Finding square roots by hand is a totally useless task … but I’ve had students ask me how people did it before calculators.

I’ve done some examples on my own, and actually found this to be pretty easy once you get used to it. And, like most math techniques, there’s a warm fuzzy when you start to get the hang of it.

Here’s an example. Suppose you want to find the square root of 523. You could write that out like this:

523.000000

Now, think of that, but with this grouping:

5 23. 00 00 00

So for this, I’m counting off pairs, from the decimal point, going both ways. If I don’t have a full pair (as on the left) I just leave it like it is.

The first step is to find the largest square root of the first term on the left. This is 2, and that’s the first digit of your answer. Now square the 2 to get 4, subtract that from 5 to get this:

2

--------------

5 23. 00 00 00

4

--------------

1 23

Note that I brought down the next pair (kind of like doing long division).

Now, here’s where it gets goofy: the first step was a normal square root, but all the following steps do the upcoming method.

Double the entry above the line, right it to the left of the 1 23, and leave a blank to the right of it, like so:

2 _

--------------

5 23. 00 00 00

4

--------------

4_|1 23

Note that there is a matching blank in the top row, and in the left column. To find the number that goes in there, figure out the digit that can go in both places so that when the new digit and the new number on the left are multiplied, the product is less than the remained in the bottom row. For example:

- Try 1: then 1 times 41 is 41
- Try 2: then 2 times 42 is 84(the right answer)
- Try 3: then 3 times 43 is 12

So after this step you should have:

2 2.

--------------

5 23. 00 00 00

4

--------------

42|1 23

84

-----

|39 00

Now, keep repeating the part about doubling what’s at the top, leaving the same blank in 2 spots, and then filling it, like so:

2 2. _

--------------

5 23. 00 00 00

4

--------------

42 |1 23

84

-----

44_|39 00

The number that will to in the blank is 8 (since 8 x 448 < 3900):

2 2 . 8

--------------

5 23. 00 00 00

4

--------------

42 |1 23

84

-----

448|39 00

35 84

-----

3 16 00

So, to three digits of accuracy, the answer is 22.8. Do note that this is getting you an answer digit by digit, so you’d have to get the 4th digit to know if you need to round up to 22.9 or down to 22.8.

You can check that on your calculator (mine is the Wolfram|Alpha app for droids.

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