Expert: Steve Holleran - 4/23/2007

Question
Hello. I am in the 10th grade and have a great understanding about mathematics. Is there any way of finding an exact square root without using a calculator? Thank you.

Answer
Hi TJ,

I'm not sure what you mean by "an exact square root" here.
The square root of any perfect square will obviously be exact, like sqrt(64) = 8, but any other number will necessarily be irrational.  So, other than leaving an expression like "sqrt(5)" , all you can do is approximate it to a certain number of decimal places.

What I'm going to describe below is an old method I learned back in the 1960's in school for finding approximations without a calculator.  I'm hoping this is what you are looking for. (If not, it's going to be a lot of unnecessary explanation!)

Let's say you want to find the sqrt of 21025.  

First, set up what looks like long division, but put in the decimal point and break up the digits into groups of two:
                        
               ---------------------
               |       2' 10' 25.

Now focus on the two.  What is the largest square less than or equal to it?  Ans--1.  Put the 1 above the 2, then double the 1 and add a blank space next to it on the outside, where the divisor would go:

                1
              --------------------
              | 2' 10' 25.
                1                  Put the 1 here also.
               ----                    Then subtract.
        2_      1  10              Bring down the next
                                   two digits together.

Now this is the weird part.  You want to put a digit in the blank space and also the same digit up top above the 10, and then multiply these so that the answer is less than but as close as you can get to 110.  For example, if you put a 2 there, you would multiply 22 by 2, a three would mean 23 by 3, etc.  The digit that works is 4: 24 * 4 = 96  (Notice that 25 * 5 = 125, too big).  Then repeat the process:


                1   4
             -------------------
             |  2' 10' 25.
                1
             ----------

    24          1  10
                   96
             ----------
                   14' 25

    28_                     Notice you double the 14
                            and add a blank.

    285            14' 25   5 will work here.285 * 5= 1425

                            Then put the 5 up top and you are done.  The sqrt of 21025 is 145.

To do one that doesn't come out exact, just add groups of zeros to the right of the decimal point.  I've tried to do sqrt(34756) below:

          1
         -------------
        | 3' 47' 56 . 00
          1
         ----------
          2 47
  2_                           8 will work.

          1   8
        -------------------
        | 3' 47 ' 56 . 00
          1
         -----------
          2  47

   28     2  24
         -----------

             23  56
  36_                        6 will work here

and you would have :

             1   8   6. (4)
         -------------------------
         |   3' 47' 56 . 00
             1
         --------------

             2  47
   28        2  24
         ----------------

                23  56
   366          21  96
         ---------------
                  1 60  00

  372_ (4)        1 48 96     


and you can keep going if you want, so to one decimal place, sqrt(34756) = 186.4

I hope this was understandable.  It's one of the more difficult things to describe even in person.

Good luck, TJ

Steve Holleran