# Approximating Square Root of Numbers by Averaging

There are several algorithms developed to manually calculate the square root of a number. My favorite is the method of averaging, which is probably the least complicated.  In this method, the number is divided by the nearest square root estimate, and then the quotient and the square root estimate are averaged. The result is the new estimate.

Sounds complicated? Not really.

Example: Approximate $\sqrt{24}$.

Note: Using a calculator, $\sqrt{24} \approx 4.8989794855$.

Solution

We know that $\sqrt{24}$ is between $4^2$ and $5^2$. Between the two, it is nearer to $5^2$. So, the first estimate is $5$. Next, we divide $24$ by the first estimate and “average” it with the quotient .

Second Estimate

Divide: $\frac{24}{5} = 4.8$.

Average: $\frac{4.8 + 5}{2} = 4.9$

Second estimate: $4.9$ (2 significant digits)

Third Estimate

Divide: $\frac{24}{4.9} = 4.898$

Average: $\frac{4.9 + 4.897}{2} = 4.899$

Third estimate: $4.899$ (4 significant digits)

Fourth Estimate

Divide: $\frac{24}{4.899} = 4.8989589$

Average: $\frac{4.899 + 4.8989589}{2}$

Fourth estimate: $4.8989794$ (8 significant digits)

Notice that the third estimate is already correct up to 8 significant digits which means that this method is quite accurate.  In the fifth, sixth,   and seventh estimates, 16, 32 and 64 digits should be kept respectively.  Of course, repeating the process over and over will obtain more accurate approximations.

Update: Don Cohen has a short Mathematica code for this method in his site.

Reference: Modern Elementary Mathematics

## 5 comments on “Approximating Square Root of Numbers by Averaging”

1. Guillermo,

The interesting thing about this method is that it is an example of iterating a function and finding infinite sequences. You can see this method using Mathematica, on my site at http://www.mathman.biz/html/sqrtbyav&iter.html with various links to other ways to use iteration to find infinite sequences with young people, and solving equations (Peter Farrell on Math 2.0 mentioned Newton’s method using iteration as another way to solve quadratic equations). This file is also on my A Map To Calculus at Pi and sqrts.

Don

• Guillermo Bautista on said:

Thank you for the information Don. I updated the blog and mentioned it above.

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