# Heron’s Method of Finding Roots

Today I’m sharing a guest post written by one of the most talented QuickerMaths.com follower Nandeesh H.N. from Kolkata. I’m quoting from his email which he has sent to get his article uploaded on QuickerMaths.com. I’m very thankful to him for sharing such a wonderful method of squaring numbers.

“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.

## Finding Square Root by Heron’s Method

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.

You can also learn the shortcut of finding the cube root of a given number

*Did you find this method interesting? If yes, say thanks to Nandeesh by posting a comment below.*

in above example we devide 78654 / 17 = 4627 is this 17 is an assumption??? or can we take any number say 13 or 19

thankyou

Zapisana w ulubione, uwielbiam witryny!

Nice Info … Thanks

Thankyou…

BTW…wat is the best method for finding the squares of number from number 50 to 100 and 100 to 200

I actually recently wrote a post about this very topic after trying out the MITx course for intro to CS. I wrote a JavaScript example which allows you to enter a number and find the square root of the number, showing the steps taken. My post is here: http://gotochriswest.com/blog/2012/10/11/javascript-herons-square-root-algorithm/

Please explain the method of squaring any number with the help of Yaavadunam method say i have to find out 979*979 and 478*478

Hi Vinod,

Please visit http://mathlearners.com/vedic-mathematics/squares/yavadunam/ for Yavadunam Process

osm..

thanks sir its very useful.

thanks sir its very useful

very useful 2 find the roots

thank u so much…….

it is a very good shortcut,because it help to prevent the use of a calculator

i loved this post…………………….very interesting, thanks:)

I really liked this method read a long time ago, so, thank you for this article. But i want to know do we have a technique in Vedic Mathematics to calculate square root???

It is available in Wikipedia:

Vedic duplex method for extracting a square root

http://en.wikipedia.org/wiki/Methods_of_computing_square_roots

Couple of methods are present in Vedic Mathematics to find square root of a number depending on the number itself.

Please visit for details.

<a href= "http://mathlearners.com/vedic-mathematics/square-roots/" title="Square roots using Vedic Mathematics"

I really liked this method read a long time ago, so, thnak you for this article. But i want to know do we have a technique in Vedic Mathematics to calculate square root???

By this method we can get closer to the answer but not the actual answer

Majority of the roots are irrational and run to unlimited number of decimals. In practice, roots correct to a finite number of decimal places are good enough. Heron’s method is a simple method of arriving at the roots of desired accuracy.

i think these tricks are no where used now ..as far as competitive exams are concerned these are useless except from few…

why would someone need to find the fourth root of 786543 .unless he is a maths freak….

you should post something which is useful from the point of view of competitive exams like IIT, AIEEE, CAT, FMS tec…

Applications of finding roots is common in calculating rates of compound interest.

For ex:

If a principal of Rs 10000 becomes Rs 23456 in 4 years, what is the rate of interest?

Rate of interest = 4th root of 2.3456 or (1/10)*4th root of 23456.

Correction:

1+Rate of interest = 4th root of 2.3456 or (1/10)*4th root of 23456.

may be for u these tricks are not of ny use,,,,bt for us its a great help…

Great, I never knew this, thanks.

Wow this is a great resource.. I’m enjoying it.. good article

good

very helpful process to find the roots.