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A Way to Estimate the Square Root of Any Number

Date: 03/16/2006 at 13:00:42
From: JP
Subject: Wow! I have found it, now why does it work?

Today in math class, we learned about finding the square root of 
certain numbers, such as the square root of 36 = 6.  I found a way to
find the approximate square root of any number, and I need to know 
why it works.

Example: the square root of 63.  To get the square root of 63 first
you take the nearest two perfect squares, 49 and 64.  Take the square
roots of those numbers, which are 7 and 8.  Set it up like this:

  49   63   64
   7    ?    8

Next we need to make a fraction.  Subtract 49 from 64 to get your
denominator, 15.  To get your numerator you subtract the first two
numbers, 63 - 49 = 14.  That leaves you with 14 over 15 or .93.  Since
the square root of 63 is higher than 7 but less than 8 your whole
number is 7.  Add .93 to 7 and get 7.93, close to the real square root
of 63.

In my math class we always question why.  So, why does this work?

Date: 03/16/2006 at 15:03:47
From: Doctor Peterson
Subject: Re: Wow! I have found it, now why does it work?

Hi, JP.

What you've discovered is called "linear interpolation". It amounts 
to approximating a point on a graph by using a straight line that is 
close to the graph.

Draw the graph of the square root, y = sqrt(x). It passes through the 
points (49,7) and (64,8), since the square roots of 49 and 64 are 7 
and 8 respectively. Between them, the graph is slightly curved.

If you draw a line between those two points, you'll find that the 
line is quite close to the graph of the square root. Here's a close-
up of the line:

  8 -------------------- o
                      /  |
                   o     |
                /  |     |1
             /     |h    |
          /        |     |
  7 -- o-----------+-----+
       |           |     |
       49          63    64

Since 63 is 14/15 of the way from 49 to 64, h is 14/15 of the way 
from 7 to 8. That means the value of y, when x is 63, is 7 14/15. 
That's your estimate of the square root.

This method is often used for quick estimates; back before 
calculators, we were taught to use it to find values from 
trigonometric tables when we needed extra accuracy.

Here's a picture of the full graph that I talked about in case you'd
like to see it, with the triangle I drew in red.  You can see that the
straight line is very close to the actual square root graph:


Did you discover this on your own, or did you have some hints that 
led you in this direction? It's a very nice thing to have found!

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum 

Date: 03/16/2006 at 21:26:41
From: JP
Subject: Wow! I have found it, now why does it work?

I discovered it on my own and my teacher was amazed at the discovery. 
She had never heard of figuring it out in that way.  Thanks for your
reply.  That makes sense.

Associated Topics:
High School Linear Equations
High School Square & Cube Roots

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