Herons Method of Finding Roots

A guest post by Nandeesh H.N. of Kolkata

"Dear Vineet,

I am really grateful to you for your blog which makes Mathematics a pleasure. Keep up your good work. As requested by you I am sending a brief note on Heron’s method of finding square root. This method can be easily extended to find any root.

Heron’s method of finding square root

If n is the square root of N, obviously, dividing N by n gives n.

We can look at the square root as the average of the factor and the quotient.

If you divide N by a number x which is not the square root, you will get the quotient different from the square root.

However the average of the factor and the quotient is closer to the actual root than the starting number x.

This is the principle of Heron’s method of finding square root of a number.

Ex: To find the square root of 500:

Let us guess that the square root is 20.

Divide 500 by 20 to get the quotient 25.

Take the average of the factor 20 and the quotient 25 which is 22.5.

This 22.5 is closer to the actual root of 500 than the initial estimate of 20.

Repeating the above process:

500/22.5 = 22.2222

Average of 22.5 and 22.2222 is 22.3611.

For more accuracy, we can repeat the step once again to get the next estimate as 22.36068.

The actual square root of 500 is 22.36068.

I love to extend this Heron’s method for finding any root of any number.

For finding cube root, divide twice and take the average of the two divisors and the final quotient.

For finding fourth root, divide thrice and take the average of the three divisors and the final quotient.

Ex: To find the cube root of say 78654.

Let the initial guess be 40.

Step 1: 78654 / 40 = 1966.35

Step 2: 1966.35 / 40 = 49.15875

The average of 40, 40 and 49.15875 is 43.05292.

You can repeat the above process with the starting number as 43 (No need to start with 43.05292).

Actual cube root of 78654 is 42.84567.

Even if you start with a very wild initial guess, you will only need a few more iterations to reach the answer.

Ex: To find the fourth root of say 78654.

Let the initial guess be 20.

Step 1: 78654 / 20 = 3932

Step 2: 3932 / 20 = 196

Step 3: 196 / 20 = 10

The average of 20, 20, 20 and 10 is 17.5.

Repeat the above process with the starting number as say 17.

Step 1: 78654 / 17 = 4627

Step 2: 4627 / 17 = 272

Step 3: 272 / 17 = 16

The average of 17, 17, 17 and 16 is 16.75.

Actual 4^th root of 78654 is 16.74674.

Ex: To find the fifth root of say 78654.

Let the initial guess be 10.

Step 1: 78654 / 10 = 7865.4

Step 2: 7865.4 / 10 = 786.54

Step 3: 786.54 / 10 = 78.654

Step 4: 78.654 / 10 = 7.8654

The average of 10, 10, 10, 10 and 7.8654 is 9.5731.

Actual 5^th root of 78654 is 9.531125.

Our first iteration itself is quite close to the actual root. Is it not great?

Tips: Start with a convenient round figure as the initial guess to make divisions easier. The next starting number can again be rounded or adjusted for easing future divisions."

This post is contributed by Nandeesh Nagarajaia. He is a Chemical Engineer who did his B.Tech from NIT Suratkal.  He is now in IT field as Assistant General Manager(Systems) in Hindustan Copper Limited. He love Maths and enjoy teaching Maths to his sons.

On behalf of all the QuickerMaths.com users, I  am highly grateful for his contribution.

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