with $7$.

$\begin{array}{ccccc}6.25& <& 7& <& 7.56\\ {2.5}^2& <& 7& <& {2.75}^2\end{array}$

Now define the new interval.

Step 4

Calculate the mean value of the interval boundaries of $I_2$.

Calculate the square of $2.625$ and compare it with $7$.

$\begin{array}{ccccc}6.89& <& 7& <& 7.56\\ {2.625}^2& <& 7& <& {2.75}^2\end{array}$

Now define the new interval.

Step 5

Calculate the mean value of the interval boundaries of $I_3$.

Calculate the square of this mean value and compare it with $7$.

Now define the new interval.

After these five steps, we know that

So the value of $\sqrt{7}$ is between $2.625$ and $2.6875$.

Final Estimate

Depending on what precision you need, you can repeat the procedure as often as you need Here, the problem setting tells us to terminate after 5 steps. So the final interval where you locate $\sqrt{7}$ is $I_4=(2.625,2.6875)$. A good estimate for $\sqrt{7}$ is now given by the "middle of the final interval", i.e., the mean value of its boundaries:

So the final estimate for $\sqrt{7}$ is $2.65625$.

For comparison: The precise value of $\sqrt{7}$ is $2.64575\dots$