====== Calculating A Square Root Using Pencil And Paper ======

| **Field** | Mathematics |
| **Went Obsolete** | In the late 1970s (or arguably 1614) |
| **Made Obsolete By** | the widespread availability of affordable pocket calculators (or the invention of the slide rule) |
| **Knowledge Assumed** | ability to estimate and to do long division and multiplication |
| **When useful** | understanding why it works is probably more useful than the method itself, but square roots are often used in engineering |

To calculate a square root using pencil and paper is very similar to [[LongDivision.html|LongDivision]]. This is easiest with an example, so for this example we'll extract the square root of 731.5. First, write down the number as pairs of digits, starting from the decimal point and going both left and right. In this case we get

          7 31.50

Obtain the first digit by estimating the single digit that, when squared, is less than or equal to the first group. In this example, the first group is just the digit 7 and so it's easy to see that the first digit is 2 (2*2 = 4, but 3*3 = 9 which is larger than 7). We write down the 2 above the group, write the square under the first group and subtract.

          2
          7 31.50
          4
          -
          3

Now, just as in long division, bring down the next group (31 in this case).

          2
          7 31.50
          4
          - --
          4 31

Double the square root we've extracted so far (2) and write that to the left on the next line

          2
          7 31.50
          4
          - --
          3 31
  4

Here's the tricky part. Figure out a single digit ? so that 4? times ? is less than or equal to 331 (the number we've brought down). We might guess 5 first and try 45 x 5 = 225. That's less, but it looks like we could use a higher number, since 225 is much less than 331. Let's go a little higher and try 46 x 6 = 276. (Note that we if we've already calculated that 45 x 5 = 225, we can save a little work by adding one more 45 and then 6. 225 + 45 + 6 = 276. Think for a minute and you should be able to see why that is so.) Since 276 is "still" less than 331, we try one more time, this time using the lazy method to calculate 47 x 7. 276 + 46 + 7 = 329. That really close to 331 and not greater than 331, so we use 7. Write that next digit on top, then subtract 329 like so:

          2  7.
          7 31.50
          4
          - --
          3 31
  47      3 29
          - --
             2

Drawing the horizontal line isn't really necessary. With paper and pencil, it just helps keep things neat, but it only clutters things up on a web page, so we'll dispense with it from here on.

          2  7.
          7 31.50
          4
          3 31
  47      3 29
              2

Again we double the square root we have so far (27 x 2 = 54) and write that on the next line and bring down the next group

          2  7.
          7 31.50
          4
          3 31
  47      3 29
             2 50
  54

It's clear that 54? times any number greater than zero will be greater than 250, so it's nice and simple to figure out that our next digit of the square root is zero, so as before we write that digit at the top of the group, multiply and subtract, and bring down the next group of digits. In this example, all of the digit groups from here on will be pairs of zeroes.

          2  7. 0
          7 31.50 00
          4
          3 31
  47      3 29
             2 50
  540           0
             2 50 00

Now that you've seen this a few times, it shouldn't be too hard to follow the next couple of steps, which we'll present without further comment.

          2  7. 0  4
          7 31.50 00
          4
          3 31
  47      3 29
             2 50
  540           0
             2 50 00
  5404       2 16 16

          2  7. 0  4  6
          7 31.50 00 00
          4
          3 31
  47      3 29
             2 50
  540           0
             2 50 00
  5404       2 16 16
               33 84 00
  54086        32 45 16
                1 38 84

You can keep going like this, computing one more digit of the square root each time, until you have as much precision as you need or want. Another, slightly different but better illustrated method[[CalculatingASquareRootUsingPencilAndPaper%3Faction=approvesites.html|(approve sites)]]

Here is one explanation for why this works http://www.qnet.fi/abehr/Achim/Calculators_SquareRoots_Expl.txt[[CalculatingASquareRootUsingPencilAndPaper%3Faction=approvesites.html|(approve sites)]]

Here's another: if y is the number we're taking the square root of, we can write y = (a + b)^2 = a^2 + 2ab + b^2. Think of a as a guess as to what the square root of our number is, and then b is the error term (the difference between the correct answer and our guess) As we refine our successive guesses, a gets closer to the real value and b approaches zero. Thus, a will be very close the the square root of the original number. If we assume that we can guess well, then we assume that b is small and so we can ignore the b^2 term entirely. For the example above, we start out with guessing that a=20 and b=0, so we get (a+b)^2 = 400. Since the difference is 731.5 - 400 = 331.5, we can simplify things by assuming that this is entirely due to the 2ab term. We could divide this and calculate 331.5/2/20 = 8.2875, but instead we approach the answer always from the low side, and guess at 2ab more or less directly by doubling our answer so far (20 x 2 = 40) and then adding a guess for b=7.